A System of Equilibria for Many-Player Games

Abstract

A rich hierarchic system of gradually strengthening natural equilibria without any artificial behavior norms was found in [1–7] for noncooperative games of two participants; these equilibria include the weakest equilibrium existing in any game and quite strong equilibria providing the game solution uniqueness. Unfortunately, it turns out that even that quite “rich” system of equilibria, called basic, does not permit one to find a unique strongest equilibrium in each game. An iterative procedure of obtaining new equilibrium notions on the basis of the basic system was developed in [8–10] in this connection. But even it does not provide finding the unique solution in any game. Possibly, this problem cannot be solved even in principle. However, in any case the main aim of the game theory is to find the richest natural system of equilibria providing the existence of a unique solution in each game or, at least, in the widest possible class of game problems. As for games with more than two participants, results of the equilibrium theory developed for games with two participants are not necessarily valid for them. Finding new equilibrium notions for games with many participants is much more complicated than finding equilibria for games with two participants. Surely, in any case, one can use the fact that any game has at least the weak active equilibrium (an A-equilibrium). Consequently, the minimum basic system of equilibria can be constructed on the basis of the single A-equilibrium and a strongly dependent equilibrium [3]. However, even by using the above-mentioned iterative procedure [8–10] of generating new equilibria, one cannot find a unique solution in any game, given a quite limited basic system. The use of the idea of passive threats lying in the basis of the statement of the Nash classical equilibrium [11] for the construction of equilibrium notions with regard to active threats (just as in the case of, for example, a weakly dependent equilibria [3]) is a weak aid for the radical solution of the problem of finding the unique strongest equilibrium in a game, since, probably, the idea of passive threats has a quite restricted application field. This becomes especially clear for the Nash equilibrium itself, since, even in games in which it exists and is unique, it can be less preferable for participants than some Nash nonequilibrium situations; this can be illustrated by the well-known game “Prisoner dilemma” [12]. Only the introduction of the new notion of a D-equilibrium [7] clarifies that situation. In the present paper, we suggest new notions of equilibria for games with many participants, which can be reduced to already known notions in the case of games with two participants. For a sample game of three participants, we show that new equilibrium notions are more meaningful than the Nash equilibrium and are preferable to the weakly dependent equilibrium [3].