on the ellipsoidal core for cooperative games under ellipsoidal uncertainty
TRANSCRIPT
On the Ellipsoidal Core
for Cooperative Games
under Ellipsoidal Uncertainty
Gerhard-Wilhelm Weber
Institute of Applied Mathematics METU AnkaraTurkey
Rodica Branzei
Faculty of Computer Science Alexandru Ioan Cuza University Iaşi Romania
S Zeynep Alparslan-Goumlk
Department of Mathematics Suumlleyman Demirel University Isparta Turkey
5th International Summer School
Achievements and Applications of Contemporary Informatics
Mathematics and Physics
National University of Technology of the Ukraine
Kiev Ukraine August 3-15 2010
Games
Games
Optimal Energy Management
according to Joint Implementation
Aims
1 The instrument must work on the micro level with minimal costs
2 It should be protected against misuse on the macro level
Ad 1 Solved by the control theoretic approach
Ad 2 Can be solved by adding constraints to the feasible set
of our control parameters
Cost games are very important in the practice of OR
Ex
airport game
unanimity game
production economy with landowners and peasants
bankrupcy game etc
There is also a cost game in environmental protection (TEM model)
The aim is to reach a state which is mentioned in Kyoto Protocol
by choosing control parameters such that
the emissions of each player become minimized
For example the value is taken as a control parameter
Cost Games
Cost Games
The central problem in cooperative game theory is how to allocate the gain
among the individual players in a ldquofairrdquo way
There are various notions of fairness and corresponding allocation rules
(solution concepts)
Here we use the notation
Any with is an allocation
So a core allocation
guarantees each coalition to be satisfied
in the sense that it gets at least what it could get on its own
( )w w N i N
( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR
( ) ( )
i
i S
x S x S N coalition
Nx R ( ) x N w
( )x Core wS N
bull Method
Determine the core by this difference and
with the value which has to lie in it
bull Approach
Steer the system
bull Main idea
Take the value as a core element
and as a control parameter
( ) ( ) w Core w
Kyoto Game
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Games
Games
Optimal Energy Management
according to Joint Implementation
Aims
1 The instrument must work on the micro level with minimal costs
2 It should be protected against misuse on the macro level
Ad 1 Solved by the control theoretic approach
Ad 2 Can be solved by adding constraints to the feasible set
of our control parameters
Cost games are very important in the practice of OR
Ex
airport game
unanimity game
production economy with landowners and peasants
bankrupcy game etc
There is also a cost game in environmental protection (TEM model)
The aim is to reach a state which is mentioned in Kyoto Protocol
by choosing control parameters such that
the emissions of each player become minimized
For example the value is taken as a control parameter
Cost Games
Cost Games
The central problem in cooperative game theory is how to allocate the gain
among the individual players in a ldquofairrdquo way
There are various notions of fairness and corresponding allocation rules
(solution concepts)
Here we use the notation
Any with is an allocation
So a core allocation
guarantees each coalition to be satisfied
in the sense that it gets at least what it could get on its own
( )w w N i N
( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR
( ) ( )
i
i S
x S x S N coalition
Nx R ( ) x N w
( )x Core wS N
bull Method
Determine the core by this difference and
with the value which has to lie in it
bull Approach
Steer the system
bull Main idea
Take the value as a core element
and as a control parameter
( ) ( ) w Core w
Kyoto Game
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Games
Optimal Energy Management
according to Joint Implementation
Aims
1 The instrument must work on the micro level with minimal costs
2 It should be protected against misuse on the macro level
Ad 1 Solved by the control theoretic approach
Ad 2 Can be solved by adding constraints to the feasible set
of our control parameters
Cost games are very important in the practice of OR
Ex
airport game
unanimity game
production economy with landowners and peasants
bankrupcy game etc
There is also a cost game in environmental protection (TEM model)
The aim is to reach a state which is mentioned in Kyoto Protocol
by choosing control parameters such that
the emissions of each player become minimized
For example the value is taken as a control parameter
Cost Games
Cost Games
The central problem in cooperative game theory is how to allocate the gain
among the individual players in a ldquofairrdquo way
There are various notions of fairness and corresponding allocation rules
(solution concepts)
Here we use the notation
Any with is an allocation
So a core allocation
guarantees each coalition to be satisfied
in the sense that it gets at least what it could get on its own
( )w w N i N
( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR
( ) ( )
i
i S
x S x S N coalition
Nx R ( ) x N w
( )x Core wS N
bull Method
Determine the core by this difference and
with the value which has to lie in it
bull Approach
Steer the system
bull Main idea
