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First World Congress of the Game Theory Society (Games 2000)
July 24-28, 2000
Basque Country University and Fundacion B.B.V.
Bilbao, Spain

Organizers
Ehud Kalai, Federico Valenciano

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Bicooperative games
by
J. M. Bilbao
Applied Mathematics II, University of Seville
Coauthors: J. R. Fernández, A. Jiménez Losada, E. Lebrón

Let N = { 1, 2, ... , n } be a set of players. Aubin (1993) Optima and equilibria: an introduction to nonlinear analysis, introduced a generalized coalition as a function c:N --> [ -1, 1] which associates each player i with his/her level of participation c(i) in [ -1, 1]. A positive level is interpreted as attraction of the player i for the coalition, and a negative level as repulsion. In particular, the set 2N is the set of functions c:N --> {0, 1}. We will introduce a natural generalization of the concept of cooperative game. First, we consider the set of all the ordered pairs of disjoint coalitions, that is, the set of all signed coalitions
3N={( S, T):S, T subset N,   S \cap T = \emptyset} .
Each signed coalition ( S, T) of 3N can be identified with the {0, -1, 1}-vector 1( S, T) defined by
1( S, T) (i)= ì
ï
í
ï
î
1,
  if   i in S
-1,
  if   i in T
0,
  otherwise,
for all i in N. The support of a signed coalition ( S, T) of N is S \cup T. A partial order \sqsub on 3N is defined by ( S1, T1) \sqsub ( S2, T2) <===> S1 subset T1   and   S2 subset T2 for all ( S1, T1) , ( S2, T2) in 3N.

Example: Felsenthal and Machover, in Ternary voting games, Int. J. Game Theory 26 (1997) 335-351, introduced ternary voting games on a finite set N. This concept is a generalization of voting games which recognizes abstention as an option alongside yes and no votes. These games are given by mappings u:3N --> { -1, 1} satisfying the following three conditions:

  1. u( N, \emptyset) = 1,

  2. u( \emptyset, N) = -1,

  3. If 1( S, T) (i) <= 1(S', T') (i) for all i in N, then u(S, T) <= u( S', T') .
A negative outcome, -1, is interpreted as defeat and a positive outcome, 1, as passage of a bill.

A bicooperative game is a pair (N, b), where N is a finite set and b:3N --> R is a function such that b( \emptyset, \emptyset) = 0.

We have two binary operations, reduced union \sqcup and intersection \sqcap , on 3N defined as
( S1, T1) \sqcup ( S2, T2)
=
( (S1 \cup S2) \( T1 \cup T2) , (T1 \cup T2) \( S1 \cup S2) ) ,
( S1, T1) \sqcap ( S2, T2)
=
(S1 \cap S2, T1 \cap T2) .

A bicooperative game c:3N --> R is bisubmodular if it satisfies c( ( S1, T1) \sqcup (S2, T2) ) +c( ( S1, T1) \sqcap (S2, T2) ) <= c( S1, T1) +c(S2, T2) for all ( S1, T1) , (S2, T2) in 3N.

A bicooperative game b:3N --> R is bisupermodular if -b is bisubmodular and bimodular if the above inequality holds with equality.

We obtain the following two properties.

Let c:3N --> R be a bisubmodular game, let (S, T) be such that S \cup T=N, and let cST:2N --> R be the cooperative game given by cST( X) = c( S \cap X, T \cap X) for all X subset N. Then the game cST is concave.

Let v:2N --> R be a convex game, and let b:3N --> R be the bicooperative game given by
b( S, T) :=v( S) - v*(T)=v( S) + v(N\T) - v(N).
Then b is bisupermodular. Moreover, P*( b) = Core(v) .

http://www.esi.us.es/~mbilbao/

Date received: May 2, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caez-97.