Short-run Players Research Articles

A reputation for honesty

We analyze situations where players build reputations for honesty rather than for playing particular actions. A patient player faces a sequence of short-run opponents. Before players act, the patient player announces their intended action after observing both a private payoff shock and a signal of what actions will be feasible that period. The patient player is either an honest type who keeps their word whenever their announced action is feasible, or an opportunistic type who freely chooses announcements and feasible actions. Short-run players only observe the current-period announcement and whether the patient player has kept their word in the past. We provide sufficient conditions under which the patient player can secure their optimal commitment payoff by building a reputation for honesty. Our proof introduces a novel technique based on concentration inequalities.

Robustness of Reputation Effects under Uncertain Monitoring

I study reputation effects under uncertain monitoring. I examine a repeated game between a long-run player and a series of short-run opponents. The long-run player can either be a strategic type or a commitment type that plays the same action in every period. The modeling innovation is that the short-run player is unsure about the monitoring structure. The uncertainty about the monitoring structure introduces new challenges to reputation building because there may not be a direct relationship between the distribution of signals and the long-run player's strategy. Thus the long-run player may not have the ability to establish a reputation for commitment. I show that, when the short-run players cannot statistically distinguish commitment action from a bad action, the standard reputation results break down. I also provide sufficient conditions under which reputation effects on long-run player’s payoffs can be extended to the current framework. When the commitment payoff is the highest payoff he can get, the conditions can be relaxed.

Using Persistence to Generate Incentives in a Dynamic Moral Hazard Problem

I study how the persistence of past choices can be used to create incentives in a continuous time stochastic game in which a large player, such as a i¬ rm, interacts with a sequence of short-run players, such as customers. The long-run player faces moral hazard and her past actions are imperfectly observed – they are distorted by a Brownian motion. Persistence refers to the fact that actions impact a payoi¬€relevant state variable, e.g. the quality of a product depends on both current and past investment choices. I obtain a characterization of actions and payoi¬€s in Markov Perfect Equilibria (MPE), for a i¬ xed discount rate. I show that the perfect public equilibrium (PPE) payoi¬€ set is the convex hull of the MPE payoi¬€ set. Finally, I derive sui¬ƒcient conditions for a MPE to be the unique PPE. Persistence creates ei¬€ective intertemporal incentives to overcome moral hazard in settings where traditional channels fail. Several applications illustrate how the structure of persistence impacts the strength of these incentives.

Using Persistence to Generate Incentives in a Dynamic Moral Hazard Problem

This paper studies how persistence can be used to create incentives in a continuous-time stochastic game in which a long-run player interacts with a sequence of short-run players. Observation of the long-run player's actions are distorted by a Brownian motion and the actions of both players impact future payoffs through a state variable. For example, a firm or worker provides customers with a product, and the quality of this product depends on both current and past investment choices by the rm. I derive general conditions under which a Markov equilibrium emerges as the unique perfect public equilibrium, and characterize the equilibrium payoff and actions in this equilibrium, for any discount rate. I develop an application of persistent product quality to illustrate how persistence creates effective intertemporal incentives in a setting where traditional channels fail, and explore how the structure of persistence impacts equilibrium behavior. This demonstrates the power of the continuous-time setting to deliver sharp insights and a tractable equilibrium characterization for a rich class of dynamic games.

Order of limits in reputations

The fact that small departures from complete information might have large effects on the set of equilibrium payoffs draws interest in the adverse selection approach to study reputations in repeated games. It is well known that these large effects on the set of equilibrium payoffs rely on long-run players being arbitrarily patient. We study reputation games where a long-run player plays a fixed stage-game against an infinite sequence of short-run players under imperfect public monitoring. We show that in such games, introducing arbitrarily small incomplete information does not open the possibility of new equilibrium payoffs far from the complete information equilibrium payoff set. This holds true no matter how patient the long-run player is, as long as her discount factor is fixed. This result highlights the fact that the aforementioned large effects arise due to an order of limits argument, as anticipated.

Learning to Coordinate in Social Networks

We study a dynamic game in which short-run players repeatedly play a symmetric, strictly supermodular game whose payoff depends on a fixed unknown state of nature. Each short-run player inherits the beliefs of his immediate predecessor in addition to observing the actions of the players in his social neighborhood in the previous stage. Due to the strategic complementary between their actions, players have the incentive to coordinate with, and learn from others. We show that in any Markov Perfect Bayesian Equilibrium of the game, players eventually reach consensus in their actions. They also asymptotically receive similar payoffs in spite of initial differences in their access to information. We further show that, if the players' payoffs can be represented by a quadratic function, then the private observations are optimally aggregated in the limit for generic specifications of the game. Therefore, players asymptotically coordinate on choosing the best action given the aggregate information available throughout the network. We provide extensions of our results to the case of changing networks and endogenous private signals.