Take the value as a core element
and as a control parameter
( ) ( ) w Core w
Kyoto Game
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Optimal Energy Management
according to Joint Implementation
Aims
1 The instrument must work on the micro level with minimal costs
2 It should be protected against misuse on the macro level
Ad 1 Solved by the control theoretic approach
Ad 2 Can be solved by adding constraints to the feasible set
of our control parameters
Cost games are very important in the practice of OR
Ex
airport game
unanimity game
production economy with landowners and peasants
bankrupcy game etc
There is also a cost game in environmental protection (TEM model)
The aim is to reach a state which is mentioned in Kyoto Protocol
by choosing control parameters such that
the emissions of each player become minimized
For example the value is taken as a control parameter
Cost Games
Cost Games
The central problem in cooperative game theory is how to allocate the gain
among the individual players in a ldquofairrdquo way
There are various notions of fairness and corresponding allocation rules
(solution concepts)
Here we use the notation
Any with is an allocation
So a core allocation
guarantees each coalition to be satisfied
in the sense that it gets at least what it could get on its own
( )w w N i N
( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR
( ) ( )
i
i S
x S x S N coalition
Nx R ( ) x N w
( )x Core wS N
bull Method
Determine the core by this difference and
with the value which has to lie in it
bull Approach
Steer the system
bull Main idea
Take the value as a core element
and as a control parameter
( ) ( ) w Core w
Kyoto Game
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Cost games are very important in the practice of OR
Ex
airport game
unanimity game
production economy with landowners and peasants
bankrupcy game etc
There is also a cost game in environmental protection (TEM model)
The aim is to reach a state which is mentioned in Kyoto Protocol
by choosing control parameters such that
the emissions of each player become minimized
For example the value is taken as a control parameter
Cost Games
Cost Games
The central problem in cooperative game theory is how to allocate the gain
among the individual players in a ldquofairrdquo way
There are various notions of fairness and corresponding allocation rules
(solution concepts)
Here we use the notation
Any with is an allocation
So a core allocation
guarantees each coalition to be satisfied
in the sense that it gets at least what it could get on its own
( )w w N i N
( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR
( ) ( )
i
i S
x S x S N coalition
Nx R ( ) x N w
( )x Core wS N
bull Method
Determine the core by this difference and
with the value which has to lie in it
bull Approach
Steer the system
bull Main idea
Take the value as a core element
and as a control parameter
( ) ( ) w Core w
Kyoto Game
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Cost Games
The central problem in cooperative game theory is how to allocate the gain
among the individual players in a ldquofairrdquo way
There are various notions of fairness and corresponding allocation rules
(solution concepts)
Here we use the notation
Any with is an allocation
So a core allocation
guarantees each coalition to be satisfied
in the sense that it gets at least what it could get on its own
( )w w N i N
( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR
( ) ( )
i
i S
x S x S N coalition
Nx R ( ) x N w
( )x Core wS N
bull Method
Determine the core by this difference and
with the value which has to lie in it
bull Approach
Steer the system
bull Main idea
Take the value as a core element
and as a control parameter
( ) ( ) w Core w
Kyoto Game
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
bull Method
Determine the core by this difference and
with the value which has to lie in it
bull Approach
Steer the system
bull Main idea
Take the value as a core element
and as a control parameter
( ) ( ) w Core w
Kyoto Game
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Only 3 time points are assumed
The 2 players have
2 alternative strategies to invest
The origin of the coordinate system
is the starting point
Each player tries
to reach the blue square
This is the level of
reductions of emissions
in a given number of steps
mentioned in Kyoto Protocol
Kyoto Game
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
o The strategy (21) leads to a greater
reduction at the beginning and a
smaller investment at the end
of the period
o The costs are lower than in the
(12) case reflecting the fact that
early innovations are favourable
o This simple model can be transferred
to a simple matrix game
our Kyoto Game
Kyoto Game
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -01 30 60 160 1 -0525 -0475
2 -01 20 60 160 -0475 1 0525
3 -01 10 60 160 -01 -01 02
The actors have not reached the limiting Kyoto level 0
at the beginning of the time period
The em-matrix has positive and negative entries which means
that we deal with both cooperative and competitive behaviour
TEM Model
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
TEM Model
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Every player is at Kyoto Level
