Cooperative Games I - imus.us.es · Introduction to Cooperative Games Preliminaries What is Game...

Cooperative Games I

Ignacio García-Jurado

Departamento de MatemáticasUniversidade da Coruña

Doc-course: The Mathematics of Games, Strategies, Cooperationand Fair Division

I. García-Jurado (UDC) Cooperative Games I Sevilla, March 16, 2011 1 / 31

Introduction to Cooperative Games Preliminaries

What is Game Theory?

Game theory can be defined as the mathematical theory ofinteractive decision situations.It officially started in 1944, when the first edition of the book“Theory of Games and Economic Behavior” by John vonNeumann and Oskar Morgenstern was launched.From its very beginning, game theory is the result of thecollaboration between mathematicians and economists.


Introduction to Cooperative Games Preliminaries

Non-Cooperative versus Cooperative Game Theory

Non-Cooperative Game Theory assumes that players’ possibilitiesfor interacting and collaborating can be fully modelled. It analyseshow players should strategically behave within the rules of thegame.Cooperative Game Theory assumes that players’ possibilities forinteracting and collaborating are too complex to be formallymodelled. It just aims to allocate among players the estimatedbenefits of their cooperation.


Introduction to Cooperative Games Models of Cooperative Game Theory

NTU-games

Definition

A non-transferable utility game (NTU-game) is a pair (N,V ), where Nis the finite set of players and V is a function that assigns, to eachcoalition S ⊂ N, a set V (S) ⊂ RS. By convention, V (∅) := {0}.Moreover, for each S ⊂ N, S 6= ∅:

1 V (S) is a non-empty and closed subset of RS.2 V (S) is comprehensive (i.e., for each pair x , y ∈ RS such that

x ∈ V (S) and y ≤ x , we have that y ∈ V (S)). Moreover, for eachi ∈ N, V ({i}) 6= R, i.e., there is vi ∈ R such that V ({i}) = (−∞, vi ].

3 The set V (S) ∩ {y ∈ RS | for each i ∈ S, yi ≥ vi} is bounded.



Bargaining problems

DefinitionA bargaining problem with finite set of players N is a pair (F ,d) whoseelements are the following:Feasible set F is a closed, convex, and comprehensive subset of RN

such that Fd := {x ∈ F | x ≥ d} is compact.Disagreement point d is an allocation in F . It is assumed that there is

x ∈ F such that x > d .

Remark(F ,d), a bargaining problem with finite set of players N, can be seenas an NTU-game (N,V ), where V (N) := F and, for each non-emptycoalition S 6= N, V (S) := {y ∈ RS : for each i ∈ S, yi ≤ di}.



TU-games

DefinitionA TU-game is a pair (N, v), where N is the finite set of players, andv : 2N → R is the characteristic function of the game, which satisfiesv(∅) = 0. In general, we interpret v(S) as the benefit that S cangenerate. We will often refer to (N, v) simply as v , and will denote byG(N) the class of TU-games with set of players N.

RemarkA TU-game (N, v) can be seen as an NTU-game (N,V ) by defining, foreach non-empty coalition S ⊂ N, V (S) := {y ∈ RS :

∑i∈S yi ≤ v(S)}.


Introduction to Cooperative Games Introduction to TU-games

Example

Divide a million. A rich person dies and leave one million euros tothree nephews, with the condition that at least two of them must agreeon how to divide this amount among them; otherwise, the million ofeuros will be given to other person. This situation can be modelled asthe TU-game (N, v), where N = {1,2,3}, and v(1) = v(2) = v(3) = 0,v(12) = v(13) = v(23) = v(N) = 1.

Example

The glove game. Three players are willing to divide the benefits ofselling a pair of gloves. Player one has a left glove, and players twoand three have one right glove each. A pair of gloves can be sold forone hundred euros. This situation can be modelled as the TU-game(N, v), where N = {1,2,3}, and v(1) = v(2) = v(3) = v(23) = 0,v(12) = v(13) = v(N) = 1.