Codes of Conduct and Bad Reputation

This paper studies bad reputation games in environments with agents that can infer an opponent's intentions. A long-run player faces a sequence of short-run players that imperfectly observe the long-run player's actions, and his type is private information. Short-run players participate in the game whenever certain actions are expected to be played. These actions, however, might be interpreted as perverse behavior, whereby pursuing a separation from the bad types leads to efficiency losses in any equilibrium without observing intentions. When players infer intentions perfectly, I first show that inefficiencies and reputation concerns due to a bad reputation disappear. Moreover, I find that stochastic renewal of the long-run player and imperfect observation of intentions together overcome inefficiencies caused by bad reputation.

Information asymmetry and reentry

We model a reputation game, in which a sequence of short-run players chooses if to interact with a long-run player. Although beliefs may be identical, choices may be different, as not-interacting can lead the long-run player to improve on effort.

Limited records and reputation bubbles

This paper offers a tractable and fully rational model to study the economics of reputation in a dynamic market with limited record-keeping, i.e., a market in which new entrants observe only the last few periods of play of the long-run player instead of the full history of the market. We show that trust is gradually granted to the opportunistic long-run player despite the fact that his type is perfectly observed by the short-run opponents, and the perfectly informed short-run players ride and drive up “reputation bubbles” at the expense of their uninformed successors. We characterize equilibrium payoffs uniformly over time, which is useful for analyzing ongoing repeated relationships where the starting moments have passed.

Learning to Coordinate in Social Networks

We study a dynamic game in which short-run players repeatedly play a symmetric, strictly supermodular game whose payoffs depend on a fixed unknown state of nature. Each short-run player inherits the beliefs of his immediate predecessor in addition to observing the actions of the players in his social neighborhood in the previous stage. Because of the strategic complementary between their actions, players have the incentive to coordinate with others and learn from them. We show that in any Markov Bayesian equilibrium of the game, players eventually reach consensus in their actions. They also asymptotically receive similar payoffs despite initial differences in their access to information. We further show that, if the players’ payoffs can be represented by a quadratic function, then the private observations are optimally aggregated in the limit for generic specifications of the game. Therefore, players asymptotically coordinate on choosing the best action given the aggregate information available throughout the network. We provide extensions of our results to the case of changing networks and endogenous private signals.

Gambling Reputation: Repeated Bargaining With Outside Options

We study the role of incomplete information and outside options in determining bargaining postures and surplus division in repeated bargaining between a long-run player and a sequence of short-run players. The outside option is not only a disagreement point, but reveals information privately held by the long-run player. In equilibrium, the uninformed short-run players' offers do not always respond to changes in reputation and the informed long-run player's payoffs are discontinuous. The long-run player invokes inefficient random outside options repeatedly to build reputation to a level where the subsequent short-run players succumb to his extraction of a larger payoff, but he also runs the risk of losing reputation and relinquishing bargaining power. We investigate equilibrium properties when the discount factor goes to 1 and when the informativeness of outside options diffuses. In both cases, bargaining outcomes become more inefficient and the limit reputation-building probabilities are interior. [PUBLICATION ABSTRACT]

A Foundation for Markov Equilibria in Sequential Games with Finite Social Memory

We study stochastic games with an infinite horizon and sequential moves played by an arbitrary number of players. We assume that social memory is finite—every player, except possibly one, is finitely lived and cannot observe events that are sufficiently far back in the past. This class of games includes games between a long-run player and a sequence of short-run players, and games with overlapping generations of players. An equilibrium is purifiable if some close-by behaviour is consistent with equilibrium when agents’ payoffs in each period are perturbed additively and independently. We show that only Markov equilibria are purifiable when social memory is finite. Thus if a game has at most one long-run player, all purifiable equilibria are Markov.

A Foundation for Markov Equilibria in Infinite Horizon Perfect Information Games

We study perfect information games with an innite horizon played by an arbitrary number of players. This class of games includes innitely repeated perfect information games, repeated games with asynchronous moves, games with long and short run players, games with overlapping generations of players, and canonical non-cooperative models of bargaining. We consider two restrictions on equilibria. An equilibrium is puriable if close by behavior is consistent with equilibrium when agents’ payos at each node are perturbed additively and independently. An equilibrium has bounded recall if there exists K such that at most one player’s strategy depends on what happened more than K periods earlier. We show that only Markov equilibria have bounded memory and are puriable. Thus if a game has at most one long-run player, all puriable equilibria are Markov.

