We formulate a notion of stable outcomes in matching problems with one-sided asymmetric information. The key conceptual problem is to formulate a notion of a blocking pair that takes account of the inferences that the uninformed agent might make. We show that the set of stable outcomes is nonempty in incomplete-information environments, and is a superset of the set of complete-information stable outcomes. We then provide sufficient conditions for incomplete-information stable matchings to be efficient. Lastly, we define a notion of price-sustainable allocations and show that the set of incomplete-information stable matchings is a subset of the set of such allocations.
This paper surveys recent work on reputations in repeated games of incomplete information, including an exposition of the entropy approach to proving both the existence of reputation effects as well as their impermanence.
We provide several generalizations of Mailath's (1987) result that in games of asymmetric information with a continuum of types incentive compatibility plus separation implies differentiability of the informed agent's strategy. The new results extend the theory to classic models in finance such as Leland and Pyle (1977), Glosten (1989), and DeMarzo and Duffie (1999), that were not previously covered.
We study stochastic games with an infinite horizon and sequential moves played by an arbitrary number of players. We assume that social memory is finiteevery 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, games with overlapping generations of players. An equilibrium is purifiable if some 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.
Different markets are cleared by different types of pricesseller specific prices that are uniform across buyers in some markets, and personalized prices tailored to the buyer in others. We examine a setting in which buyers and sellers make investments before matching in a competitive market. We introduce the notion of premuneration valuesthe values to the transacting agents prior to any transferscreated by a buyer-seller match. Personalized price equilibrium outcomes are independent of premuneration values and exhibit inefficiencies only in the event of coordination failures, while uniform-price equilibria depend on premuneration values and in general feature inefficient investments even without coordination failures. There is thus a trade-off between the costs of personalizing prices and the inefficient investments under uniform prices. We characterize the premuneration values under which uniform-price equilibria similarly exhibit inefficiencies only in the event of coordination failures. Keywords: Directed search, matching, premuneration value, prematch investments, search. JEL codes: C78, D40, D41, D50, D83.
Premuneration Values and Investments in Matching Markets obtains clean comparative statics with respect to premuneration values by imposing additional structure (in particular, by only having one-sided investments).
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, Ely, Mailath, and Samuelson, 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.
We prove the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. As a consequence, the perfect monitoring folk theorem is shown to be behaviorally robust under almost-perfect almost-public monitoring. That is, the same specification of behavior continues to be an equilibrium when the monitoring is perturbed from perfect to highly-correlated private. Keywords: Repeated games, bounded recall strategies, folk theorem, imperfect monitoring. JEL codes: C72, C73.
The proof of Theorem 3 can be found here.
This article gives a brief introduction to reputation effects. A canonical model is described, and the reputation bound result of Fudenberg and Levine (1989, 1992) and the temporary reputation result of Cripps, Mailath, and Samuelson (2004, Imperfect Monitoring and Impermanent Reputations, and 2007, Disappearing Private Reputations in Long-Run Relationships) are discussed. Keywords: commitment, incomplete information, reputation bound, reputation effects. JEL Classification Numbers: C70, C78.
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 that when each agent's signal space is finite, the agents will commonly learn its value, i.e., 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. Keywords: Common learning, common belief, private signals, private beliefs. JEL Classification Numbers: D82, D83.
See Cripps, Ely, Mailath, and
Samuelson (2012) for common learning with intertemporally dependent
This paper investigates the Harsanyi (1973)-purifiability
of mixed strategies in the repeated prisoners' dilemma with perfect monitoring.
We perturb the game so that in each period, a player receives a private
payoff shock which is independently and identically distributed across
players and periods. We focus on the purifiability of one-period memory
mixed strategy equilibria used by Ely and Valimaki (2002) in their study
of the repeated prisoners' dilemma with private monitoring. We find that
any such strategy profile is not the limit of one-period memory equilibrium
strategy profiles of the perturbed game, for almost all noise distributions.
However, if we allow infinite memory strategies in the perturbed game,
then any completely-mixed equilibrium is purifiable.Keywords: Purification,
belief-free equilibria, repeated games. JEL Classification Numbers:
We study market breakdown in a finance context under extreme
adverse selection with and without competitive pricing. Adverse selection
is extreme if for any price there are informed agent types with whom uninformed
agents prefer not to trade. Market breakdown occurs when no trade is the
only equilibrium outcome. We present a necessary and sufficient condition
for market breakdown. If the condition holds, then trade is not viable.
