Mathematical Social Sciences 37 1999 281–305
A new axiomatization of the Owen value for games with coalition structures
´ Gerard Hamiache
´ ´
G .R.E.Q.A.M., Centre de la Vieille Charite, 2 rue de la Charite, 13002 Marseille, France
Received 9 September 1997; received in revised form 28 February 1998; accepted 31 March 1998
Abstract
This paper presents a new axiomatization of the Owen value for games with coalition structures. The driving force of this new axiomatization is a consistency axiom based on an associated game
which is not a reduced game. We show that the Owen value is the only one satisfying, in addition with other rather ‘standard’ properties, a consistency requirement which makes it unnecessary to
perform a particular kind of manipulation.
1999 Elsevier Science B.V. All rights reserved. Keywords
: Owen value; Coalition structure; Associated game; Consistency; Iterative functional equation
JEL classification : C71
1. Introduction
The Owen 1977 value is a generalization of the Shapley 1953 value that, to some extent, answers the question: how the Shapley value of a game should be modified in
order to reflect membership of players to a priori unions. These unions are the expression of affinities existing between players. Each player is a member of one and
only one such union. Formally, these affinities induce a partition of the set of players called a coalition structure.
Another answer to that question was presented in the pioneering work of Aumann and `
Dreze 1974. These two authors consider that there are no lateral payments between coalitions of the coalition structure. Their -value coincides, in each coalition
separately, with the regular Shapley value of the corresponding restricted game.
Tel.: 133-4-9114-0770; fax: 133-4-9190-0227. E-mail address
: hamiacheehess.cnrs-mrs.fr G. Hamiache 0165-4896 99 – see front matter
1999 Elsevier Science B.V. All rights reserved.
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282 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
In this paper a new axiomatization of the Owen value for games with coalition structure is offered. Besides Owen 1977, more axiomatizations can be found in the
literature Hart and Kurz, 1983; Peleg, 1989; Winter, 1992. It is worth mentioning the interesting survey on games with coalition structures presented in Greenberg 1994.
An axiomatization unveils hidden properties of solution concepts, and reveals relations with other concepts which would not be suspected otherwise. For example, the
present work shows that the Owen value is the only one satisfying the property that players neither lose nor gain when performing a manipulation of a particular kind.
The driving force of the present axiomatization is an associated consistency axiom based on an associated game, that is not a reduced game. The main difference between
the two approaches is that the associated consistency requires the definition of a single game while the usual consistency requires the definition of a new game for each of the
coalitions. In this sense the present consistency axiom is less demanding.
In Section 2, we present the formal framework and review the Owen value. In Section 3, the new axiomatization is presented.
2. The Owen value
