Integral games

Abstract

GAMES with an infinite number of players where the pay-offs depend only on the “popularity” of the strategies, are considered. A proof is given of the convexity of the set of equilibrium plans in such games. A finite algorithm for finding this set is presented for the case where the set of players admits of a finite subdivision. We frequently encounter situations in which the number of participants is very great and an individual participant cannot affect the total situation, it can be varied only by the combined behaviour of large groups of participants. Examples of such situations are the distributions of purchasers over shops, passengers over transport routes, drivers over routes, students in institutes of higher education, referenda and the like. Such situations can be described and analyzed in terms of non-atomic games, that is, games in which a positive non-atomic measure is defined on the set of players. In characteristic form such games were described by Owen [1]. In the normal form non-atomic games were defined by Schmeidler [2], who proved the existence ofNash equilibrium [3]. In the present paper we consider games where the pay-offs of the players depend on the “popularity” of their chosen strategies. We describe an important particular case of such games and an algorithm for finding all the equilibrium situations in this case.