Games with Many Players as Models of Large Economies

In this chapter we model economies as coalitional games, provide some new results, and review results showing that under apparently mild conditions games with many players share properties of markets. The examples include economies with pure or local public goods, economies with large or small clubs, coalition production economies, and economies with ever-increasing returns to firm size. The framework presented, of parameterized collections of games, includes cooperative games as discussed in Chapters 5 and 8 of this volume. For the case of quasi-linear utilities, the framework also incorporates games derived from the club and local public good economies treated in Chapter 7 of this volume.

Roughly, the key assumptions we require on economies and their derived games are that

1. The economies are essentially superadditive, that is, a group of players can freely break into smaller groups.

2. Gains to forming increasingly larger coalitions become small as coalitions become large.

The second condition simply rules out unboundedly large average payoffs as population size increases. We stress that the conditions permit both situations in which optimality requires that the total player set be partitioned into relatively small groups as well as some situations with a fixed, finite number of firms, clubs, or jurisdictions. We review results showing that, under our two conditions, economies and the games they generate have nonempty approximate cores and that the games are approximated by market games, that is, games generated by private goods economies in which all players have concave, quasi linear monotonic increasing objective functions (utility functions or production functions).