Common Learning and Cooperation in Repeated Games

Abstract

We study repeated games in which players learn the unknown state of the world by observing a sequence of noisy private signals. We find that for generic signal distributions, the folk theorem obtains using ex-post equilibria. In our equilibria, players commonly learn the state, that is, the state becomes asymptotic common knowledge.

Communication and Community Enforcement

“…and Olszewski (2006) in the context of repeated games with almostperfect monitoring, and they were extended to anonymous random matching games by Deb, Sugaya, and Wolitzky (2020) and to ex post equilibria in games with incomplete information by Sugaya and Yamamoto (2020).…”

“…23 Intuitively, even if at some point in the game player i comes to believe with probability one that the set of players who cooperated in period 1 was v, she continues to entertain the possibility that the set of period 1 cooperators was actually some v 0 ≠ v, and she keeps track of a state x v 0 i ∈ fG, Bg for each possible set v 0 . The interpretation of player i's state is as follows: as in Hörner and Olszewski (2006), Deb, Sugaya, and Wolitzky (2020), and Sugaya and Yamamoto (2020, player i can be viewed as the arbiter of player i 1 1's payoff, meaning that player i 1 1's equilibrium continuation payoff is high when player i is in the good state G, and player i 1 1's equilibrium continuation payoff is low when player i is in the bad state B. Specifically, x v i 5 G means that, if in the coming block the players reach agreement that the set of period 1 cooperators was v, then player i prescribes a high continuation payoff for player i 1 1 (which is delivered both by player i cooperating with player i 1 1 herself and by player i instructing other players to cooperate with player i 1 1); similarly, x v i 5 B means that, if agreement is reached that the set of period 1 cooperators was v, then player i prescribes a low continuation payoff for player i 1 1 (and thus defects against player i 1 1 herself while also instructing others to defect against player i 1 1).…”

Learning Efficiency of Multi-Agent Information Structures

“…In this case, players' past actions can reveal information about their private signals. Basu, Chatterjee, Hoshino, and Tamuz (2020) and Sugaya and Yamamoto (2020) study such settings and construct equilibria that lead to common learning. An interesting open question is to analyze the speed of common learning and how this is affected by players' strategic incentives.…”

Risk-Averse Equilibrium for Games

“…These games have received a great deal of attention as they more accurately model real-world situations where not all parameters are known precisely, with later works such as Wiseman (2005) addressing how players sequentially refine their equilibria as they learn the distributions and the more recent Mertikopoulos and Zhou (2019) addressing how players learn their payoffs with continuous action sets. Another recent work (Sugaya and Yamamoto (2019)) considers the more specific question of how firms in a duopoly should play when the payoff distributions are based on the market state, a random variable with possibly unknown distribution. Despite all the work that has gone into expected utility as the objective value players wish to maximize, it is still questionable whether this is a good assumption.…”