We study the extent to which equilibrium payo®s of discounted repeated games can be obtained by 1 { memory strategies. First, we present robust examples of games in which there is a subgame perfect equilibrium payo® pro¯le that cannot be obtained by any 1 { memory subgame perfect equilibrium. Then, a complete characterization of 1 { memory simple strategies is provided, and it is employed to establish the following in games with more than two players each having connected action spaces: 1. all subgame perfect equilibrium payo®s can be approximately supported by an " { sub- game perfect equilibrium strategy of 1 { memory, 2. all strictly enforceable subgame perfect equilibrium payo®s can be approximately sup- ported by a 1 { memory subgame equilibrium, and 3. the subgame perfect Folk Theorem holds for 1 { memory strategies. While no further restrictions are needed for the third result to hold in 2 { player games, an additional restriction is needed for the ¯rst two: players must have common punishments.
