Sophisticated Reasoning, Learning, and Equilibrium in Repeated Games with Imperfect Feedback

Number: 702
Year: 2023
Author(s): Pierpaolo Battigalli, Davide Bordoli

We analyze the infinite repetition with imperfect feedback of a simultaneous or sequential game, assuming that players are strategically sophisticated---but possibly impatient---expected-utility maximizers. Sophisticated strategic reasoning in the repeated game is combined with belief updating to provide a foundation for a refinement of self-confirming equilibrium. In particular, we model strategic sophistication as rationality and common strong belief in rationality. Then, we combine belief updating and sophisticated reasoning to provide sufficient conditions for a kind of learning--that is, the ability, in the limit, to exactly forecast the sequence of future observations--thus showing that impatient agents end up playing a sequence of self-confirming equilibria in strongly rationalizable conjectures of the one-period game. We also provide a converse of this result. Irrespective of whether individuals value the future, if they are able to learn then they will play in the limit a self-confirming equilibrium in strongly rationalizable conjectures of the continuation (infinitely repeated) game.


Keywords: Self-confirming equilibrium; Common strong belief in rationality; Learning; Repeated games