cooperation under interval uncertainty

6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011

Cooperative Game Theory. Operations ResearchGames. Applications to Interval GamesLecture 3: Cooperation under Interval Uncertainty

Sırma Zeynep Alparslan GokSuleyman Demirel University

Faculty of Arts and Sciences

Department of Mathematics

Isparta, Turkey

email:[email protected]

August 13-16, 2011


Outline

Introduction

Preliminaries on classical games in coalitional form

Cooperative games under interval uncertainty

On two-person cooperative games under interval uncertainty

Some economic examples

References


Introduction

Introduction

This lecture is based on the paper

Cooperation under interval uncertaintyby Alparslan Gok, Miquel and Tijs

which was published in

Mathematical Methods of Operations Research.


Introduction

Introduction

Classical cooperative game theory deals with coalitions whocoordinate their actions and pool their winnings.One of the problems is how to divide the rewards or costs amongthe members of the formed coalition.Generally, the situations are considered from a deterministic pointof view.However, in most economical situations rewards or costs are notknown precisely, but it is possible to estimate intervals to whichthey belong.


Introduction

I define a cooperative interval game

I introduce the notion of the core set of a cooperative intervalgame and various notions of balancedness

I focus on two-person cooperative interval games and extend tothese games well-known results for classical two-personcooperative games

I define and analyze specific solution concepts on the class oftwo-person interval games, e.g., mini-core set, the ψα-values




I A cooperative n-person game in coalitional form is an orderedpair < N, v >, where N = {1, 2, ..., n} (the set of players) andv : 2N → R is a map, assigning to each coalition S ∈ 2N areal number, such that v(∅) = 0.

I v is the characteristic function of the game.

I v(S) is the worth (or value) of coalition S .



A payoff vector x ∈ Rn is called an imputation for the game< N, v > (the set is denoted by I (v)) if

I x is individually rational: xi ≥ v({i}) for all i ∈ N

I x is efficient (Pareto optimal):∑n

i=1 xi = v(N)

The core (Gillies (1959)) of a game < N, v > is the set

C (v) =

{x ∈ I (v)|

∑i∈S

xi ≥ v(S) for all S ∈ 2N \ {∅}

}.

The idea of the core is by giving every coalition S at least theirworth v(S) so that no coalition has an incentive to split off.

For a two-person game I (v) = C (v).



An n-person game < N, v > is called a balanced game if for eachbalanced map λ : 2N \ {∅} → R+ we have∑

S

λ(S)v(S) ≤ v(N).

Theorem (Bondareva (1963) and Shapley (1967)):

Let < N, v > be an n-person game. Then the following twoassertions are equivalent:

(i) C (v) 6= ∅,(ii) < N, v > is a balanced game.



π(N): the set of all permutations σ : N → N

The marginal vector mσ(v) ∈ Rn has as i-th coordinate (i ∈ N)

mσi (v) = v(Pσ(i) ∪ {i})− v(Pσ(i))

Pσ(i) ={r ∈ N|σ−1(r) < σ−1(i)

}The Shapley value (Shapley (1953)) Φ(v) of a game is the averageof the marginal vectors of the game:

Φ(v) =1

n!

∑σ∈π(N)

mσ(v)



For a two-person game < N, v >:

I

m(12)(v) = (v({1}), v({1, 2})− v({1}))

m(21)(v) = (v({1, 2})− v({1}), v({2}))

I

Φi (v) = v({i}) +v({1, 2})− v({1})− v({2})

2(i = {1, 2})

Shapley value is the standard solution, in the middle of the core.Marginal vectors are the extreme points of the core whose averagegives the Shapley value.



I A game < N, v > is superadditive if v(S ∪ T ) ≥ v(S) + v(T )for all S ,T ∈ 2N with S ∩ T = ∅.

I A two-person cooperative game < N, v > is superadditive ifand only if v({1}) + v({2}) ≤ v({1, 2}) holds.

I A two-person cooperative game < N, v > is superadditive ifand only if the game is balanced.

For further details on Cooperative game theory see Tijs (2003) andBranzei Dimitrov and Tijs (2008).




A cooperative n-person interval game in coalitional form is anordered pair < N,w >:

I N := {1, 2, . . . , n} is the set of players

I w : 2N → I (R) is the characteristic function which assigns toeach coalition S ∈ 2N a closed interval w(S) ∈ I (R)

I I (R) is the set of all closed intervals in R such thatw(∅) = [0, 0]

The worth interval w(S) is denoted by [w(S),w(S)].

Here, w(S) is the lower bound and w(S) is the upper bound.

If all the worth intervals are degenerate intervals, i.e.,w(S) = w(S), then the interval game < N,w > corresponds tothe classical cooperative game < N, v >, where v(S) = w(S).



Let < N,w > be an interval game; then v is called a selection ofw if v(S) ∈ w(S) for each S ∈ 2N .

The set of selections of w is denoted by Sel(w).

The imputation set of an interval game < N,w > is defined by

I (w) := ∪{I (v)|v ∈ Sel(w)}.

