Cooperative Game Theory

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

Cooperative Game Theory. Operations Research
Games. Applications to Interval Games
Lecture 1: Cooperative Game Theory

Sırma Zeynep Alparslan G¨k
o
S¨leyman Demirel University
u
Faculty of Arts and Sciences
Department of Mathematics
Isparta, Turkey
zeynepalparslan@yahoo.com

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

Game theory is a mathematical theory dealing with models of
conﬂict and cooperation.
Game Theory has many interactions with economics and with
other areas such as Operations Research and social sciences.
A young ﬁeld of study:
The start is considered to be the book Theory of Games and
Economic Behaviour by von Neumann and Morgernstern.
Game theory is divided into two parts: non-cooperative and
cooperative.

Introduction

Introduction

Cooperative game theory deals with coalitions who coordinate their
actions and pool their winnings.
Natural questions for individuals or businesses when dealing with
cooperation are:
Which coalitions should form?
How to distribute the collective gains (rewards) or costs
among the members of the formed coalition?


Cooperative game theory

A cooperative n-person game in coalitional form (TU
(transferable utility) game) is an ordered pair < N, v >, where
N = {1, 2, ..., n} (the set of players) and v : 2N → R is a
map, assigning to each coalition S ∈ 2N a real number, such
that v (∅) = 0.
v is the characteristic function of the game.
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 the
player 3 possesses a right-hand glove. A single glove is worth
nothing 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

G N : the set of coalitional games with player set N
G N forms a (2|N| − 1)-dimensional linear space equipped with
the usual operators of addition and scalar multiplication of
functions.
A basis of this space is supplied by the unanimity games uT
(or < N, uT >), T ∈ 2N {∅}, which are deﬁned by

1, if T ⊂ S
uT (S) :=
0, otherwise.

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


Example

Since the unanimity games is the basis of coalitional games, each
cooperative 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
x is individually rational: xi ≥ v ({i}) for all i ∈ N
n
x is eﬃcient: i=1 xi = v (N)
Example (Glove game continues): The imputation set of the glove
game 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 system
x1 + 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 )| xi ≥ v (S) for all S ∈ 2N {∅} .
i∈S

The idea of the core is by giving every coalition S at least their
worth v (S) so that no coalition has an incentive to split off.
If C (v ) = ∅, then elements of C (v ) can easily be obtained,
because the core is defined with the aid of a finite system of
linear inequalities (optimization-linear programming (see
Dantzig(1963))).
The core is a convex set and the core is a polytope (see
Rockafellar (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 N
S∈2N {∅} λ(S)e = e .
Here, e S is the characteristic vector for coaliton S with

1, if i ∈ S
eiS :=
0, if i ∈ N S.

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

λ(S)v (S) ≤ v (N).
S

Balanced games

Example
For N = {1, 2, 3}, the set B = {{1, 2} , {1, 3} , {2, 3}} is balanced
1
and corresponds to the balanced map λ with λ(S) = 2 if |S| = 2.
That is
1 1 1
v ({1, 2}) + v ({1, 3}) + v ({2, 3}) ≤ v (N)
2 2 2
Let us show it:

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

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 is
1
λ({1, 2}) = λ({1, 3}) = λ({2, 3}) = 2 .

Balanced games

Balanced game

The importance of a balanced game becomes clear by the following
theorem 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 are
equivalent:
(i) C (v ) = ∅,
(ii) < N, v > is a balanced game.


Marginal contribution

Let v ∈ G N . For each i ∈ N and for each S ∈ 2N with i ∈ S, the
marginal 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 P σ (i) := r ∈ N|σ −1 (r ) < σ −1 (i) consists of all
predecessors of i with respect to the permutation σ.
Let v ∈ G N and σ ∈ Π(N).
The marginal contribution vector mσ (v ) ∈ Rn with respect to σ
and v has the i-th coordinate the value
miσ (v ) := v (P σ (i) ∪ {i}) − v (P σ (i)) for each i ∈ N.


Example
Let < N, v > be the three-person game with v ({i}) = 0 for each
i ∈ 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, where
σ : N → N is identiﬁed with (σ(1), σ(2), σ(3)).
 
