Abstract
We examine games played by a single large player and a large number of opponents who are small, but not anonymous. If the play of the small players is observed with noise, and if the number of actions the large player controls is bounded as the number of small players grows, the equilibrium set converges to that of the game where there is a continuum of small players. This paper extends previous work on the negligibility of small players by dropping the assumption that small players' actions are “anonymous.” That is, we allow each small player's actions to be observed separately, instead of supposing that the small players' actions are only observed through their effect on an aggregate statistic.Journal of Economic LiteratureClassification Numbers: C72, C73.
Highlights
We examine games played by a single large player and a large number of opponents who are small, but not anonymous
This paper examines a game played by a single large player and a number of small opponents
All that matters for these results is the expected distribution of the large player’s actions contingent on various observations, so the propositions apply to models where the large players has several possible “types” as in the reputation effects literature
Summary
This paper examines a game played by a single large player and a number of small opponents. As we increase the number of small players, we hold fixed the players’ action sets, the utility functions ui(ai, b) , and the distribution over individual outcomes ρi( yi, ai ). Our main result is that when small players are asymptotically negligible and the large player observes only the small players’ outcomes, she can do no better in the limit than the Stackelberg payoff. This is analogous to a similar result in Levine and Pesendorfer (1995), who restrict the large player to play strategies that depend only on the average play of the small players.
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