convex games and bargaining sets · 2010. 9. 9. · j.m. izquierdo & c. rafels | convex games...
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Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Convex games and bargaining sets
J.M. Izquierdo & C. Rafels
Universitat de Barcelona
SCW2010 - Moscow
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 1/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Outline
1 Introduction
2 Preliminaries
3 Max-payoff vectors: a necessary condition
4 Characterization result
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 2/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Outline
1 Introduction
2 Preliminaries
3 Max-payoff vectors: a necessary condition
4 Characterization result
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 3/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
IntroductionConvex games
We study cooperative situations among agents (cooperativeTU games) where marginal contributions of agents grow ascoalitions players add also grow.
N = {1, 2, . . . , n} is the set of players.
v(S) is the worth of coalition S ⊆ N .
For all i ∈ N and for all S ⊆ T ⊆ N \ {i},
v(S ∪ {i})− v(S) ≤ v(T ∪ {i})− v(T ).
It is said the game is convex (Shapley, 1971)
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 4/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
IntroductionConvex games
We study cooperative situations among agents (cooperativeTU games) where marginal contributions of agents grow ascoalitions players add also grow.
N = {1, 2, . . . , n} is the set of players.
v(S) is the worth of coalition S ⊆ N .
For all i ∈ N and for all S ⊆ T ⊆ N \ {i},
v(S ∪ {i})− v(S) ≤ v(T ∪ {i})− v(T ).
It is said the game is convex (Shapley, 1971)
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 4/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
IntroductionSome characterizations of a convex game
Some characterizations of a convex game are:
v is convex ⇔ marginal worth vectors are in the coreShapley (1971) and Ichiisi (1981)
⇔ the core and the Weber set coincideWeber (1988)
⇔ the cores of the game and subgamesare stable sets Einy and Shitovitz (1996)
⇔ the Weber set is a subset ofthe DM bargaining set (balanced games)
Izquierdo & Rafels (2008)
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 5/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Introduction
Can we characterize the convexity of a game by comparing thecore and the bargaining set?
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 6/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Outline
1 Introduction
2 Preliminaries
3 Max-payoff vectors: a necessary condition
4 Characterization result
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 7/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
DefinitionsConvex games
A cooperative TU-game a pair (N , v) whereN = {1, 2, . . . , n} is the set of players and
v is the characteristic function, Sv→ v(S), v(∅) = 0.
A game is 0-monotonic if for all S ⊆ T ⊆ N ,
v(S) +∑
i∈T\S
v(i) ≤ v(T ).
A game is superadditive if for all S , T ⊆ N , S ∩ T ) 6= ∅v(S) + v(T ) ≤ v(S ∪ T ).
A game v is convex if, for all i ∈ N and for allS ⊆ T ⊆ N \ {i}
v(S ∪ {i})− v(S) ≤ v(T ∪ {i})− v(T ).
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Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
DefinitionsCore
The set of preimputations of a game v is
I ∗(v) = {x ∈ RN | x(N) = v(N)}
The set of imputations of a game v is
I (v) = {x ∈ RN | x(N) = v(N) and xi ≥ v(i), for all i ∈ N}
The core of a game v is
C (v) = {x ∈ I (v) | x(S) ≥ v(S), for all S ⊆ N}
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 9/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
DefinitionsBargaining set
A payoff vector x ∈ RN is in the bargaining set if for everyobjection to x there is a counterobjection.
Several definitions:
Davis and Maschler (1963)
Mas-Colell (1989)
Zhou (1994)
Shimomura (1997)
Granot (2010)
Sudholter and Potters (2001).
