This paper provides a dual characterization of the limit set of perfect public equilibrium payoﬀs in stochastic games (in particular, repeated games) as the discount factor tends to one. As a ﬁrst corollary, the folk theorems of Fudenberg, Levine and Maskin (1994), Kandori and Matsushima (1998) and Hörner, Sugaya, Takahashi and Vieille (2011) obtain. As a second corollary, in the context of repeated games, it follows that this limit set of payoﬀs is a polytope (a bounded polyhedron) when attention is restricted to equilibria in pure strategies. We provide a two-player game in which this limit set is not a polytope when mixed strategies are considered.
Hörner, Johannes; Takahashi, Satoru; and Vieille, Nicolas, "On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games" (2012). Cowles Foundation Discussion Papers. 2207.