Mathematical Social Sciences 39 2000 175–194 www.elsevier.nl locate econbase
Stable effectivity functions and perfect graphs
1
Endre Boros , Vladimir Gurvich
RUTCOR , Rutgers, The State University of New Jersey, 640 Bartholomew Road, Piscataway,
NJ 08854-8003, USA
Received 30 October 1995; received in revised form 30 January 1999; accepted 28 February 1999
Abstract
We consider the problem of characterizing the stability of effectivity functions EFF, via a correspondence between game theoretic and well-known combinatorial concepts. To every EFF
we assign a pair of hypergraphs, representing clique covers of two associated graphs, and obtain some necessary and some sufficient conditions for the stability of EFFs in terms of graph-
properties. These conditions imply, for example, that to check the stability of an EFF is an NP-complete problem. We also translate some well-known conjectures of graph theory into game
theoretic language and vice versa.
2000 Elsevier Science B.V. All rights reserved.
Keywords : Core; Kernel; Stable effectivity function; Perfect graph; Kernel-solvable graph
1. Introduction
Let us consider multiplayer games, in which I and A denote finite sets of players and outcomes, respectively. Subsets of players are called coalitions, while subsets of
outcomes are called blocks. An effectivity function or EFF in short is a mapping
I A
:2 3 2 ∞ h0, 1j representing the ‘power’ of the players, in very general terms.
Namely, K, B 5 1 for K I and B A, i.e. K is said to be effective for B, exactly when the coalition K is able to guarantee that an outcome from the block B will be
realized, see, for example, Moulin 1983, Peleg 1984. Obviously, the players of a
Corresponding author. Tel.: 11-732-445-3235; fax: 11-732-445-5472. E-mail address
: borosrutcor.rutgers.edu E. Boros
1
On leave from the International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow, Russia.
0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved. P I I : S 0 1 6 5 - 4 8 9 6 9 9 0 0 0 1 7 - 7
176 E
. Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
coalition K may choose to cooperate in many different ways, hence the same coalition may be effective for many different blocks of the outcomes.
We shall consider monotone effectivity functions, i.e. for which K9, B9 K, B holds for all K9 K and B9 B. It is quite natural to assume both that a larger set of
players have a greater influence, and that it is easier to guarantee the final outcome to fall into a larger set. Hence monotonicity is a natural assumption for effectivity functions
corresponding to games.
A simple example could be a voting game, i.e. in which players are voters, and outcomes are candidates. In this case the effectivity function describes the power of
certain groups of voters coalitions to eliminate some of the candidates from the final race.
Another example is given by game forms kI, A, X, gl, where X 5 3
X , and the
i [I i
finite set X represents the possible strategies of the player i [ I, and where the
i
mapping g:X∞A defines the game form, i.e. determines the outcome for each situation in which all the players have chosen a strategy. A coalition K I is effective for a block
B A whenever the players of K can choose a strategy vector x [ 3 X such that
K i [K
i
the restriction gx , p takes values only from B, i.e. if the outcome of the game falls
K
into the set B no matter what strategy the players outside of K follow. Not all effectivity functions correspond, of course, to game forms. Besides the
monotonicity, there are other fairly natural properties satisfied by effectivity functions of game forms. One of them, the so called superadditivity states that if two disjoint
coalitions K and K are, respectively, effective for the blocks B and B , then, since the
1 2
1 2
players of both coalitions can independently practice their power, the union coalition K K must also be effective for the intersection B B . Moulin and Peleg 1982
1 2
1 2
proved that monotonicity and superadditivity together with some boundary conditions characterize the effectivity functions corresponding to game forms.
The individual preferences of the players over the different outcomes are represented by a utility function, i.e. by a mapping of the form u:I 3 A∞
R, where the real value ui, a for a player i [ I and outcome a [ A is interpreted as the profit of player i in the case
of outcome a is realized. For a coalition K I and outcome a [ A let PRK, a, u 5
ha9 [ Auui, a9 . ui, a for all players i [ Ij denote the set of outcomes preferred over a unanimously by all players of K. We shall say that a coalition K can reject an
outcome a if K, PRK, a, u 5 1, i.e. if the players in K have the power guaranteeing that the outcome of the game will be preferred by all of them over a.
For a given effectivity function and utility u, the core , u is defined as the set of outcomes not rejected by any of the coalitions. One of the central problems of game
theory is to find conditions guaranteeing that the core , u is not empty. A somewhat surprising fact is that the non-emptiness of , u can be guaranteed in certain cases,
even if the utility function is not specified. In other words, one can find conditions based only on the structure of the effectivity function, which imply that , u ± 5 for all
utility functions u. Effectivity functions for which this happens are called stable.
In this paper we study the stability of effectivity functions, present some necessary and some sufficient conditions for stability, and show, among other results that testing
the stability of a given effectivity function is an NP-complete problem.
E . Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
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2. Results
