One Memory in Repeated Games

We study the extent to which equilibrium payoffs of discounted repeated games can be obtained by 1- memory strategies. To this end, we provide a complete characterization of the 1- memory simple strategies and use it in games with a connected action space to show that: 1. all subgame perfect equilibrium payoffs can be approximately supported by an e - sub- game perfect equilibrium strategy of 1-memory, 2. all strictly enforceable subgame perfect equilibrium payoffs can be approximately sup- ported by a 1- memory subgame equilibrium, and 3. the subgame perfect Folk theorem holds for 1-memory strategies. Furthermore, we present two robust examples of games in which there is a subgame perfect equilibrium payoff that cannot be obtained by any 1-memory subgame perfect strategy.