This paper investigates the Harsanyi-puriﬁability of mixed strategies in the repeated prisoners’ dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoﬀ shock which is independently and identically distributed across players and periods. We focus on the puriﬁability of a class of one-period memory mixed strategy equilibria used by Ely and Välimäki in their study of the repeated prisoners’ dilemma with private monitoring. We ﬁnd that the strategy proﬁle is puriﬁable by perturbed-game ﬁnite-memory strategies if and only if it is strongly symmetric, in the sense that after every history, both players play the same mixed action. Thus “most” strategy proﬁles are not puriﬁable by ﬁnite memory strategies. However, if we allow inﬁnite memory strategies in the perturbed game, then any completely-mixed equilibrium is puriﬁable.
Bhaskar, V.; Mailath, George J.; and Morris, Stephen, "Purification in the Infinitely-Repeated Prisoners’ Dilemma" (2004). Cowles Foundation Discussion Papers. 1727.