This example is an extraordinary case
The memory parameter φ is chosen very big
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 0 03 1 001 1 -05 -05
2 0 05 1 001 -05 1 -05
3 0 02 1 001 -05 -05 1
TEM Model
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Every actor reaches the Kyoto Level without any control parameter
But it takes a 100 year time period which is a too long time
TEM Model
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
player i Ei (0) Mi (0) Mi λi 160 x em matrix
1 -1 03 1 082 1 -07 -03
2 06 01 1 025 -08 1 -02
3 05 02 1 04 -09 -01 1
TEM Model
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
TEM Model
Here a chaotic behaviour occurs after a 20 year time period
the first bifurcation can be observed after 20 time steps
This behavior requests the determination of optimal control parameters
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
TEM Model
ui(t) control parameters
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
The figure shows the behavior
of the 3 players
after the control parameters added
It is seen that every actor can reach
the Kyoto Level in a short time period
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )
( )
0k k k
k
kM
u
E E E
M M M
TEM Model
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
time-continExpression data
time-discr
kkk EE M1
)(Μ jik em M
0)0( EE
Ex Euler Runge-Kutta
( ) ( )E M E E C E
environmental effects
TEM Model Gene-Environment Networks
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
( ) ΜkM E
E
TEM Model Gene-Environment Networks
( )NI R
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
( ) ΜkM E
E
TEM Model Gene-Environment Networks
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Identify groups (clusters) of jointly acting
genetic and environmental variables
stable clustering
disjoint
overlapping
Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
2) Interaction of Genetic Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
3) Interaction of Environmental Clusters
Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
3) Interaction of Genetic amp Environmental Clusters
Determine the degree of connectivity
Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Clusters and Ellipsoids
Genetic clusters C1C2hellipCR
Environmental clusters D1D2hellipDS
Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)
Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)
Gene-Environment Networks Ellipsoidal Calculus
Clusters Pre-Coalitions
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
r 1
Gene-Environment Networks Ellipsoidal Calculus
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
r=1
Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
The Regression Problem
Maximize (overlap of ellipsoids)
T R
r
R
r
rrrr EEXX1 1 1
)()()()( ˆˆ
measurement
prediction
Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Measures for the size of intersection
bull Volume rarr ellipsoid matrix determinant
bull Sum of squares of semiaxes rarr trace of configuration matrix
bull Length of largest semiaxes rarr eigenvalues of configuration matrix
semidefinite programming
interior point methods
rr E
r
Gene-Environment Networks Ellipsoidal Calculus
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborativecooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval Games
cooperative
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval Games
cooperative
Robust Optimization
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval Gamescooperative
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
I Ni i
i N
w I I I w N I w i i NR
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborative
( ) | ( ) ( ) ( )
Ni i
i N
w I I I w N I w i i NRI
= 12
2 ( ) ( ) [0
0]
lt gt
N
NnN I ww
N IGw
R
cooperative
1 2 1 2
1 2
1 2
( )( ) ( ) ( )
( )( ) ( )
N
lt N gt
lt N gt
S S S
S S
w w
w w IG
w
w w
w
w w
w
Interval Glove Game
0 euro 0 euro
10 - 20 euro
(13) (23) (123) [10 20]
( ) [00] else
( ) = ([00][00][10 20])
= 1 23
= 1 2
L R
w w w
w S
Co
N
L
re w
L R
Interval core
( ) | ( ) ( )
ii S
Core w I w I w S S N SI
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborative
2
1
1 2
( ) = | ( ) ( ) 1
( ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
cooperative
( ) ( ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 ( ( )
( ) (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
0 euro 0 euro
( ) euro
(13) (23) (123) ( )
( ) 0 else
= 123
= 12
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore value
E
E
Kyoto Game
(individual roles in TEM Model)
(individual role in TEM Model)
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoidal core
iv
Ellipsoid Glove Game
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Ellipsoid Games Interval GamesC ollaborativecooperative
= 12 2
)
( 0
lt gt
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Malacca Police Game
R
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Some results related with the ellipsoidal core
rer
rr
r
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Some results related with the ellipsoidal core
rer
rr
r
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
Thank you very much for your attention
gwebermetuedutr
httpwww3iammetuedutriamimages773Willi-CVpdf
References
rr
Appendix
rr
Appendix