Cripps, Ely, Mailath and Samuelson (2008) showed that if there are finitely many states, and the signals are i.i.d and finite, then individual learning is sufficient for common learning. In this note we describe what is commonly learned when this sufficient condition does not hold.

We analyze the social and private learning at the symmetric equilibria of a queueing game with strategic experimentation. An infinite sequence of agents arrive at a server that processes them at an unknown rate. The number of agents served at each date is either a geometric random variable in the good state or zero in the bad state. The queue lengthens with each new arrival and shortens if the agents are served or choose to quit the queue. Agents can observe only the evolution of the queue after they arrive; they, therefore, solve a strategic experimentation problem when deciding how long to wait to learn about the probability of service. The agents, in addition, benefit from an informational externality by observing the length of the queue and the actions of other agents. They also incur a negative payoff externality, as those at the front of the queue delay the service of those at the back. We solve for the long‐run equilibrium behavior of this queue and show there are typically mass exits from the queue, even if the server is in the good state.

Two axiomatic characterizations are provided of belief updating. A class of updating processes, termed quasi-Bayesian updating, is characterized by four axioms. These include the divisibility property: that the update does not alter if a composite signal is broken into several parts and several updates. Bayesian updating itself is also characterized using further axioms. Quasi-Bayes updating is applied to a model of sequential sampling. In this model updating that overreacts to new information leads to increased information acquisition.

We consider a model of price competition and collusion with private monitoring. The sellers have homogeneous good, and the buyer has a unit demand every period over an infinite horizon. The buyer’s valuation is his private information, and the sellers start with a common prior. The seller only knows when the buyer comes to him and what price he charges. The buyer knows the price of the seller he visits. We characterize sequential equilibria and focus on equilibria that maximize the sellers’ payoffs. When the sellers know the buyer’s valuation, the buyer’s payoff can be any number between his outside option and the upper bound. With incomplete information, the sellers repeat the static optimum in the best equilibria for the sellers. If the virtual surplus is positive, the buyer buys every period with probability one, and he never buys otherwise. Market sharing is limited by the seller’s incentives to charge the equilibrium price.

Under appropriate assumptions (private values and uniform punishments), the Nash equilibria of a Bayesian repeated game without discounting are payoff-equivalent to tractable, completely revealing, equilibria and can be achieved as interim cooperative solutions of the initial Bayesian game. This characterization does not apply to discounted games with patient players. In a class of public good games, the set of Nash equilibrium payoffs of the undiscounted game can be empty, while limit (perfect Bayesian) Nash equilibrium payoffs of the discounted game, as players become infinitely patient, do exist. These equilibria share some features with the ones of multi-sided reputation models. JEL-Code: C730, C720, C710, D820, H410.

Consider two agents who learn the value of an unknown parameter by observing a sequence of private signals. Will the agents commonly learn the value of the parameter, i.e., will the true value of the parameter become approximate common-knowledge? If the signals are independent and identically distributed across time (but not necessarily across agents), the answer is yes (Cripps et al., Econometrica, 76(4):909–933, 2008). This paper explores the implications of allowing the signals to be dependent over time. We present a counterexample showing that even extremely simple time dependence can preclude common learning, and present sufficient conditions for common learning.

There are N projects of unknown quality. We solve the problem of choosing the best n<N projects from this set when there is a finite time to allocate to learning their quality.

Consider two agents who learn the value of an unknown parameter by observing a sequence of private signals. The signals are independent and identically distributed across time but not necessarily across agents. We show that when each agent's signal space is finite, the agents will commonly learn the value of the parameter, that is, that the true value of the parameter will become approximate common knowledge. The essential step in this argument is to express the expectation of one agent's signals, conditional on those of the other agent, in terms of a Markov chain. This allows us to invoke a contraction mapping principle ensuring that if one agent's signals are close to those expected under a particular value of the parameter, then that agent expects the other agent's signals to be even closer to those expected under the parameter value. In contrast, if the agents' observations come from a countably infinite signal space, then this contraction mapping property fails. We show by example that common learning can fail in this case.

For games of public reputation with uncertainty over types and imperfect public monitoring, Cripps et al. [Imperfect monitoring and impermanent reputations, Econometrica 72 (2004) 407–432] showed that an informed player facing short-lived uninformed opponents cannot maintain a permanent reputation for playing a strategy that is not part of an equilibrium of the game without uncertainty over types. This paper extends that result to games in which the uninformed player is long-lived and has private beliefs, so that the informed player's reputation is private. The rate at which reputations disappear is uniform across equilibria and reputations also disappear in sufficiently long discounted finitely repeated games.

