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Working paper

Nash-2 equilibrium: selective farsightedness under uncertain response

This paper provides an extended analysis of an equilibrium concept for non-cooperative games with boundedly rational players: a Nash-2 equilibrium. Players think one step ahead and account all profitable responses of player-specific subsets of opponents because of both the cognitive limitations to predict everyone's reaction and the inability to make more deep and certain prediction even about a narrow sample of agents. They cautiously reject improvements that might lead to poorest profit after some possible reasonable response. For $n$-person games we introduce a notion of 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 Nash-2 equilibrium. Nash-2 equilibrium sets in the models of price and quantity competition, and in Tullock's rent-seeking model with 2 players are obtained. It is shown that such a farsighted behavior may provide a strategic support for tacit collusion.