If the condition fails, then trade can occur under competitive pricing.
There are environments in which the condition holds and others in which
it fails. Keywords: Adverse selection, market breakdown, separation,
competitive pricing. JEL Classification Numbers: D40, D82, D83, G12,
For games of public reputation with uncertainty over types and imperfect public monitoring, we showed in Imperfect Monitoring and Impermanent Reputations 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. Keywords: Reputation, Imperfect Monitoring, Repeated Games, Commitment, Private Beliefs. .JEL classification numbers: C70, C78.
We present a model incorporating both social and economic components, and analyze their interaction. The notion of a social asset, an attribute that has value only because of the social institutions governing society, is introduced. In the basic model, agents match on the basis of income and unproductive attributes. An attribute has value in some equilibrium social institutions (matching patterns), but not in others. We then show that productive attributes (such as education) can have their value increased above their inherent productive value by some social institutions, leading to the notion of the social value of an asset. Keywords: Social assets, social capital, social arrangements, nonmarket interactions, social norms. JEL classification numbers: D20, D31, D5, J41, Z13.
Some private-monitoring games, that is, games with no public histories, have histories that are almost public. These games are the natural result of perturbing public-monitoring games towards private monitoring (our earlier paper, Repeated Games with Almost-Public Monitoring, studies a special case where the set of private signals coincides with the set of public signals). We explore the extent to which it is possible to coordinate continuation play in such games. It is always possible to coordinate continuation play by requiring behavior to have bounded recall (i.e., there is a bound L such that in any period, the last L signals are sufficient to determine behavior). We show that, in games with general almost-public private monitoring, this is essentially the only behavior that can coordinate continuation play. Keywords: repeated games, private monitoring, almost-public monitoring, coordination, bounded recall. JEL Classification Numbers: C72, C73, D82.
An earlier version of this paper was titled "Finite state strategies and coordination in repeated games with private monitoring."
We examine contemporaneous perfect epsilon-equilibria, in which a player's actions after every history, evaluated at the point of deviation from the equilibrium, must be within epsilon of a best response. This concept implies, but is not implied by Radner's ex ante perfect epsilon-equilibrium. A strategy profile is a contemporaneous perfect epsilon-equilibrium of a game if it is a subgame perfect equilibrium in a game achieved by perturbing payoffs by at most epsilon/2, with the converse holding for pure equilibria. Keywords: Epsilon equilibrium, ex ante payoff, multistage game, subgame perfect equilibrium. JEL classification numbers: C70, C72, C73.
We posit that the value of a manager's human capital depends on the firm's business strategy. The resulting interaction between business strategy and managerial incentives affects the organization of business activities, both the internal organization of the firm and the determination of firm boundaries. We illustrate the impact of this interaction on firm boundaries in a dynamic agency model. There may be disadvantages in merging two firms even when such a merger allows the internalization of externalities between the two firms. Merging, by making unprofitable certain decisions, increases the cost of inducing managerial effort. This incentive cost is a natural consequence of the manager's business-strategy-specific human capital.
We study transactions that require investments before trading in a competitive market, when forward contracts fixing the transaction price are absent. We show that, despite the market being perfectly competitive and subject to arbitrarily little uncertainty, the inability to jointly determine investment levels and prices may make it impossible for buyers and sellers to predict the prices at which they will trade, leading to inefficient levels of investment and trade.
We study the long-run sustainability of public reputations in games with imperfect public monitoring. It is impossible to maintain a permanent reputation for playing a strategy that does not 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. JEL Classification Numbers: C70, C78. Keywords: Reputation, Imperfect Monitoring, Repeated Games, Commitment, Stackelberg Types.
For a useful clarification of the proof of Theorem 1, click here.
The case of private reputations with a long-lived uninformed player is studied in Disappearing Private Reputations in Long-Run Relationships, which is also the paper to read for the case of public reputations with an uninformed long-lived player. (The result in the Econometrica paper for an uninformed long-lived player does not imply the corresponding result for an uninformed short-lived player, while the result in “Disappearing Private Reputations” does. See the latter paper for details.)