Example

The Parliament of Aragón. This example illustrates that TU-games can also be usedto model coalitional bargaining situations in which players negotiate with somethingmore abstract than money. In this case we consider the Parliament of Aragón, one ofthe regions in which Spain is divided. After the elections which took place in May1991, its composition was: PSOE (Socialist Party) had 30 seats, PP (ConservativeParty) had 17 seats, PAR (Regional Party of Aragón) had 17 seats, and IU (a coalitionmainly formed by communists) had 3 seats. In a Parliament, the most relevantdecisions are taken making use of the simple majority rule. Measuring the power ofthe different parties in a Parliament, which is an interesting problem, can be thoughtas “dividing” the power among them. A coalition is said to have the power if it collectsmore than half of the seats of the Parliament, thirty-four seats in this example. So, thissituation can be modelled as the TU-game (N, v), where N = {1, 2, 3, 4} (1=PSOE,2=PP, 3=PAR, 4=IU), and v(S) = 1 if there is T ∈ {{1, 2}, {1, 3}, {2, 3}} with T ⊂ S,v(S) = 0 otherwise. Notice that we indicate that S has more than half of the seatsmaking v(S) = 1.



Example

The visiting professor. Three research groups belonging to the universities of Milano(group one), Genova (group two) and Santiago de Compostela (group three) plan toinvite a Japanese professor to give a course on game theory. To minimize the cost,they coordinate the courses, so that the professor makes a tour visiting Milano,Genova and Santiago de Compostela. Then the groups want to allocate the cost ofthe tour among them. For that purpose they have estimated the travel cost (in euros)of the visit for all the possible coalitions of groups: c(1) = 1500, c(2) = 1600,c(3) = 1900, c(12) = 1600, c(13) = 2900, c(23) = 3000, c(N) = 3000. TakeN = {1, 2, 3}. Notice that (N, c) is a TU-game; however, it is what we usually call acost game, in the sense that c(S) (for every S) does not represent the benefits that Scan generate, but the costs it must support. The saving game associated to thissituation (displaying the benefits generated by each coalition) is (N, v) where, forevery S ⊂ N,

v(S) =∑i∈S

c(i)− c(S).

Thus, v(1) = v(2) = v(3) = 0, v(12) = 1500, v(13) = 500, v(23) = 500,v(N) = 2000.



Superadditive GamesDefinitionTake a TU-game v ∈ G(N). We say that v is superadditive if, for allS,T ⊂ N, with S ∩ T = ∅,

v(S ∪ T ) ≥ v(S) + v(T ).

We denote by SG(N) the set of superadditive TU-games with set ofplayers N.

Note that a game is superadditive when players have true incentivesfor cooperation, in the sense that the union of any two disjoint groupsof players never diminishes the total benefits. In fact, most part of thetheory of TU-games is really developed for superadditive games,although it is formally presented, for simplicity, for the whole class ofTU-games. Notice that all the games in the examples above, with theonly exception of (N, c) in the visiting professor game, aresuperadditive.



Solutions for TU-Games

The main goal of the theory of TU-games is to select, for everyTU-game, an allocation, or a set of allocations, which is admissible forthe players. In making this, there are two main approaches.

The first approach is based on stability: it aims to find a set ofallocations which is stable, in the sense that it can be expectedthat the final agreement adopted by the players lies within this set;this is the approach underlying, for instance, the core (Gillies(1953)), the stable sets (von Neumann and Morgenstern (1944))and the bargaining set (Aumann and Maschler (1964)).The second approach is based on fairness: it aims to propose forevery TU-game one allocation which represents a fair compromisefor the players; this is the approach underlying, for instance, theShapley value (Shapley (1953)), the nucleolus (Schmeidler(1969)) and the Tijs value (Tijs (1981)).


The core of a TU-game and related concepts The core

Imputations and the core

DefinitionLet v ∈ G(N) be a TU-game. An imputation of v is an x ∈ RN suchthat:∑

i∈N xi = v(N),xi ≥ v(i), for all i ∈ N.

We denote by I(v) the set of imputations of v .

DefinitionLet v ∈ G(N) be a TU-game. The core of v , that we denote by C(v), isthe following set:

C(v) = {x ∈ I(v) |∑i∈S

xi ≥ v(S) for all S ⊂ N}.



ExampleConsider the TU-game in the visiting professor game, which is givenby: N = {1,2,3}, and v(1) = v(2) = v(3) = 0, v(12) = 1500,v(13) = 500, v(23) = 500, v(N) = 2000. Its set of imputations isdepicted below.



ExampleConsider the TU-game in the visiting professor game, which is givenby: N = {1,2,3}, and v(1) = v(2) = v(3) = 0, v(12) = 1500,v(13) = 500, v(23) = 500, v(N) = 2000. Now the core of this game isdisplayed.