The core set of an interval game < N,w > is defined by

C (w) := ∪{C (v)|v ∈ Sel(w)}.

An interval game < N,w > is strongly balanced if for eachbalanced map λ it holds that∑

S

λ(S)w(S) ≤ w(N).



Proposition: Let < N,w > be an interval game. Then, thefollowing three statements are equivalent:

(i) For each v ∈ Sel(w) the game < N, v > is balanced.

(ii) For each v ∈ Sel(w), C (v) 6= ∅.(iii) The interval game < N,w > is strongly balanced.

Proof: (i)⇔ (ii) follows from Bondareva and Shapley theorem.(i)⇔ (iii) follows using the inequalities:

w(N) ≤ v(N) ≤ w(N)∑λ(S)w(S) ≤

∑λ(S)v(S) ≤

∑λ(S)w(S)



Balancedness and related topicsLet < N,w > be a two-person interval game:

I the pre-imputation set

I ∗(w) :={x ∈ R2|x1 + x2 ∈ w(1, 2)

}I the imputation set

I (w) :={x ∈ R2|x1 ≥ w(1), x2 ≥ w(2), x1 + x2 ∈ w(1, 2)

}I the mini-core set

MC (w) :={x ∈ R2|x1 ≥ w(1), x2 ≥ w(2), x1 + x2 ∈ w(1, 2)

}I the core set

C (w) :={x ∈ R2|x1 ≥ w(1), x2 ≥ w(2), x1 + x2 ∈ w(1, 2)

}



Example 1

The mini-core set and the core set of a strongly balanced game

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mini-core set

core set



w(1) + w(2) = 3 + 5 ≤ w(1, 2) = 10 (a strongly balanced game)

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mini-core set

core set



Balancedness and related topics

s1 ∈ w(1) = [w(1),w(1)], s2 ∈ w(2) = [w(2),w(2)]

t ∈ w(1, 2) = [w(1, 2),w(1, 2)]

w s1,s2,t : the selection of w corresponding to s1, s2 and t

C (w) = ∪{C (w s1,s2,t)|(s1, s2, t) ∈ w(1)× w(2)× w(1, 2)

}MC (w) = ∪

{C (w s1,s2,t)|s1 ∈ [w(1),w(1)], s2 ∈ [w(2),w(2)]

}MC (w) ⊂ ∪

{C (w s1,s2,t)|s1 ∈ w(1), s2 ∈ w(2), t ∈ w(1, 2)

}The mini-core set is interesting because for each s1, s2 and t allpoints in MC (w) with x1 + x2 = t are also in C (w s1,s2,t).



Superadditivity

Let A and B be two intervals. We say that A is left to B, denotedby A≺B, if for each a ∈ A and for each b ∈ B, a ≤ b.

A two-person interval game < N,w > is called superadditive, if

w(1) + w(2)≺w(1, 2)



If < N,w > is a superadditive game, then for each s1, s2 and t wehave s1 + s2 ≤ t. So, each selection w s1,s2,t of w is balanced.

If w(1) + w(2) ≤ w(1, 2) is satisfied, then each selection w s1,s2,t ofw is superadditive.

A two-person interval game < N,w > is superadditive if and onlyif < N,w > is strongly balanced.



ψα-values and their axiomatization

α = (α1, α2) ∈ [0, 1]× [0, 1]: the optimism vector

sα11 (w) := α1w(1) + (1− α1)w(1)

sα22 (w) := α2w(2) + (1− α2)w(2)



I If α = (1, 1), then sα11 (w) = w(1), sα2

2 (w) = w(2) which arethe optimistic (upper) points.

I If α = (0, 0), then sα11 (w) = w(1), sα2

2 (w) = w(2) which arethe pessimistic (lower) points.

I If α = (12 ,12), then sα1

1 (w) = w(1)+w(1)2 , sα2

2 (w) = w(2)+w(2)2

which are the middle points of the intervals w(1) and w(2),respectively.



A map F : IG {1,2} → K(R2) assigning to each interval game wa unique curve

F (w) : [w(1, 2),w(1, 2)]→ R2

for t ∈ [w(1, 2),w(1, 2)], i ∈ {1, 2}, in K(R2) is called a solution.

IG {1,2}: the family of all interval games with player set {1, 2}



F has the following properties:

(i) efficiency (EFF), if for all w ∈ IG {1,2} andt ∈ [w(1, 2),w(1, 2)];

∑i∈N F (w)(t)i = t.

(ii) α-symmetry (α-SYM), if for all w ∈ IG {1,2} andt ∈ [w(1, 2),w(1, 2)] with sα1

1 (w) = sα22 (w);

F (w)(t)1 = F (w)(t)2.

(iii) covariance with respect to translations (COV), if for allw ∈ IG {1,2}, t ∈ [w(1, 2),w(1, 2)] and a = (a1, a2) ∈ R2

F (w + a)(a1 + a2 + t) = F (w)(t) + a.