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 10/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
DefinitionsBargaining set
An objection to x ∈ RN is a coalition S ⊆ N and a payoffvector y ∈ RS such that
yi > xi for all i ∈ S and y(S) = v(S)
A counterobjection to some objection(S , y) is a coalitionT ⊆ N and a payoff vector z ∈ RT such that
zi > yi for all i ∈ T∩S , zi > xi , for all i ∈ T\S and z(T ) = v(T )
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 11/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
DefinitionsBargaining sets
DM MC Z ZSH MCSH
M(i)1 (v) MB(v) Z(v) ZSh(v) MBSh(v)
x I (v) I ∗(v) I (β) I (v) I (v)
OBJ
S yk > xk yk ≥ xk yk > xk yk > xk yk > xk
S ∈ Γij ONE STRICT
COUNT OBJ
S ∩ T zk ≥ yk zk ≥ yk zk > yk zk > yk zk > yk
T \ S zk ≥ xk zk ≥ xk zk > xk zk > xk zk > xk
ONE STRICT
T ∈ Γji S ∩ T 6= ∅ S ∩ T 6= ∅S \ T 6= ∅ S \ T 6= ∅T \ S 6= ∅ T \ S 6= ∅
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 12/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Outline
1 Introduction
2 Preliminaries
3 Max-payoff vectors: a necessary condition
4 Characterization result
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 13/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Max-payoff vectors: a necessary conditionmarginal worth vectors
A marginal worth vector of the game v relative toθ = (i1, . . . , in), mθ(v), is defined as:
mθik
(v) := v({i1, . . . ik})− v({i1, . . . ik−1}), for all k = 1, . . . , n.
mθii(v) = v({i1}),
mθi2
(v) = v({i1, i2})− v({i1}),
mθi3
(v) = v({i1, i2, i3})− v({i1, i2})...mθ
in(v) = v({i1, . . . , in})− v({i1, . . . , in−1})
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Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Max-payoff vectors: a necessary conditionmarginal worth vectors
A marginal worth vector of the game v relative toθ = (i1, . . . , in), mθ(v), is defined as:
mθik
(v) := v({i1, . . . ik})− v({i1, . . . ik−1}), for all k = 1, . . . , n.
mθii(v) = v({i1}),
mθi2
(v) = v({i1, i2})− v({i1}),
mθi3
(v) = v({i1, i2, i3})− v({i1, i2})...mθ
in(v) = v({i1, . . . , in})− v({i1, . . . , in−1})
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 14/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Max-payoff vectors: a necessary conditionmax-payoff vectors
A max-payoff vector xθ(v) of v relative to θ = (i1, . . . , in) isdefined by
xθik
:= maxQ⊆Pθ
ik
{v({ik} ∪ Q)− xθ(Q)}, for all k ∈ {1, . . . , n − 1},
xθin
:= v(N)− xθ(N \ {in}).
xθii
(v) = v({i1}),
xθi2
(v) = max{v({i2}), v({i1, i2})− xθi1
(v)},
xθi3
(v) = max{v({i3}), v({i1, i3})− xθi1
(v), v({i2, i3})− xθi2
(v),
v({i1, i2, i3})− xθi1
(v)− xθi2
(v)}...xθin
(v) = v(N)− xθ(N \ {in})
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Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Max-payoff vectors: a necessary conditionmax-payoff vectors
A max-payoff vector xθ(v) of v relative to θ = (i1, . . . , in) isdefined by
xθik
:= maxQ⊆Pθ
ik
{v({ik} ∪ Q)− xθ(Q)}, for all k ∈ {1, . . . , n − 1},
xθin
:= v(N)− xθ(N \ {in}).
xθii
(v) = v({i1}),
xθi2
(v) = max{v({i2}), v({i1, i2})− xθi1
(v)},
xθi3
(v) = max{v({i3}), v({i1, i3})− xθi1
(v), v({i2, i3})− xθi2
(v),
v({i1, i2, i3})− xθi1
(v)− xθi2
(v)}...xθin
(v) = v(N)− xθ(N \ {in})
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Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Max-payoff vectors: a necessary conditionAn example
Consider the 2× 2 glove market defined by matrix
3 41 1 12 1 1
v({i}) = 0v({1, 3}) = v({1, 4}) = 1v({2, 3}) = v({2, 4}) = 1v(S) = 1, if |S | = 3v(N) = 2
Take the ordering θ∗ = (1, 3, 4, 2),
1 3 4 2mθ∗(v) 0 1 0 1xθ∗(v) 0 1 1 0
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Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Max-payoff vectors: a necessary conditionProperties of max-payoff vectors
Max-payoff vector. Property 1
Given θ = (i1, i2, . . . , in) and S ⊆ N ,
if xθ(S) < v(S), then in ∈ S
Max-payoff vector. Property 2
If v is convex xθ(v) = mθ(v), for all ordering θ = (i1, i2, . . . , in)
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 17/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Max-payoff vectors: a necessary conditionThe Theorem
Theorem 1 (Izquierdo and Rafels, 2010a)
For any arbitrary balanced game v ∈ BN we have:
