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# Repeated Games with Incomplete Information on One Side: The Case of Different Discount Factors

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## Abstract

Two players engage in a repeated game with incomplete information on one side, where the underlying stage-games are zero-sum. In the case where players evaluate their stage-payoffs by using different discount factors, the payoffs of the infinitely repeated game are typically non zero-sum. However, if players grow infinitely patient, then the equilibrium payoffs will sometimes approach the zero-sum result, depending on the asymptotic relative patience of the players. We provide sufficient conditions that ensure a zero-sum limit. Moreover, we provide examples of games violating these conditions that possess "cooperative" equilibria whose payoffs are bounded away from the zero-sum payoffs set.
... First, we weaken their assumption of perfect monitoring to private monitoring. This extension is interesting in light of the results in Lehrer and Yariv (1999), where it is shown that conclusions of Lehrer and Pauzner (1999) fail in the context of repeated games with incomplete information on one side. Specifically, one-sided incomplete information and private monitoring are two ways of extending the perfect monitoring, complete information case by introducing some degree of incomplete information. ...
... Specifically, one-sided incomplete information and private monitoring are two ways of extending the perfect monitoring, complete information case by introducing some degree of incomplete information. In contrast to what happens in the onesided incomplete information case of Lehrer and Yariv (1999), we show that the conclusions of Lehrer and Pauzner (1999) extend to the case of private monitoring. 2 Second, we weaken the assumption that each player has a constant discount factor. In particular, our setting allows for the case in which there are two distinct discount factors, one player uses the lower one in odd periods and the higher one in even periods, and vice versa for the other player. ...
... The stage game: A two-player, zero-sum private monitoring game having the one-sided incomplete information setting of Lehrer and Yariv (1999) as well as the private monitoring framework considered here as special cases, and to obtain conditions under which the results in Lehrer and Pauzner (1999) do, and do not, extend. We leave this for further research. ...
Article
We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.
... The results we present in this paper are also based on this assumption. Different approaches can be found in Fudenberg et al. (1990), Lehrer and Pauzner (1999) and Lehrer and Yariv (1999). ...
... On the other hand, if all players are uncertain about either of game properties, such game is said to be of incomplete information (Rasmusen, 1994;Aumann et al., 1995). One can also distinguish an intermediate situation, in which players having a complete information about the game they play are playing together with players having only a partial information about certain game properties (Lehrer & Yariv, 1999;Laraki, 2002). ...
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Repeated games are an important mathematical formalism to model and study long-term economic interactions between multiple self-interested parties (individuals or groups of individuals). They open attractive perspectives in modeling long-term multiagent interactions. This overview paper discusses the most important results that actually exist for repeated games. These results arise from both economics and computer science. Contrary to a number of existing surveys of repeated games, most of which originated from the economic research community, we are first to pay a special attention to a number of important distinctive features proper to artificial agents. More precisely, artificial agents, as opposed to the human agents mainly aimed by the economic research, are usually bounded whether in terms of memory or performance. Therefore, their decisions have to be based on the strategies defined using finite representations. Furthermore, these strategies have to be efficiently computed or approximated using a limited computational resource usually available to artificial agents.
... Sorin (1999) provides a synthesis of a number of the results in this literature. Finally, in a recent paper, equilibrium payo®s in discounted repeated zero-sum games with incomplete information have been studied by Lehrer and Yariv (1999), who show that as both players become in¯nitely and equally patient the equilibrium payo®s converge to those with no discounting, whereas if the informed player is in¯nitely more patient than the uninformed an example is given to show that this is not true. ...
Article
The paper analyzes the Nash equilibria of two-person discounted repeated games with one-sided incomplete information and known own payoffs. If the informed player is arbitrarily patient, relative to the uninformed player, then the characterization for the informed player's payoffs is essentially the same as that in the undiscounted case. This implies that even small amounts of incomplete information can lead to a discontinuous change in the equilibrium payoff set. For the case of equal discount factors, however, and under an assumption that strictly individually rational payoffs exist, a result akin to the Folk Theorem holds when a complete information game is perturbed by a small amount of incomplete information.
... The most thorough analysis of repeated games with different discount factors with complete information that we are aware of is Lehrer and Pauzner (1999) who characterize the equilibrium payoffs in two-player games and show that the set of feasible payoffs in the repeated game is typically larger than the convex hull of the underlying stage-game payoffs. Lehrer and Yariv (1999) analyze the case of two-player zero-sum repeated game with one-sided incomplete information regarding the payoff matrix in which the discount factors are common knowledge. The analysis closest to ours is Blonsky and Probst (2008), who deal with a two-player game with incomplete information regarding the discount factors. ...
Article
In repeated games, cooperation is possible in equilibrium only if players are sufficiently patient, and long-term gains from cooperation outweigh short-term gains from deviation. What happens if the players have incomplete information regarding each other's discount factors? In this paper we look at repeated games in which each player has incomplete information regarding the other player's discount factor, and ask when full cooperation can arise in equilibrium. We provide necessary and sufficient conditions that allow full cooperation in equilibrium that is composed of grim trigger strategies, and characterize the states of the world in which full cooperation occurs. We then ask whether these "cooperation events" are close to those in the complete information case, when the information on the other player's discount factor is "almost" complete.
... In this subsection, we explore how to modify the punishment paths to make these play paths renegotiation proof, while continuing to restrict attention to pure strategy SPEs. 13 renegotiation-proofness for all (δ I , δ P ) ∈Ω CC wherê ...
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In the infinitely repeated Prisoners' Dilemma with side payments, we characterize the Pareto frontier of the set of subgame perfect equilibrium payoffs for all possible combinations of discount factors. Play paths implementing Pareto dominant equilibrium payoffs are uniquely determined in all but the first period. Full cooperation does not necessarily implement these payoffs even when it maximizes total stage game payoffs. Rather, when the difference in players' discount factors is sufficiently large, Pareto dominant equilibrium payoffs are implemented by partial cooperation supported by repeated payments from the impatient to the patient player. When both players are sufficiently patient, such payoffs, while implemented via full cooperation, are supported by repeated payments from the impatient to the patient player. We characterize conditions under which public randomization has no impact on the Pareto dominant equilibrium payoffs and conditions under which such payoffs are robust to renegotiation.
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