cooperative game theory

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 Games

Lecture 1: Cooperative Game Theory

Sırma Zeynep Alparslan GokSuleyman Demirel UniversityFaculty of Arts and SciencesDepartment of Mathematics

Isparta, [email protected]

August 13-16, 2011


Outline

Introduction

Introduction to cooperative game theory

Basic solution concepts of cooperative game theory

Balanced games

Shapley value and Weber set

Convex games

Population Monotonic Allocation Schemes (pmas)

References


Introduction

Introduction

I Game theory is a mathematical theory dealing with models ofconflict and cooperation.

I Game Theory has many interactions with economics and withother areas such as Operations Research and social sciences.

I A young field of study:The start is considered to be the book Theory of Games andEconomic Behaviour by von Neumann and Morgernstern.

I Game theory is divided into two parts: non-cooperative andcooperative.


Introduction

Introduction

Cooperative game theory deals with coalitions who coordinate theiractions and pool their winnings.Natural questions for individuals or businesses when dealing withcooperation are:

I Which coalitions should form?

I How to distribute the collective gains (rewards) or costsamong the members of the formed coalition?



Cooperative game theory

I A cooperative n-person game in coalitional form (TU(transferable utility) game) is an ordered pair < N, v >, whereN = {1, 2, ..., n} (the set of players) and v : 2N → R is amap, assigning to each coalition S ∈ 2N a real number, suchthat v(∅) = 0.

I v is the characteristic function of the game.

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



Example (Glove game)

N = {1, 2, 3}. Players 1 and 2 possess a left-hand glove and theplayer 3 possesses a right-hand glove. A single glove is worthnothing and a right-left pair of glove is worth 10 euros.Let us construct the characteristic function v of the game< N, v >.

v(∅) = 0, v({1}) = v({2}) = v({3}) = 0,

v({1, 2}) = 0, v({1, 3}) = v({2, 3}) = 10, v(N) = 10.



Cooperative game theory

I GN : the set of coalitional games with player set N

I GN forms a (2|N| − 1)-dimensional linear space equipped withthe usual operators of addition and scalar multiplication offunctions.

I A basis of this space is supplied by the unanimity games uT

(or < N, uT >), T ∈ 2N \ {∅}, which are defined by

uT (S) :=

{1, if T ⊂ S0, otherwise.

The interpretation of the unanimity game uT is that a gain (orcost savings) of 1 can be obtained if and only if all players incoalition T are involved in cooperation.



Example

Since the unanimity games is the basis of coalitional games, eachcooperative game can be written in terms of unanimity games.Consider the game < N, v > with N = {1, 2}, v({1}) = 3,v({2}) = 4 and v(N) = 9.Here v = 3u{1} + 4u{2} + 2u{1,2}.Let us check it

v({1}) = 3u{1}({1}) + 4u{2}({1}) + 2u{1,2}({1}) = 3 + 0 + 0 = 3.

v({2}) = 3u{1}({2}) + 4u{2}({2}) + 2u{1,2}({2}) = 0 + 4 + 0 = 4.

v({1, 2}) = 3u{1}({1, 2})+4u{2}({1, 2})+2u{1,2}({1, 2}) = 3+4+2 = 9.




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:∑n

i=1 xi = v(N)

Example (Glove game continues): The imputation set of the glovegame LLR is the triangle with vertices

f 1 = (10, 0, 0), f 2 = (0, 10, 0), f 3 = (0, 0, 10)

I (v) = conv {(10, 0, 0), (0, 10, 0), (0, 0, 10)}

(solution of the linear systemx1 + x2 + x3 = 10, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0).



The core (Gillies (1959))

The core 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.

I If C (v) 6= ∅, then elements of C (v) can easily be obtained,because the core is defined with the aid of a finite system oflinear inequalities (optimization-linear programming (seeDantzig(1963))).

I The core is a convex set and the core is a polytope (seeRockafellar (1970)).



Example (Glove game continues)...

The core of the LLR game consists of one point (0, 0, 10).

C (v) = {(0, 0, 10)}

(solution of the linear system x1 + x2 + x3 = 10,

x1 + x2 ≥ 0, x1 + x3 ≥ 10, x2 + x3 ≥ 10, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.)