1. If C (v) = ZSh(v), then xθ(v) ∈ C (v), for all θ ∈ ΘN
2. If C (v) =MBSh(v), then xθ(v) ∈ C (v), for all θ ∈ ΘN .
Hint: If not, by Property 1, we can raise an objection to xθ(v)
through a coalition of maximal excess S∗ with in ∈ S∗, giving as
much as possible to player in.
The condition is not sufficient: for the game associated tothe 2× 2 glove market, xθ(v) ∈ C (v), for all θ ∈ ΘN , but thecore is strictly included in the Shimonura bargaining sets.
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 18/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Outline
1 Introduction
2 Preliminaries
3 Max-payoff vectors: a necessary condition
4 Characterization result
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 19/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Characterization resultBargaining sets and the core
The coincidence of the core with the bargaining set
CONVEX GAMES AVERAGE MONOTONIC GAMES ASSIGNMENT GAMES
DAVIS-MASCHLER X X XMAS-COLELL X X X
MAS-COLELLSh X x xZHOU , ZHOUSh X x x
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 20/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
CharacterizationBargaining sets and the core
Theorem 2 (Izquierdo and Rafels, 2010b)
Let v ∈ GN . Then,
1. v is convex ⇔ ZSh(v) = C (v) and v is superadditive;
2. v is convex ⇔MBSh(v) = C (v) and v is 0-monotonic.
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 21/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
CharacterizationBargaining sets and the core
sketch of the proof
if v is convex then C (v) = ZSH(v) =MBSh(v))
Let x ∈ I (v) \ C (v) and let S∗ ⊆ N be a minimal coalition of thelargest excess,
S∗ ∈ argmax{v(S)− x(S)}
where, if S S∗, v(S)− (S) < v(S∗)− x(S∗).
Compute the excess game
wx(S) := maxR⊆S{v(R)− x(R)}, for all S ⊆ N
and take the Shapley value of wx , namely Φ(wx).
define the objection (S∗, y) with yi = xi + Φi , for all i ∈ S∗.
this objection cannot be countered 2
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 22/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Characterization resultBargaining sets and the core
sketch of the proof
If C (v) =MBSH(v) and v is 0-monotonic ⇒ v is convex)
By Theorem 1 xθ(v) ∈ C (v), for all θ.
Since the game is not convex there exists
θ∗ = (i1, . . . , ik∗−1, ik∗ , ik∗+1, . . . , in)
such that
? xθ∗(v) 6= mθ∗(v), (Property 2)
xθ∗ik∗> mθ∗
ik∗≥ v({ik∗}), and xθ∗
ik= mθ∗
ik, k = 1, . . . k∗−1.
? For all S ⊆ {i1, . . . , ik∗} such that |S | ≤ k ∗ −1, thesubgame vS is convex
J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 23/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
Characterization resultBargaining sets and the core
The vector x 6∈ C (v) but we prove x ∈MBSH(v). 2J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 24/25
Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References
References
Davis, M., Maschler, M., 1963. Existence of stable payoffconfigurations for cooperative games. B. Am. Math Soc. 69,106–108.
Einy, E., Shitovitz, B., 1996. Convex Games and Stable Sets,Games Econ. Behav. 16(2), 192–201.
Ichiishi, T., 1981. Super-modularity: applications to convexgames and to the greedy algorithm in LP, J Econ. Theory 25,283–286.
Izquierdo, J.M., Rafels, C., 2010a. On the coincidence betweenthe Shimomura’s bargaining sets and the core, Working paperseries E10/241, Faculty of Economics, University of Barcelona.
Shimomura, K., 1997. Quasi-Cores in Bargaining Sets. Int. J.Game Theory 26, 283–302.
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