σ σ σ σ
 m1 (v ) m2 (v ) m3 (v ) 
 
(123) 
 0 3 7 

(132) 
 0 5 5 
.
(213) 
 3 0 7 

(231) 
 3 0 7 

(312)  5 5 0 
(321) 3 7 0


The Shapley value (Shapley (1953)) and the Weber set
(Weber (1988))
The Shapley value φ(v ) of a game v ∈ G N is the average of the
marginal vectors of the game
1 σ (v ).
φ(v ) := n! σ∈Π(N) m

This value associates to each n-person game one (payoﬀ) vector in
Rn .
The Shapley value of the previous example is
1 7 10 13
φ(v ) = (14, 20, 26) = ( , , ).
3! 3 3 3
The Weber set (Weber (1988)) W (v ) of v is deﬁned as the convex
hull of the marginal vectors of v .
Theorem: Let v ∈ G N . Then C (v ) ⊂ W (v ).


Example (LLR game)
Marginal vectors can be observed from the following table
 
σ σ σ σ
 m1 (v ) m2 (v ) m3 (v ) 
 
(123) 
 0 0 10 

(132) 
 0 0 10 
.
(213) 
 0 0 10 

(231) 
 0 0 10 

(312)  10 0 0 
(321) 0 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 , 1 , 3 ).
6 6
2

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

Convex games

Convex games

< N, v > is convex if and only if the supermodularity
condition v (S ∪ T ) + v (S ∩ T ) ≥ v (S) + v (T ) for each
S, T ∈ 2N holds (desirable for reward games).
< N, v > is called concave (or submodular) if and only if
v (S ∪ T ) + v (S ∩ T ) ≤ v (S) + v (T ) for all S, T ∈ 2N
(desirable for cost games).
CG N –The family of all convex games with player set N.

Convex games

Theorem (characterizations of convex games): Let v ∈ G N . The
following 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} we
have

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

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

Convex games

For convex games the gain made when individuals or groups
join larger coalitions is higher than when they join smaller
coalitions.
A convex game is balanced and the core of the convex games
is nonempty.
The Shapley value is a core element if the game is convex.



For a game v ∈ G N and a coalition T ∈ 2N {∅}, the subgame
with player set T , (T , vT ), is the game vT deﬁned by
vT (S) := v (S) for all S ∈ 2T .

A game v ∈ G N is called totally balanced if (the game and) all its
subgames are balanced.

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


Let v ∈ G N . A scheme a = (aiS )i∈S,S∈2N {∅} of real numbers is a
pmas 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.

Interpretation: in larger coalitions, higher rewards (or in larger
coalitions lower costs).
It is known that for v ∈ CG N the (total) Shapley value
generates population monotonic allocation schemes. Further,
in a convex game all core elements generate pmas.


Example
Let < 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.
 

 1 2 3 

N 
 29 34 39 

{1, 2} 
 20 30 ∗ 

Φ({1, 3} , v ) → {1, 3} 
 15 ∗ 35 .

{2, 3} 
 ∗ 20 30 

{1} 
 10 ∗ ∗ 

{2}  ∗ 20 ∗ 
{3} ∗ ∗ 30


For detailed information about Cooperative game theory see
Introduction to Game Theory by Tijs
and
Models in Cooperative Game Theory by Branzei, Dimitrov
and Tijs.

References

References
[1]Bondareva O.N., Certain applications of the methods of linear
programming to the theory of cooperative games, Problemly
Kibernetiki 10 (1963) 119-139 (in Russian).
[2]Branzei R., Dimitrov D. and Tijs S., Models in Cooperative
Game Theory, Springer (2008).
[3]Dantzig G. B., Linear Programming and Extensions, Princeton
University 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 of
games IV, Annals of Mathematical Studies 40. Princeton
University 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 Research
Logistics Quarterly 14 (1967) 453-460.
[8]Shapley L.S., A value for n-person games, Annals of
Mathematics Studies, 28 (1953) 307-317.
[9]Sprumont Y., Population monotonic allocation schemes for
cooperative games with transferable utility, Games and Economic
Behavior, 2 (1990) 378-394.
[10] Tijs S., Introduction to Game Theory, SIAM, Hindustan Book
Agency, India (2003).
[11] von Neumann J. and Morgenstern O. , Theory of Games and
Economic 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

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