Reputation with Analogical Reasoning*

Abstract We consider a repeated interaction between a long-run player and a sequence of short-run players, in which the long-run player may either be rational or may be a mechanical type who plays the same (possibly mixed) action in every stage game. We depart from the classical model in assuming that the short-run players make inferences by analogical reasoning, meaning that they correctly identify the average strategy of each type of long-run player, but do not recognize how this play varies across histories. Concentrating on 2 × 2 games, we provide a characterization of equilibrium payoffs, establishing a payoff bound for the rational long-run player that can be strictly larger than the familiar “Stackelberg” bound. We also provide a characterization of equilibrium behavior, showing that play begins with either a reputation-building or a reputation-spending stage (depending on parameters), followed by a reputation-manipulation stage.

A Foundation for Markov Equilibria with Finite Social Memory

We study stochastic games with an infinite horizon and sequential moves played by an arbitrary number of players. We assume that social memory is finite—every player, except possibly one, is finitely lived and cannot observe events that are sufficiently far back in the past. This class of games includes games between a long-run player and a sequence of short-run players and games with overlapping generations of players. Indeed, any stochastic game with infinitely lived players can be reinterpreted as one with finitely lived players: Each finitelylived player is replaced by a successor, and receives the value of the successor’s payoff. This value may arise from altruism, but the player also receives such a value if he can “sell” his position in a competitive market. In both cases, his objective will be to maximize infinite horizon payoffs, though his information on past events will be limited. An equilibrium is purifiable if close-by behavior is consistent with equilibrium when agents’ payoffs in each period are perturbed additively and independently. We show that only Markov equilibria are purifiable when social memory is finite. Thus if a game has at most one long-run player, all purifiable equilibria are Markov.

Impermanent types and permanent reputations

We study the impact of unobservable stochastic replacements for the long-run player in the classical reputation model with a long-run player and a series of short-run players. We provide explicit lower bounds on the Nash equilibrium payoffs of a long-run player, both ex-ante and following any positive probability history. Under general conditions on the convergence rates of the discount factor to one and of the rate of replacement to zero, both bounds converge to the Stackelberg payoff if the type space is sufficiently rich. These limiting conditions hold in particular if the game is played very frequently.

Simple Bounds on the Value of a Reputation

We introduce entropy techniques to study the classical reputation model in which a long-run player faces a series of short-run players. The long-run player's actions are possibly imperfectly observed. We derive explicit lower and upper bounds on the equilibrium payoffs to the long-run player.

Is Reputation Good or Bad? An Experiment

We investigate the impact of reputation in a laboratory experiment. We do so by varying whether the past choices of a long-run player are observable by the short-run players. Our framework allows for reputation to have either a beneficial or a harmful effect on the long-run player. We find that reputation is seldom harmful and its beneficial effects are not as strong as theory suggests. When reputational concerns are at odds with other-regarding preferences, we find the latter overwhelm the former. (JEL C91, D12, D82, D83, Z13)

Information value and externalities in reputation building

In sequential equilibrium theory, reputation building is independent of whether the reputation builder is matched with one long-run partner or a series of short-run “strangers”. We observe, however, that reputation builders are significantly more challenged by long-run players in both laboratory chain store and buyer–seller games. Reputation builder behavior is more predictable than implied by equilibrium, and so reputation information has more economic value than implied by equilibrium. For short-run players, this reputation information value is an externality. For long-run players, the value of the information is internalized and so they have greater incentive to challenge the reputation builder.

A Foundation for Markoiv Equilibria in Infinite Horizon Perfect Information Games

We study perfect information games with an infinite horizon played by an arbitrary number of players. This class of games includes infinitely repeated perfect information games, repeated games with asynchronous moves, games with long and short run players, games with overlapping generations of players, and canonical non-cooperative models of bargaining. We consider two restrictions on equilibria. An equilibrium is purifiable if close by behavior is consistent with equilibrium when agents' payoffs at each node are perturbed additively and independently. An equilibrium has bounded recall if there exists K such that at most one player's strategy depends on what happened more than K periods earlier. We show that only Markov equilibria have bounded memory and are purifiable. Thus if a game has at most one long-run player, all purifiable equilibria are Markov.