Balanced games

Balanced game

A map λ : 2N \ {∅} → R+ is called a balanced map if∑S∈2N\{∅} λ(S)eS = eN .

Here, eS is the characteristic vector for coaliton S with

eSi :=

{1, if i ∈ S0, if i ∈ N \ S .

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).


Balanced games

Example

For N = {1, 2, 3}, the set B = {{1, 2} , {1, 3} , {2, 3}} is balancedand corresponds to the balanced map λ with λ(S) = 1

2 if |S | = 2.That is

1

2v({1, 2}) +

1

2v({1, 3}) +

1

2v({2, 3}) ≤ v(N)

Let us show it:

λ({1, 2})e{1,2} + λ({1, 3})e{1,3} + λ({2, 3})e{2,3} = eN

λ({1, 2})(1, 1, 0) + λ({1, 3})(1, 0, 1) + λ({2, 3})(0, 1, 1) = (1, 1, 1).

Solution of the above system isλ({1, 2}) = λ({1, 3}) = λ({2, 3}) = 1

2 .


Balanced games

Balanced game

The importance of a balanced game becomes clear by the followingtheorem which characterizes games with a non-empty core.

Theorem (Bondareva (1963) and Shapley (1967)): Let < N, v >be an n-person game. Then the following two assertions areequivalent:

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



Marginal contribution

Let v ∈ GN . For each i ∈ N and for each S ∈ 2N with i ∈ S , themarginal contribution of player i to the coalition S is

Mi (S , v) := v(S)− v(S \ {i}).

Let Π(N) be the set of all permutations σ : N → N of N.The set Pσ(i) :=

{r ∈ N|σ−1(r) < σ−1(i)

}consists of all

predecessors of i with respect to the permutation σ.Let v ∈ GN and σ ∈ Π(N).The marginal contribution vector mσ(v) ∈ Rn with respect to σand v has the i-th coordinate the valuemσ

i (v) := v(Pσ(i) ∪ {i})− v(Pσ(i)) for each i ∈ N.



Example

Let < N, v > be the three-person game with v({i}) = 0 for eachi ∈ N, v({1, 2}) = 3, v({1, 3}) = 5, v({2, 3}) = 7, v(N) = 10.Then the marginal vectors are given in the following table, whereσ : N → N is identified with (σ(1), σ(2), σ(3)).

σ(123)(132)(213)(231)(312)(321)

mσ1 (v) mσ

2 (v) mσ3 (v)

0 3 70 5 53 0 73 0 75 5 03 7 0

.



The Shapley value (Shapley (1953)) and the Weber set(Weber (1988))

The Shapley value φ(v) of a game v ∈ GN is the average of themarginal vectors of the game

φ(v) := 1n!

∑σ∈Π(N) mσ(v).

This value associates to each n-person game one (payoff) vector inRn.The Shapley value of the previous example is

φ(v) =1

3!(14, 20, 26) = (

7

3,

10

3,

13

3).

The Weber set (Weber (1988)) W (v) of v is defined as the convexhull of the marginal vectors of v .Theorem: Let v ∈ GN . Then C (v) ⊂W (v).



Example (LLR game)

Marginal vectors can be observed from the following table

σ(123)(132)(213)(231)(312)(321)

mσ1 (v) mσ

2 (v) mσ3 (v)

0 0 100 0 100 0 100 0 10

10 0 00 10 0

.

The Weber set is W (v) = conv {(0, 0, 10), (10, 0, 0), (0, 10, 0)}.The Shapley value is φ(v) = ( 1

6 ,16 ,

23 ).

Note that {(0, 0, 10)} = C (v) ⊂W (v) and φ(v) ∈W (v).


Convex games

Convex games

I < N, v > is convex if and only if the supermodularitycondition v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T ) for eachS ,T ∈ 2N holds (desirable for reward games).

I < N, v > is called concave (or submodular) if and only ifv(S ∪ T ) + v(S ∩ T ) ≤ v(S) + v(T ) for all S ,T ∈ 2N

(desirable for cost games).

I CGN–The family of all convex games with player set N.


Convex games

Theorem (characterizations of convex games): Let v ∈ GN . Thefollowing five assertions are equivalent:

(i) < N, v > is convex.

