A Nash equilibrium of a game describes, by definition, a situation in which no player can improve his objective functional by a unilateral deviation from the equilibrium strategy. It has to be emphasized, however, that joint deviations by more than one player could lead to such improvements. In particular, Nash equilibria are usually not Pareto efficient. This observation raises the question of whether there exist efficient Nash equilibria at all and whether there are any general methods for constructing such equilibria. In the present chapter we present one such method which is based on the use of so-called trigger strategies.

Trigger strategies are non-Markovian, that is, they determine the control variable at time t as a function of the entire history of the endogenous variables up to time t and not just as a function of the state at time t. In the first section of this chapter we discuss a few concepts which are important for the analysis of non-Markovian equilibria. In the second section we introduce the main building blocks for trigger strategies: target paths, threats, and punishment. The basic idea of the scenario under consideration is that the agents agree to follow a certain target path and sustain their agreement by threatening to punish any defector. We start by discussing trigger strategies in the framework of an infinite horizon differential game and assume that there is a fixed positive delay δ between the deviation of a defector from the target path and the start of punishment by his opponents.