Individual Learning and Cooperation in Noisy Repeated Games

Abstract

We investigate whether two players in a long-run relationship can maintain cooperation when the details of the underlying game are unknown. Specifically, we consider a new class of repeated games with private monitoring, where an unobservable state of the world influences the payoff functions and/or the monitoring structure. Each player privately learns the state over time, but cannot observe what the opponent learns. We show that there are robust equilibria where players eventually obtain payoffs as if the true state were common knowledge and players played a "belief-free" equilibrium. The result is applied to various examples, including secret pricecutting with unknown demand.

Tacit Collusion in Oligopoly

“…28 In environments with incomplete information about the pro…tability of a market, recent work shows that collusive payo¤s may still be possible without communication at the implementation stage. See, e.g., Yamamoto (2012) and Schenone (2011). 29 In this literature, in some cases, the necessary conditions 27 Abreu, Pearce and Stacchetti (1990) rely on a di¤erent method to solve for the equilibria of dynamic games, using the notion of self-generation, which allows applications to asymmetric games.…”

“…28 See Ely and Valimaki (2002) and Ely,Horner,and Olszewski 29 The results of Yamamoto (2012) imply that under some conditions a cartel can selfenforce some degree of collusion even if …rms do not know how pro…table the market is. The environment is a two-player game with a public randomization device and private monitoring.…”

Common Learning and Cooperation in Repeated Games

“…In other words, the opponents' deviation cannot manipulate player i's state learning. This condition is similar to the statewise full-rank condition of Yamamoto (2014). Condition 5(ii) generalizes the correlated learning condition (Condition 3) to the private-monitoring case.…”

“…We will provide a set of conditions under which the folk theorem is established using ex-post equilibria. Our first condition is the statewise full-rank condition of Yamamoto (2014), which requires that there be an action profile such that each player i can learn the true state ω from her private signal z i :…”

“…(Wiseman (2005), Wiseman (2012), Fudenberg and Yamamoto (2010), Fudenberg and Yamamoto (2011a)). An exception is Yamamoto (2014), who considers the case in which players learn from private signals only. The difference from this paper is that he focuses on belief-free equilibria, which are a subset of sequential equilibria.…”