Dynamic Incentives in Incompletely Specified Environments

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Consider a repeated interaction where it is unknown which of various stage games will be played each period. This framework captures the logic of intertemporal incentives even though numeric payoffs to any strategy profile are indeterminate. A natural solution concept is ex post perfect equilibrium (XPE): strategies must form a subgame-perfect equilibrium for any realization of the sequence of stage games. When (i) there is one long-run player and others are short-run, and (ii) public randomization is available, we can adapt the standard recursive approach to determine the maximum sustainable gap between reward and punishment. This leads to an explicit characterization of what outcomes are supportable in equilibrium, and an optimal penal code that supports them. Any non-XPE-supportable outcome fails to be an SPE outcome for some (possibly ambiguous) specification of the stage games. Unlike in standard repeated games, restrictions (i) and (ii) are crucial.