ExampleConsider the glove game, which is given by: N = {1,2,3}, andv(1) = v(2) = v(3) = v(23) = 0, v(12) = v(13) = v(N) = 1. It is easyto check that its core is the set {(1,0,0)}.

ExampleConsider the divide a million game, which is given by: N = {1,2,3},and v(1) = v(2) = v(3) = 0, v(12) = v(13) = v(23) = v(N) = 1. It iseasy to check that its core is empty.


The core of a TU-game and related concepts Balancedness and the core

Balancedness

DefinitionA family of coalitions F ⊂ 2N \ {∅} is said to be balanced if there existsa corresponding family of positive real numbers (called balancingcoefficients) {yS | S ∈ F} such that, for all i ∈ N,∑

S∈F ,i∈S

yS = 1.

DefinitionA game v ∈ G(N) is said to be balanced if, for every balanced family Fwith balancing coefficients {yS | S ∈ F}, it holds that∑

S∈F

ySv(S) ≤ v(N).



The Bondareva-Shapley theoremDefinitionA game v ∈ G(N) is said to be balanced if, for every balanced family Fwith balancing coefficients {yS | S ∈ F}, it holds that∑

S∈F

ySv(S) ≤ v(N).

The fact that v is balanced can be roughly interpreted as that itsintermediate coalitions are not too powerful. Hence, the balancednessof a game should be, in some sense, related to the stability of thecoalitional bargaining situation modelled by it. This is precisely whatthe Bondareva-Shapley theorem asserts.

Theorem(Bondareva-Shapley). Let v ∈ G(N) be a TU-game. Then C(v) 6= ∅ ifand only if v is balanced.



Bondareva-Shapley theorem: a sketch of the proofSuppose that v is balanced and consider the following linearprogramming problem (P):

Minimise∑

i∈N xi ,

provided that∑

i∈S xi ≥ v(S), ∀S ∈ 2N \ {∅}.

Clearly, C(v) 6= ∅ if and only if there exists x̄ an optimal solution of (P)with

∑i∈N x̄i = v(N). The dual of (P) is the following linear

programming problem (D):

Maximise∑

S∈2N\{∅} ySv(S),

provided that∑

S∈2N\{∅},i∈S yS = 1, ∀i ∈ N,

yS ≥ 0, ∀S ∈ 2N \ {∅}.

The balancedness of v implies that C(v) 6= ∅.


The core of a TU-game and related concepts Domination and the core

DominationDefinitionLet v ∈ G(N) be a TU-game and take a non-empty S ⊂ N andx , y ∈ I(v). We say that x dominates y through S if:

xi > yi for all i ∈ S,∑i∈S xi ≤ v(S).

We say that x dominates y if there exists a non-empty S ⊂ N such thatx dominates y through S. Finally, y is said to be an undominatedimputation of v if there does not exist another imputation x ∈ I(v) suchthat x dominates y .

PropositionLet v ∈ G(N) be a TU-game.

1 If x ∈ C(v), then x is undominated.2 If, moreover, v ∈ SG(N), then

C(v) = {x ∈ I(v) | x is undominated }.I. García-Jurado (UDC) Cooperative Games I Sevilla, March 16, 2011 19 / 31

The core of a TU-game and related concepts Stable sets and the core

Stable sets

Definition(von Neumann and Morgenstern). Let v ∈ G(N) be a TU-game. Asubset M of I(v) is said to be a stable set if it satisfies the following twoproperties:

Internal stability. For every x , y ∈ M, x does not dominate y and ydoes not dominate x .External stability. For every x ∈ I(v) \M, there exists y ∈ M suchthat y dominates x .

If M is a stable set of v , then C(v) ⊂ M.If C(v) is a stable set of v , then it is the unique stable set of v .A game can have many stable sets. In 1968 W. Lucas constructeda 10-person game without stable sets.


The Shapley value of a TU-game Definition and axiomatic characterization

Values for TU-games

Now we study the Shapley value, the most important solutionconcept for TU-games dealing with fairness.Shapley wants to propose, for every TU-game, one allocationwhich is a fair compromise for the players.To that aim, Shapley gives the concept of value of a TU-game, avalue being a map φ which associates to every v ∈ G(N) anelement φ(v) ∈ RN ; he also introduces some properties that a fairvalue should satisfy.Finally, Shapley proves that his properties characterize a uniquevalue and finds an explicit expression for it. This value is what wecall the Shapley value.



