Unilaterally competitive games with more than two players
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Unilaterally competitive games with more than two players Takuya Iimura1 Accepted: 28 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove some interesting properties of unilaterally competitive games when there are more than two players. We show that such games possess: (1) a Nash equilibrium, (2) maximin-solvability, (3) strong solvability in the sense of Nash, and (4) weak acyclicity, all in pure strategies of finite or infinite games. Keywords Unilaterally competitive games · Existence of a pure strategy equilibrium · Maximin · Strongly solvable games · Weak acyclicity JEL Classification C72 (Noncooperative game)
1 Introduction The concept of a unilaterally competitive (UC) game is due to Kats and Thisse (1992). UC games are multi-person, finite or infinite, generally nonzero-sum games, in which each player improves his own payoff by a unilateral deviation if and only if all the opponents decrease their own payoffs. Such a situation naturally arises in the (broadly defined) relative payoff games, in which each player maximizes his own relative payoff that is negatively affected by any opponent’s absolute payoff. In the extreme case of no payoff externalities in the absolute payoffs, the relative payoff game is necessarily UC. Also, it is an ordinal potential game (Monderer and Shapley 1996). As we shall see in Sect. 3, however, some sort of this relative payoff game admits a continuous deformation from a potential game to a non-potential game while staying in the domain of UC games, as we continuously deform its base absolute payoff game. Thus, UC games are not confined to potential games. Like the mixed-extension of finite two-person zero-sum games, multi-person games that satisfy the UC condition in pure strategies have many nice properties, as follows. First, if nonempty, the set of pure Nash equilibria (hereafter equilibria) * Takuya Iimura [email protected] 1
Faculty of Economics and Business Administration, Tokyo Metropolitan University, Tokyo 192‑0397, Japan
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of UC games has payoff equivalence and interchangeability (Kats and Thisse 1992). Nash (1951) calls a game solvable if its set of equilibria is nonempty and interchangeable. Thus, the UC games are solvable in the sense of Nash if there exists an equilibrium. Second, if nonempty, the set of equilibria of a UC game coincides with the set of profiles of maximin strategies (De Wolf 1999). Thus, UC games are maximin-solvable if there exists an equilibrium, in the sense that they can be solved by finding maximin strategies. Here, the existence clauses are significant since there are UC games having no equilibrium, e.g., two-person Matching Pennies played in pure strategies. In this paper, we show, among other things, that UC games always possess an equilibrium when there are more than two players. Thus, we shall establish the solvability in the sense of Nash and maximin-solvablity for UC games with more than two players. This result is derived from the following two specific
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