Values of Games with Infinitely Many Players

Abraham Neyman

The Shapley value is one of the basic solution concepts of cooperative gaem theory. It can be viewed as a sort of average or expected outcome, or as an a priori evaluation of the players' expected payoffs. The value has a very wide range of applications, particularly in economics and political science (see chapters 32, 33 and 34 in this Handbook). In many of these applications it is necessary to consider games that involve a large number of players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single-handed influence. A typical example is provided by voting among stockholders of a corporation, with a few major stockholders and an "ocean" of minor stockholders. In economics, one considers an oligopolistic sector of firms embedded in a large population of "perfectly competitive" consumers. In all of these cases, it is fruitful to model the game as one with a continuum of players. In general, the continuum consists of a non-atomic part (the "ocean"), along with (at most countably many) atoms. The continuum provides a convenient framework for mathematical analysis, and approximates the results for large finite games well. Also, it enables a unified view of games with finite, countable or oceanic player-sets, or indeed any mixture of these.

June, 2001
Published in: 
Handbook of Game Theory, with Economic Applications, Vol. III, R. J. Aumann and S. Hart (eds.), Elsevier North-Holland (2002), 2121-2167.