Fairness and valuesDefinitionLet v ∈ G(N) be a TU-game.

i ∈ N is said to be a null player of v if, for every S ⊂ N,v(S ∪ {i}) = v(S).Two players i , j ∈ N are said to be interchangeable in v if, for everycoalition S ⊂ N \ {i , j}, v(S ∪ {i}) = v(S ∪ {j}).

Efficiency (EFF). φ satisfies EFF if, for all v ∈ G(N),∑i∈N φi(v) = v(N).

The Null Player Property (NPP). φ satisfies NPP if, for everyv ∈ G(N) and every null player i ∈ N, φi(v) = 0.Anonimity (AN). φ satisfies AN if, for every v ∈ G(N) and everyi , j ∈ N, interchangeable players in v , it holds that φi(v) = φj(v).Additivity (ADD). φ satisfies ADD if, for every v ,w ∈ G(N), it holdsthat φ(v + w) = φ(v) + φ(w).



The Shapley valueTheoremThere exists a unique value Φ satisfying EFF, NPP, AN and ADD. Thisvalue, that is called the Shapley value, is given by

Φi(v) =∑

S⊂N\{i}

s!(n − s − 1)!

n!(v(S ∪ {i})− v(S)) (1)

for all v ∈ G(N) and all i ∈ N, n and s denoting the cardinalities of Nand S, respectively.

Sketch of the proof. For every non-empty S ⊂ N, define uS ∈ G(N)(the unanimity game of S) by uS(T ) = 1 if S ⊂ T , and uS(T ) = 0otherwise. Now take into account that every v ∈ G(N) can be seen asa vector in R2n−1 and that the set {uS}S∈2N\∅ is a basis of R2n−1.Finally, it is clear that EFF, NPP and AN characterize a unique solutionfor every unanimity game.


The Shapley value of a TU-game Some comments on the Shapley value

Heuristic interpretation

The Shapley value can be seen as the expected reward vector inthe following situation: a) players agree to meet at a certainlocation, b) all possible arrival orderings are equally probable, andc) on arrival every player receives a reward equal to itscontribution to the coalition of players who arrived before.In other words, the proposal of the Shapley value for gamev ∈ G(N) can also be written as

Φi(v) =1n!

∑π∈Π(N)

v(Bπ(i) ∪ {i})− v(Bπ(i)),

for all i ∈ N, where Π(N) is the set of permutations of N, and Bπ(i)is the set {j ∈ N | π(j) < π(i)} (for all π ∈ Π(N) and all i ∈ N).



ExampleThe proposal of the Shapley value for the visiting professor game isΦ(v) = (5000/6,5000/6,2000/6).

1 2 3123 0 1500 500132 0 1500 500213 1500 0 500231 1500 0 500312 500 1500 0321 1500 500 0

5000 5000 2000

According to this allocation of the savings, the players have to pay(4000/6,4600/6,9400/6). Note that the latter vector is precisely Φ(c).Finally, observe that Φ(v) ∈ C(v).



ExampleThe proposal of the Shapley value for the divide a million game is(1/3,1/3,1/3). Note that, although the core of this game is empty, theShapley value proposes an allocation for it. Remember that the coredealt with stability and that the Shapley value deals with fairness.

ExampleThe proposal of the Shapley value for the glove game is the vector(2/3,1/6,1/6). Remember that the core of this game is {(1,0,0)}.

ExampleThe proposal of the Shapley value for the Parliament of Aragón is(1/3,1/3,1/3,0). This is a measure of the power of the four politicalparties in this Parliament. Note that IU is a null player and that theother three parties are interchangeable if we only take into accounttheir voting power.



The Shapley value and the core

DefinitionLet v ∈ G(N) a TU-game. We say that v is convex if, for everyS,T ⊂ N \ {i} with S ⊂ T ,

v(T ∪ {i})− v(T ) ≥ v(S ∪ {i})− v(S).

It can be checked that every convex game is superadditive. Moreover,Shapley demonstrated the following result.

TheoremLet v ∈ G(N) be a TU-game.

If v is superadditive then Φ(v) ∈ I(v).If v is convex then Φ(v) ∈ C(v).


The nucleolus of a TU-game An introduction to the nucleolus

The definition of excessThe nucleolus is a value for TU-games based on the idea of minimizingthe dissatisfaction of the most dissatisfied groups. To that aim, theconcept of excess is defined.

