Strategic Behavior in Non-Atomic Games

Typically, economic situations featuring a large number of agents are not modelled with a finite normal form game, rather by a non-atomic game. Consequently, the possibility of strategic interaction may be completely ignored. In order to restore strategic interaction among agents we propose a refinement of Nash equilibrium, strategic equilibrium, for non-atomic games with a continuum of agents, each of whose payo® depends on what he chooses and a societal choice. Given a non-atomic game, we consider a perturbed game in which every player believes that he alone has a small, but positive, impact on the societal choice. A strategy profile 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 every strategic equilibrium must be a Nash equilibrium of the original non-atomic game, thus, our concept of strategic equilibrium is indeed a refinement of Nash equilibrium. Next, we show that the concept of strategic equilibrium is the natural extension of Nash equilibrium infinite normal form games, to non-atomic games: That is, given any finite normal form game, we consider its non- atomic version, and prove that a strategy profile, in the non-atomic version of the given finite normal form game, is a strategic equilibrium if and only if the associated strategy profile in the finite form game is a Nash equilibrium. Finally, applications of strategic equilibrium is presented examples in which the set of strategic equilibria, in contrast with the set of Nash equilibria, does not contain any implausible Nash equilibrium strategy profiles. These examples are: a game of proportional voting, a game of allocation of public resources, and finally non-atomic Cournot oligopoly.