Mathematical Social Sciences 39 2000 263–275 www.elsevier.nl locate econbase
Exact stability and its applications to strong solvability
J. Abdou
´ ´
CERMSEM , Universite de Paris 1, 12, Place du Pantheon, 75005 Paris, France
Received 1 October 1997; received in revised form 1 January 1999; accepted 1 April 1999
Abstract
We introduce the exact core and the biexact core of a strategic game form. Those are solutions which lie between the usual b -core and the set of strong equilibrium outcomes. We define the
corresponding notion of exact and biexact stability. We prove that a game form is exactly stable if and only if it is exact, tight and subadditive and that it is biexactly stable if and only if in addition
it is biexact. As an application, we study the exactness of rectangular game forms. We prove that an exact rectangular game form is essentially a one-player game form. In particular any strongly
solvable rectangular game form is essentially a one-player game form. This generalizes a result of Ichiishi, 1985.
2000 Elsevier Science B.V. All rights reserved.
1. Introduction
This paper contributes to the study of strongly solvable finite game forms via two new solution concepts namely the exact core and the biexact core and the related notions of
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exact and biexact stability. A game form henceforth GF is said to be strongly solvable if for any preference assignment to the players, the resulting game has a pure strong
equilibrium. A complete and useful characterization of such game forms seems to be out of reach for the time being. But some results are already available. Necessary conditions
for strong solvability, as a by-product of strong implementation theory, can be found in Peleg 1984 and Abdou and Keiding 1991. Since a strong equilibrium outcome is in
the b -core then a necessary condition for strong solvability is stability and the latter can be characterized by two structural properties: subadditivity and tightness. Recently Li
1991 has made a step further: tightness is replaced by a stronger property: exact
Tel.: 133-1-46-33-3448; fax: 133-1-46-33-3448. E-mail address
: abdouuniv-paris.1.fr J. Abdou
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Or strongly consistent: but since the words ‘‘consistence’’ and ‘‘consistency’’ are awfully polysemic we prefer ‘‘solvable’’, ‘‘solvability’’ which belong to the same paradigm as ‘‘solution’’.
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 3 7 - 2
264 J
. Abdou Mathematical Social Sciences 39 2000 263 –275
tightness. None of the mentioned conditions is sufficient for strong solvability. However for the two-player case necessary and sufficient conditions for strong solvability have
been established in Abdou 1995. Note that in that case, a new notion is needed in order to accomplish the characterization: joint exactness. The present paper carries out a
similar analysis for the general case. We introduce two new solution concepts called respectively exact core and biexact core which both refine the core and contain the
strong equilibrium outcomes. Therefore new necessary conditions for strong solvability are exact stability and biexact stability. It turns out that exactly stable game forms are
those which are subadditive and exactly tight. Thus our results imply that of Li 1991. The new ingredient between stability and exact stability or equivalently between
tightness and exact tightness is a property that we call exactness. Though the latter appears in this study as coupled with tightness, it has its own interpretation: a game form
is exact if and only if the exact core correspondence coincides with the b-core correspondence. Biexact stability is more stringent. A game form is biexactly stable if
and only if it is subadditive, tight and biexact. This in its turn extends the result of Abdou 1995 where joint exactness appears as a particular case of biexactness. The
second contribution of this paper is a characterization of exact and rectangular game forms.The class of rectangular game forms has been introduced by Gurvich Gurvich,
1975, 1989. A game form is rectangular if the inverse image of any alternative is a cartesian product of strategy subsets. Any free extensive game form, i.e. with the
endpoints of the underlying tree as outcomes, is rectangular. Thus the class of rectangular game forms is quite large. If we add exactness then this class shrinks
drastically: We prove that an exact rectangular game form is essentially a one-player game form. Note that we do not even need tightness for this ‘‘impossibility result’’. In
particular any strongly solvable rectangular game form is essentially a one-player game form. This generalizes a result of Ichiishi 1985 where the same is stated for free
extensive game forms.
2. Definitions and preliminaries
