Mathematical Social Sciences 37 1999 189–204
A necessary and sufficient condition for the convexity in oligopoly games
Jingang Zhao
Department of Economics , Ohio State University, 1945 North High Street, Columbus, OH 43210-1172,
USA Received 23 October 1997; received in revised form 5 March 1998; accepted 30 March 1998
Abstract
This paper establishes a necessary and sufficient condition for the convexity or supermodulari- ty in oligopoly games.
1999 Elsevier Science B.V. All rights reserved.
Keywords : Convex games; Supermodularity; Oligopoly industry; a -core; b-core
JEL classification : C71; D43; L13
1. Introduction
Convex games are equivalent to supermodular functions, which represent the rank function of matroids in combinatorics. Such convexity or supermodularity arises not
only in economic problems like coalition production economies, cost allocation problems, and oligopoly problems, but also in optimization problems like network
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problems. The literature on convex games is characterized by two features. First,
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convexity was used in the early works to establish the geometric structure of its core allocations i.e. the extreme points of the core of a convex game are the marginal value
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vectors of the game, then many weak versions or extensions of convexity followed. Because these properties and extensions follow from the original convexity, they all can
Tel.: 11-614-292-6523; fax: 11-614-292-3906; e-mail: Zhao.18Osu.Edu
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See Fujishige, 1991; McLean and Sharkey, 1993 for short survey.
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See Edmonds, 1970; Maschler et al., 1972; Shapley, 1971.
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˜ These include average convexity Inarra and Usategui, 1993; Sprumont, 1990, k-convexity Driessen, 1986,
odd submodularity Qi, 1988, permutational submodularity Granot and Huberman, 1982, semiconvex ¨
games Driessen and Tijs, 1985, extension to convex set functions Rosenmuller and Weidner, 1974, and extension to games without sidepayments Sharkey, 1981; Vilkov, 1977.
0165-4896 99 – see front matter
1999 Elsevier Science B.V. All rights reserved. P I I : S 0 1 6 5 - 4 8 9 6 9 8 0 0 0 1 9 - 5
190 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
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be understood as providing necessary conditions for convexity. Second, few studies established convexity, and these works can be understood as providing sufficient
conditions for convexity. In contrast to the existing literature, this article establishes a necessary and sufficient
condition for convexity in oligopoly games, based on the primitive parameters in an oligopoly industry. Our result also makes a step forward in solving the problem of ‘‘how
to split joint profits among firms engaged in a merger.’’ When firms engaged in a merger are asymmetric, the most difficult and most
important issue on the negotiation table is how to share the benefits of the merger. For example, when all n firms are merging into a monopolist, the issue becomes whether or
not to agree on a specific split of monopoly profit among the n members. For such a
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merger i.e. the monopoly merger to take place, the associated core has to be nonempty otherwise, the blocking coalitions will not join the merger. When the associated
transferable utility game is convex, the issue of splitting profits can be perfectly answered by using either Shapley value Shapley, 1953 or nucleolus Schmeidler,
1969, because they are both in the core.
The rest of the paper is organized as follows. Section 2 provides the definitions and notations, Section 3 presents the main results and examples, Section 4 establishes the
proofs, Section 5 discusses some extensions and provides some concluding remarks.
2. Definitions and notations