ψα-values and their axiomatization

a ∈ IG {1,2} is defined by

a({1}) = [a1, a1], a({2}) = [a2, a2]

a({1, 2}) = [a1 + a2, a1 + a2]

w + a ∈ IG {1,2} is defined by

(w + a)(s) = w(s) + a(s)

fors ∈ {{1} , {2} , {1, 2}}



For each w ∈ IG {1,2} and t ∈ [w(1, 2),w(1, 2)]we define the map

ψα : IG {1,2} → K(R2)

such that

ψα(w)(t) := (sα11 (w) + β, sα2

2 (w) + β),

where

β =1

2(t − sα1

1 (w)− sα22 (w)).



Proposition: The ψα-value satisfies the properties EFF, α-SYMand COV.Proof:

(i) EFF property is satisfied since

ψα(w)(t)1 + ψα(w)(t)2 = sα11 (w) + sα2

2 (w) + 2β = t.

(ii) α-SYM property is satisfied since sα11 (w) = sα2

2 (w) implies

ψα(w)(t)1 = sα11 (w) + β = sα2

2 (w) + β = ψα(w)(t)2.



(iii) COV property is satisfied since

ψα(w + a)(a1 + a2 + t) = (sα11 (w + a) + β, sα2

2 (w + a) + β)

Then,

ψα(w+a)(a1+a2+t) = (sα11 +β, sα2

2 +β)+(a1, a2) = ψα(w)(t)+a.

Note that

β = β =1

2(t − sα1

1 (w + a)− sα22 (w + a)),

where t = a1 + a2 + t.



Theorem: The ψα-value is the unique solution satisfying EFF,α-SYM and COV properties.

Proof: Suppose F satisfies the properties above. Show F = ψα.

Let a = (sα11 (w), sα2

2 (w)). Then, sα(w − a) = (0, 0).By α-SYM and EFF, for each t = t − a1 − a2

F (w − a)(t) = (12 t,12 t) = ψα(w − a)(t).

By COV of F and ψα,F (w)(t) = F (w − a)(t) + a = ψα(w − a)(t) + a = ψα(w)(t).

From Proposition it follows ψα satisfies EFF, α-SYM and COV.



Marginal curves and the Shapley-like solution

The marginal curves for a two-person game < N,w > are definedby

mσ,α(w) : [w(1, 2),w(1, 2)]→ R2,

wherem(1,2),α(w)(t) = (sα1

1 (w), t − sα11 (w)),

m(2,1),α(w)(t) = (t − sα22 (w), sα2

2 (w)).

The Shapley-like solution ψα is equal to

ψα(w) =1

2(m(1,2),α(w) + m(2,1),α(w)).



Example 2

A bankruptcy situation with two claimants with demands d1 = 70and d2 = 90 on (uncertain) estate E = [100, 120].

w(∅) = [0, 0],w(1) = [(E − d2)+, (E − d2)+] = [10, 30]

w(2) = [(E − d1)+, (E − d1)+] = [30, 50],w(1, 2) = [100, 120].

w(1) + w(2) = 30 + 50 ≤ w(1, 2) = 100

(a strongly balanced game)



ψ(0,0)(w)(t) = (10 + β, 30 + β) with β =1

2(t − 40) and t ∈ [100, 120]

tβ

ψ(0,0)(w)(t)

100 106 110 114 12030 33 35 37 40

(40, 60) (43, 63) (45, 65) (47, 67) (50, 70)



The mini-core set and the ψ(0,0)-values of the game < N,w >,where L = {ψα(w)(t)|t ∈ [100, 120]}.

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Example 3

A reward game with N = {1, 2} which is not strongly balanced.

w(∅) = [0, 0],w(1) = [1, 3],w(2) = [2, 5],w(1, 2) = [6, 10].

ψ(0,0)(w)(t) = (1 + β, 2 + β) with 3 + 2β = t and t ∈ [6, 10].

tβ

ψ(0,0)(w)(t)

6 7 8 9 10112 2 21

2 3 312

(212 , 3

12) (3, 4) (31

2 , 412) (4, 5) (41

2 , 512)



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The mini-core set and the ψα-values of an interval game, whereL = {ψα(w)(t)|t ∈ [6, 10]}.


References

References

[1] Alparslan Gok S.Z., Miquel S. and Tijs S., “Cooperation underinterval uncertainty”, Mathematical Methods of OperationsResearch, Vol. 69 (2009) 99-109.[2] Bondareva O.N., “Certain applications of the methods of linearprogramming to the theory of cooperative games”, ProblemlyKibernetiki 10 (1963) 119-139 (in Russian).[3] Branzei R., Dimitrov D. and Tijs S., “Models in CooperativeGame Theory”, Springer-Verlag Berlin (2008).[4] Gillies D.B., “Some Theorems on n-person Games”, Ph.D.Thesis, Princeton University Press (1953).


References

[5] Shapley L.S., “A value for n-person games”, Annals ofMathematics Studies 28 (1953) 307-317.[6] Shapley L.S., “On balanced sets and cores”, Naval ResearchLogistics Quarterly 14 (1967) 453-460.[7] Tijs S., “Introduction to Game Theory”, SIAM, HindustanBook Agency, India (2003).

cooperation under interval uncertainty

Education

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