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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.

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... 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. ...

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). ...

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. ...

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. ...

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 ...

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.

We analyze how agents use side payments to induce cooperation in the infinitely repeated Prisoners' Dilemma. For every pair of discount factors, we characterize the Pareto frontier of the set of sub-game perfect equilibrium payoffs. Play paths imple-menting Pareto dominant equilibrium payoffs are uniquely determined in all but the first period. It is an intuitive conjecture that whenever full cooperation maximizes the aggregate instantaneous payoff, it necessarily implements Pareto dominant equi-librium outcomes, supported by repeated bribes from the patient to the impatient player. However, this conjecture is valid only when one player is sufficiently impa-tient, yet the difference in the players' time preferences is moderate. In contrast, when the difference in discount factors is sufficiently large, all Pareto dominant equi-librium payoffs are achieved by partial cooperation supported by repeated bribes from the impatient player to the patient player. Finally, when both players are sufficiently patient, Pareto dominant equilibrium outcomes are achieved through full cooperation, albeit supported by repeated bribes from the impatient to the patient player. We show that public randomization has no impact upon the Pareto frontier, and characterize conditions for renegotiation-proofness.

We study collective decisions by time-discounting individuals choosing a common consumption stream. We show that with any heterogeneity in time preferences, every Pareto efficient and non-dictatorial method of aggregating utility functions must be time inconsistent. We also show that decisions made via non-dictatorial voting methods are intransitive.

In the repeated Prisoner's Dilemma, when every player has a different discount factor, the grim-trigger strategy is an equilibrium if and only if the discount factor of each player is higher than some threshold. What happens if the players have incomplete information regarding the discount factors? In this work we look at repeated games in which each player has incomplete information regarding the other player's discount factor, and ask when a pair of grim-trigger strategies is an equilibrium. We provide necessary and sufficient conditions for such strategies to be an equilibrium. We characterize the states of the world in which the strategies are not triggered, i.e., the players cooperate, in such equilibria (or $\epsilon$-equilibria), and ask whether these "cooperation events" are close to those in the complete information case, when the information is "almost" complete, in several senses. Comment: Thesis submitted in partial fulfillment of requirements for the M. Sc. degree in the School of Mathematical Sciences, Tel-Aviv University. The research work for this thesis has been carried out at Tel-Aviv University under the supervision of Prof. Eilon Solan

This paper studies reputation effects in games with a single long-run player whose choice of stage-game strategy is imperfectly
observed by his opponents. We obtain lower and upper bounds on the long-run player's payoff in any Nash equilibrium of the
game. If the long-run player's stage-game strategy is statistically identified by the observed outcomes, then for generic
payoffs the upper and lower bounds both converge, as the discount factor tends to 1, to the long-run player's Stackelberg
payoff, which is the most he could obtain by publicly committing himself to any strategy.

A single, long-run player plays a simultaneous-move stage game against a sequence of opponents who only play once, but observe all previous play. If there is a positive prior probability that the long-run player will always play the pure strategy he would most like to commit himself to (his Stackleberg strategy), then his payoff in any Nash equilibrium exceeds a bound that converges to the Stackleberg payoff as his discount factor approaches one. When the stage game is not simultaneous move, this result must be modified to account for the possibility that distinct strategies of the long-run player are observationally equivalent. Copyright 1989 by The Econometric Society.

This paper provides a survey on studies that analyze the macroeconomic effects of intellectual property rights (IPR). The first part of this paper introduces different patent policy instruments and reviews their effects on R&D and economic growth. This part also discusses the distortionary effects and distributional consequences of IPR protection as well as empirical evidence on the effects of patent rights. Then, the second part considers the international aspects of IPR protection. In summary, this paper draws the following conclusions from the literature. Firstly, different patent policy instruments have different effects on R&D and growth. Secondly, there is empirical evidence supporting a positive relationship between IPR protection and innovation, but the evidence is stronger for developed countries than for developing countries. Thirdly, the optimal level of IPR protection should tradeoff the social benefits of enhanced innovation against the social costs of multiple distortions and income inequality. Finally, in an open economy, achieving the globally optimal level of protection requires an international coordination (rather than the harmonization) of IPR protection.

Unlike in the traditional theory of games of incomplete information, the players here arenot Bayesian, i.e. a player does not necessarily have any prior probability distribution as to what game is being played. The game is infinitely repeated. A player may be absolutely uninformed, i.e. he may know only how many strategies he has. However, after each play the player is informed about his payoff and, moreover, he has perfect recall. A strategy is described, that with probability unity guarantees (in the sense of the liminf of the average payoff) in any game, whatever the player could guarantee if he had complete knowledge of the game.

A two-person game is of conflicting interests if the strategy to which player one would most like to commit herself holds player two down to his minimax payoff. Suppose there is a positive prior probability that player one is a "commitme nt type" who will always play this strategy. Then player one will get a t least her commitment payoff in any Nash equilibrium of the repeated game if her discount factor approaches one. This result is robust against further perturbations of the informational structure and in striking contrast to the message of the Folk theorem for games with incomplete information. Copyright 1993 by The Econometric Society.

The authors consider a repeated game between two long-run players, one of whom is relatively patient. Each player has a small amount of uncertainty about the other's strategy. Given a weak assumption about the support of this uncertainty, the more patient player obtains (in any Nash equilibrium) approximately the highest payoff consistent with the individual rationality of the other player, if the latter is patient enough. If the less patient player is relatively impatient, any Nash equilibrium gives the more patient player at least the Stackelberg payoff: this generalizes K. M. Schmidt's (1993) result, which applies only to games of conflicting interests.

Report of the U.S. Arms Control and Disarmament Agency ST-116

- J P Mayberry