We explain what reputation effects are, how they arise and the factors that limit or strenghten them. Signalling activity has an increased importance in a dynamic setting because signals sent will affect current and future behavior of other parties; this is called the reputation effect. The literature on reputation has two main themes. The first is that introducing a small amount of incomplete information in a dynamic game can dramatically change the set of equilibrium payoffs: introducing something to signal can have big implications in a dynamic model. These kind of results can also be interpreted as providing a robustness check. Dynamic and repeated games typically have many equilibria and reputation results allow us to determine which equilibria continue to be played when a game is " close " to complete information. The second theme of the literature on reputations is that introducing incomplete information in a dynamic game may introduce new and important signalling dynamics in the players' strategies. Thus reputation effects tells us something about behavior. This theme is particularly important in applications to macroeconomics and to industrial organization, for example. For either of these themes to be relevant it is necessary to have a dynamic game with incomplete information, so work on reputation has been influenced by, and influences, the larger literature on repeated and dynamic games of incomplete information. An excellent detailed treatment of reputation can be found in Mailath and Samuelson (2006). Most of the results below will be described in the context of a simple infinitely repeated trading game. The row player is a seller who can produce high or low quality. The column player is a buyer. Producing high quality is always expensive for the seller, so she would rather produce low quality, the buyer, however, only wants to buy a high quality product. The only non-standard element is that the buyer regrets not buying a high quality product. The trading game (Figure 1) has a unique equilibrium (L, N).

Abstract The reputation literature, following Fudenberg and Levine (1989)’s seminal pa- per, often considers a perturbation of a repeated game in which player one may, with small probability, be a “commitment” type that always plays a “commitment” strategy. Reputation results, in which player one obtains the same payo¤ as if she could commit publicly tothat strategy, typically assume that player one is in…nitely more patient than her opponent. Schmidt (1993) obtained a reputation result when player one is much more (but not in…nitely more) patient for perturbed repeated games of con‡icting interests, i.e., games in which an opponents’ best reply to the best commitment strategy of player one yields the opponent the worst, i.e., min- max, payo¤. We obtain a reputation result with equal discounting for perturbed repeated games of strictly con‡icting interest, which require, in addition to the con- ‡icting interest assumption, that player one obtain the best possible payo¤ from the best reply to the commitment strategy. We are grateful to the NSF and the Hammer Fund for …nancial support.

We study the long-run sustainability of private reputations in the presence of imperfect public monitoring. A long-lived player has private reputations if the uninformed players’ beliefs are private. We show that even in this case it is impossible to maintain a permanent reputation for playing a strategy that does not eventually play an equilibrium of the game without uncertainty about types. Thus, a player cannot indefinitely sustain a reputation for non-credible behavior in the presence of imperfect monitoring. Journal of Economic Literature Classification Numbers C70, C78.

This paper studies a game of strategic experimentation in which the players have access to two-armed bandits where the risky arm distributes lumpsum payoffs according to a Poisson process with unknown intensity. Because of free-riding, there is an inefficiently low level of experimentation in any equilibrium where the players use stationary Markovian strategies. We characterize the unique symmetric Markovian equilibrium of the game, which is in mixed strategies. A variety of asymmetric pure-strategy equilibria is then constructed for the special case where there are two players and the arrival of the first lump-sum fully reveals the quality of the risky arm. Equilibria where players switch finitely often between the roles of experimenter and free-rider all lead to the same pattern of information acquisition; the efficiency of these equilibria depends on the way players share the burden of experimentation among them. We show that at least for relatively pessimistic beliefs, even the worst asymmetric equilibrium is more efficient than the symmetric one. In equilibria where players switch roles infinitely often, they can acquire an approximately efficient amount of information, but the rate at which it is acquired still remains inefficient.

I show that if σ is the correlated equilibrium distribution generated by an extreme Nash equilibrium of a bimatrix game, then σ is an extreme point of the set of correlated equilibria of the bimatrix game. I also describe a perturbation of a correlated equilibrium derived from an extreme Nash equilibrium that will generate further extreme points of the set of correlated equilibrium distributions. I use these to provide simple conditions for the set of correlated equilibrium distributions to be different from the set of correlated equilibrium distributions derived from Nash equilibria.

This paper considers the situation where two products are sold by the same seller, but to disjoint sets of potential buyers. Externalities may arise from each market outcome to the other. The paper examines the nature of the seller’s optimal mechanism, and, for example in the case of positive externalities, it is shown that the allocation decision in either market depends on the highest types in both markets. The optimal mechanism can be implemented by an indirect mechanism that essentially charges winning bidders for the value of their externalities. The analysis is applied to the sale of public sector franchises including exploration and development rights for oil and gas tracts.

This paper introduces a dynamic model of the wealth distribution with aggregate risk in the capital market; the model combines credit rationing and portfolio selection decisions. In a closed economy the long-run behaviour of wealth is independent of the initial income distribution when there is aggregate uncertainty, although further restrictions are necessary when there is no aggregate uncertainty. There can be credit rationing at the long-run equilibrium. In poor economies aggregate risk in the capital market slows growth, whereas in richer economies a risky capital market is good for income growth.