Working Papers
By Year:
Paper #  Author  Title  

17011 
Yuichi Yamamoto 
We Can Cooperate Even When the Monitoring Structure Will Never Be Known  
This paper considers infinitehorizon stochastic games with hidden states and hidden actions. The state changes over time, players observe only a noisy public signal about the state each period, and actions are private information. In this model, uncertainty about the monitoring structure does not
disappear. We show how to construct an approximately efficient equilibrium in a repeated Cournot game. Then we extend it to a general case and obtain the folk theorem using expost equilibria under a mild condition. Download Paper


16012 
Yuichi Yamamoto 
Stochastic Games With Hidden States (Third Version)  
This paper studies infinitehorizon stochastic games in which players observe payoffs and noisy public information about a hidden state each period.We find that, very generally, the feasible and individually rational payoff set is invariant to the initial prior about the state in the limit as the discount factor goes to one. This result ensures that players can punish or reward the opponents via continuation payoffs in a flexible way. Then we prove the folk theorem, assuming that public randomization is available. The proof is
constructive, and introduces the idea of random blocks to design an effective punishment mechanism. Download Paper


15019 
Yuichi Yamamoto 
"Stochastic Games with Hidden States, Second Version"  
This paper studies infinitehorizon stochastic games in which players ob serve payoffs and noisy public information about a hidden state each period. Public randomization is available. We find that, very generally, the feasible and individually rational payoff set is invariant to the initial prior about the state in the limit as the discount factor goes to one. We also provide a re cursive characterization of the equilibrium payoff set and establish the folk theorem. Download Paper


15007 
Yuichi Yamamoto 
"Stochastic Games with Hidden States"  
This paper studies infinitehorizon stochastic games in which players observe noisy public information about a hidden state each period. We find that if the game is connected, the limit feasible payoff set exists and is invariant to the initial prior about the state. Building on this invariance result, we provide a recursive characterization of the equilibrium payoff set and establish the folk theorem. We also show that connectedness can be replaced with an even weaker condition, called asymptotic connectedness. Asymptotic connectedness is satisfied for generic signal distributions, if the state evolution is irreducible. Download Paper


13038 
Yuichi Yamamoto 
"Individual Learning and Cooperation in Noisy Repeated Games"' (second version)  
We investigate whether two players in a longrun 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 learned. We show that there are robust equilibria in which players eventually obtain payoffs as if the true state were common knowledge and players played a “belieffree” equilibrium. We also provide explicit equilibrium constructions in various economic examples. Download Paper


12044 
Yuichi Yamamoto 
"Individual Learning and Cooperation in Noisy Repeated Games"  
We investigate whether two players in a longrun 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 “belieffree” equilibrium. The result is applied to various examples, including secret pricecutting with unknown demand. Download Paper


12005 
Yuichi Yamamoto 
"Characterizing BeliefFree ReviewStrategy Equilibrium Payoffs under Conditional Independence"  
This paper proposes and studies a tractable subset of Nash equilibria, belieffree reviewstrategy equilibria, in repeated games with private monitoring. The payoff set of this class of equilibria is characterized in the limit as the discount factor converges to one for games where players observe statistically independent signals. As an application, we develop a simple sufficient condition for the existence of asymptotically efficient equilibria, and establish a folk theorem for Nplayer prisoner’s dilemma. All these results are robust to a perturbation of the
signal distribution, and hence remain true even under almostindependent monitoring. Download Paper
