A single, long-run player plays a simultaneous-move stage game against a sequence of opponents who only play once, but observe all previous play. If there is a positive prior probability that the long-run player will always play the pure strategy he would most like to commit himself to (his Stackleberg strategy), then his payoff in any Nash equilibrium exceeds a bound that converges to the Stackleberg payoff as his discount factor approaches one. When the stage game is not simultaneous move, this result must be modified to account for the possibility that distinct strategies of the long-run player are observationally equivalent.

The authors study repeated games in which players observe a public outcome that imperfectly signals the actions played. They provide conditions guaranteeing that any feasible, individually rational payoff vector of the stage game can arise as a perfect equilibrium of the repeated game with sufficiently little discounting. The central condition requires that there exist action profiles with the property that, for any two players, no two deviations--one by either player--give rise to the same probability distribution over public outcomes. The results apply to principal-agent, partnership, oligopoly, and mechanism-design models, and to one-shot games with transferable utilities.

4 Celentani (1996) uses multiple types of short-run players to get around the problem of unobservable off-path behavior of the long-run player in repeated games with observable actions.His approach can be extended to short run players who live for more than one periodo

We study how an agent learns from endogenous data when their prior belief is misspecified. We show that only uniform Berk–Nash equilibria can be long‐run outcomes, and that all uniformly strict Berk–Nash equilibria have an arbitrarily high probability of being the long‐run outcome for some initial beliefs. When the agent believes the outcome distribution is exogenous, every uniformly strict Berk–Nash equilibrium has positive probability of being the long‐run outcome for any initial belief. We generalize these results to settings where the agent observes a signal before acting.

Perturbed utility functions—the sum of expected utility and a non-linear perturba-tion function—provide a simple and tractable way to model various sorts of stochastic choice. We provide easily understood conditions that characterize this representation by generalizing the acyclicity condition used in revealed preference theory. We show how to relax Luce’s IIA condition to model cases where the agent finds it harder to discriminate between items in larger menus, and how to extend the perturbation-function approach to model choice overload and nested decisions. We also show that these representations correspond to a form of ambiguity-averse preferences for an agent who is uncertain about her true utility