Nash-2 Equilibrium: Selective Farsightedness Under Uncertain Response

Abstract

This paper provides an extended analysis of an equilibrium concept for non-cooperative games with boundedly rational players: Nash-2 equilibrium. Players think one step ahead and account for all profitable responses of player-specific subsets of opponents because of both the cognitive limitations on predicting everyone’s reaction and the inability to make deeper and certain predictions. They cautiously reject improvements that might lead to worse profits after some reasonable response. For n-person games we introduce the notion of a reflection network consisting of direct competitors to express the idea of selective farsightedness. For almost every 2-person game with a complete reflection network, we prove the existence of a Nash-2 equilibrium. Nash-2 equilibrium sets are obtained in models of price and quantity competition, and in Tullock’s rent-seeking model with two players. It is shown that such farsighted behavior may provide strategic support for tacit collusion. The analyses of n-person Prisoner’s dilemma and oligopoly models with a star reflection structure demonstrate some possibilities of strategic collusion and a large variety of potentially stable outcomes.