Strategic behavior in non-atomic games ¬リニ

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a strategy is a strategic equilibrium if it is a limit point of a sequence of Nash equilibria of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. Moreover, it is established that under the mass action interpretation of Nash [9], the concept of strategic equilibrium is the extension of Nash equilibrium in finite normal form games, to non-atomic games by also proving the following: Given any finite normal form game, we consider its symmetric replicas and non-atomic version and show that every strategic equilibrium can be approached by a sequence Nash equilibria of replica games, and every limit point of such a sequence constitutes a strategic equilibrium. Finally, this notion is applied to non-atomic games of voting and it is displayed that if the perturbed game is such that the societal choice does not involve discontinuities of high magnitudes, then each player choosing his favorite candidate is the unique strategic equilibrium. ∗We are grateful to Narayana Kocherlakota, Andy McLennan, Aldo Rusticini and Jan Werner for their valuable advice and support. We thank Pedro Amaral, Kemal Badur, Han Ozsoylev, and especially David Schmeidler for helpful comments and suggestions. We benefited from discussions in the Mathematical Economics Workshop at the University of Minnesota. All remaining errors are ours.