(ii) For all S1,S2,U ∈ 2N with S1 ⊂ S2 ⊂ N \ U we have

v(S1 ∪ U)− v(S1) ≤ v(S2 ∪ U)− v(S2).

(iii) For all S1,S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N \ {i} wehave

v(S1 ∪ {i})− v(S1) ≤ v(S2 ∪ {i})− v(S2).

(iv) Each marginal vector mσ(v) of the game v with respect tothe permutation σ belongs to the core C (v).

(v) W (v) = C (v).


Convex games

I For convex games the gain made when individuals or groupsjoin larger coalitions is higher than when they join smallercoalitions.

I A convex game is balanced and the core of the convex gamesis nonempty.

I The Shapley value is a core element if the game is convex.




For a game v ∈ GN and a coalition T ∈ 2N \ {∅}, the subgamewith player set T , (T , vT ), is the game vT defined byvT (S) := v(S) for all S ∈ 2T .

A game v ∈ GN is called totally balanced if (the game and) all itssubgames are balanced.

The class of totally balanced games includes the class of gameswith a population monotonic allocation scheme (pmas) (Sprumont(1990)).



Let v ∈ GN . A scheme a = (aiS)i∈S,S∈2N\{∅} of real numbers is apmas of v if

(i)∑

i∈S aiS = v(S) for all S ∈ 2N \ {∅},(ii) aiS ≤ aiT for all S ,T ∈ 2N \ {∅} with S ⊂ T and for each

i ∈ S .

I Interpretation: in larger coalitions, higher rewards (or in largercoalitions lower costs).

I It is known that for v ∈ CGN the (total) Shapley valuegenerates population monotonic allocation schemes. Further,in a convex game all core elements generate pmas.



ExampleLet < N, v > be the 3-person game with v({1}) = 10,v({2}) = 20, v({3}) = 30,v({1, 2}) = v({1, 3}) = v({2, 3}) = 50, v(N) = 102.Then a pmas is the (total) Shapley value.

Φ({1, 3} , v)→

N{1, 2}{1, 3}{2, 3}{1}{2}{3}

1 2 329 34 3920 30 ∗15 ∗ 35∗ 20 30

10 ∗ ∗∗ 20 ∗∗ ∗ 30

.



For detailed information about Cooperative game theory see

I Introduction to Game Theory by Tijs

and

I Models in Cooperative Game Theory by Branzei, Dimitrovand Tijs.


References

References

[1]Bondareva O.N., Certain applications of the methods of linearprogramming to the theory of cooperative games, ProblemlyKibernetiki 10 (1963) 119-139 (in Russian).[2]Branzei R., Dimitrov D. and Tijs S., Models in CooperativeGame Theory, Springer (2008).[3]Dantzig G. B., Linear Programming and Extensions, PrincetonUniversity Press (1963).[4]Gillies D. B., Solutions to general non-zero-sum games, In:Tucker, A.W. and Luce, R.D. (Eds.), Contributions to theory ofgames IV, Annals of Mathematical Studies 40. PrincetonUniversity Press, Princeton (1959) pp. 47-85.[6] Rockafellar R.T., Convex Analysis, Princeton University Press,Princeton, (1970).


References

References

[7]Shapley L.S., On balanced sets and cores, Naval ResearchLogistics Quarterly 14 (1967) 453-460.[8]Shapley L.S., A value for n-person games, Annals ofMathematics Studies, 28 (1953) 307-317.[9]Sprumont Y., Population monotonic allocation schemes forcooperative games with transferable utility, Games and EconomicBehavior, 2 (1990) 378-394.[10] Tijs S., Introduction to Game Theory, SIAM, Hindustan BookAgency, India (2003).[11] von Neumann J. and Morgenstern O. , Theory of Games andEconomic Behavior, Princeton Univ. Press, Princeton NJ (1944).[12] Weber R., Probabilistic values for games, in Roth A.E. (ed.),The Shapley Value: Essays in Honour of Lloyd S. Shapley,Cambridge University Press, Cambridge (1988) 101-119.

cooperative game theory

Education

game n

suchthat v

setc v

eachcooperative game

characteristic function

elements of c v

unanimity game ut

nexample glove game