Definition

Let v ∈ G(N) be a TU-game and take x ∈ RN and S ∈ 2N \ ∅. Theexcess of x with respect to S, denoted by e(S, x), is given by

e(S, x) =∑i∈S

xi − v(S).

The excess of x , denoted by e(x), is the vector in R2n−1 containing theexcesses of x with respect to all the non-empty coalitions disposed innon-decreasing order. More precisely, if i ∈ {1, . . . ,2n − 2} and ei(x)denotes the i-th component of e(x), then ei(x) ≤ ei+1(x).


The nucleolus of a TU-game An introduction to the nucleolus

The nucleolus

Definition

Let v ∈ G(N) be a TU-game such that I(v) 6= ∅. The nucleolus maps vto N (v) ∈ RN , where N (v) is the unique point of the set

{x ∈ I(v) | e(y) ≤L e(x) for all y ∈ I(v)}, (2)

≤L denoting the lexicographic order on R2n−1. N (v) is said to be thenucleolus of v .

It can be check that N is well-defined, in the sense that the set in (2)contains in fact a unique point for those v ∈ G(N) with I(v) 6= ∅.Moreover, it can be easily proved that if C(v) 6= ∅ then N (v) ∈ C(v).There are several procedures to compute the nucleolus of a TU-game,but its computation can be quite hard.


Bibliography

Bibliography

R.J. Aumann and M. Maschler (1964). “The bargaining set for cooperative games”. In M. Dresher, L.S. Shapley and A.W. Tucker(eds.), Advances in game theory. Princeton University Press.O. Bondareva (1963). “Certain applications of the methods of linear programming to the theory of cooperative games”. ProblemyKibernetiki 10, 119-139.D.B. Gillies (1953). “Some theorems on n-person games”. PhD dissertation. Princeton University.J. González-Díaz, I. García-Jurado, M.G. Fiestras Janeiro (2010). “An introductory course on mathematical game theory”.Graduate Studies in Mathematics 115. American Mathematical Society.Y. Kannai (1992). “The Core and balancedness”. In R.J. Aumann and S. Hart (eds.), Handbook of Game Theory Vol. I.North-Holland.J. von Neumann and O. Morgenstern (1944). “Theory of games and economic behavior”. Princeton University Press.M.J. Osborne and A. Rubinstein (1994). “A course on game theory”. The MIT Press.G. Owen (1995). “Game theory”. Academic Press.M. Maschler (1992). “The bargaining set, kernel, and nucleolus”. In R.J. Aumann and S. Hart (eds.), Handbook of Game TheoryVol. I. North-Holland.S. Moretti and F. Patrone (2008). “Transversality of the Shapley value”. Top 16, 1-41.D. Schmeidler (1969). “The nucleolus of a characteristic function game”. SIAM Journal on Applied Mathematics 17, 1163-1170.L.S. Shapley (1953). “A value for n-person games”. In H.W. Kuhn and A.W. Tucker (eds.), Contributions to the theory of games II.Princeton University Press.L.S. Shapley (1967). “On balanced sets and cores”, Naval Research Logistics Quarterly 14, 453-460.L.S. Shapley (1971). “Cores of convex games”, International Journal of Game Theory 1, 11-26.S.H. Tijs (1981). “Bounds for the core and the τ -value”. In O. Moeschlin and D. Pallaschke (eds.), Game theory andmathematical economics. North Holland.


Exercises

Exercises

1 Compute the Shapley value of the TU-game (N, v) given by:N = {1,2,3}, v(1) = v(2) = v(3) = 0, v(12) = 2,v(13) = v(23) = 4, v(N) = 8. Depict C(v) and obtain its extremepoints. Show that Φ(v) ∈ C(v).

2 Compute the Shapley value of the TU-game (N, v) given by:N = {1,2,3}, v(1) = v(2) = v(3) = 0, v(12) = 4, v(13) = 8,v(23) = 10, v(N) = 10. Give an imputation of v which dominates(2,4,4). Use the Bondareva-Shapley theorem to prove that C(v)is an empty set. If w(13) = w(23) = 4 and w(S) = v(S) for allother coalition S, depict C(w) and obtain its extreme points.

3 Prove that a TU-game (N, v) which satisfies thatI v(S) < 0 for all S ⊂ N, andI v(S)2 =

∑i∈S v(i)2 for all S ⊂ N,

has a non-empty core.


Cooperative Games I - imus.us.es · Introduction to Cooperative Games Preliminaries What is Game...

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