The social context can have a large impact on economic decisions. The theoretical challenge is to formulate a model that encompasses both social and economic decisions in a meaningful manner. We discuss the addition of social context to neoclassical economic models using social institutions. We also discuss the relationship between social institutions, social capital, and the social value of assets (introduced in an early version of Mailath and Postlewaite (2006)). JEL Classification Codes: Z13. Keywords: Social Capital, Social Assets, Social Institutions.
We consider a dynamic general equilibrium asset pricing model with heterogeneous agents and asymmetric information. We show how agents' different methods of gathering information affect their chances of survival in the market depending upon the nature of the information and the level of noise in the economy.
We describe the maximum efficient subgame perfect equilibrium payoff for a player in the repeated Prisoners' Dilemma, as a function of the discount factor. For discount factors above a critical level, every efficient, feasible, individually rational payoff profile can be sustained. For an open and dense subset of discount factors below the critical value, the maximum efficient payoff is not an equilibrium payoff. When a player cannot achieve this payoff, the unique equilibrium outcome achieving the best efficient equilibrium payoff for a player is eventually cyclic. There is an uncountable number of discount factors below the critical level such that the maximum efficient payoff is an equilibrium payoff.
We present three examples of finitely repeated games with public monitoring that have sequential equilibria in private strategies, i.e., strategies that depend on own past actions as well as public signals. Such private sequential equilibria can have features quite unlike those of the more familiar perfect public equilibria: (i) making a public signal less informative can create Pareto superior equilibrium outcomes; (ii) the equilibrium final-period action profile need not be a stage game equilibrium; and (iii) even if the stage game has a unique correlated (and hence Nash) equilibrium, the first-period action profile need not be a stage game equilibrium. Journal of Economic Literature classification numbers: C72, C73. Keywords: Private strategies, repeated games, public perfect equilibria.
In repeated games with imperfect public monitoring, players can use public signals to coordinate their behavior perfectly, and thus support cooperative outcomes with the threat of punishments. In this paper, we perturb public-monitoring games towards private monitoring (keeping the set of signals the same) so that the histories, while no longer public, are almost public. Even when the histories are almost public, their private nature may lead players to have sufficiently different views of the world that coordination on punishments is no longer possible. For example, though grim trigger is a public perfect equilibrium (PPE) in games with public monitoring, it often fails to be an equilibrium in arbitrarily close games with private monitoring. If a PPE has players' behavior conditioned only on finite histories, then it induces an equilibrium in all close-by games with private monitoring. This implies a mutual minmax folk theorem for repeated games with almost-public almost-perfect monitoring. Journal of Economic Literature Classification Numbers: C72, C73.
The proof of the almost-public almost-perfect folk theorem (Theorem 6.1) is incorrect (the profile is incorrectly asserted to have bounded recall), and so Theorem 6.2 does not follow from Theorem 6.1. We do not know if Theorem 6.1 is true. Mailath and Samuelson (2006) presents a weaker result: Proposition 13.6.1 is a mutual minmax folk theorem (see also the accompanying discussion). Mailath and Olszewski (2011) provides an analysis of bounded recall folk theorems in perfect monitoring games, and a proof of Theorem 6.2 for generic stage games, and the appropriate version for nongeneric games.
Reference item 23, "Finite state strategies and coordination in repeated games with private monitoring," has been revised and published under a new title. That paper, Coordination Failure in Repeated Games with Almost Public Monitoring, studies a larger class of games with almost public monitoring (where the set of private signals need not coincide with the set of public signals) and explores the extent to which it is possible to coordinate play in games with almost-public monitoring.
Do investors making complementary investments face the correct incentives, especially when they cannot contract with each other prior to their decisions? We present a two-sided matching model in which buyers and sellers make investments prior to matching. Once matched, buyer and seller bargain over the price, taking into account outside options. Efficient decisions can always be sustained in equilibrium. We characterize the inefficiencies that can arise in equilibrium, and show that equilibria will be constrained efficient. We also show that the degree of diversity in a large market has implications for the extent of any inefficiency. Journal of Economic Literature Classification Numbers: C78, D41, D51.
There is a companion paper dealing with finite economies, Efficient Non-Contractible Investments in Finite Economies.
Economists typically analyze individuals' market behavior in isolation from their nonmarket decisions. While this research strategy has generally been successful, it can lead to systematic errors when agents' nonmarket behavior affects their market choices. In this paper we analyze how individuals' investment behavior changes as a result of nonmarket behavior. Specifically, we analyze a model in which individuals must decide how to allocate their initial endowment between two random investments, where the returns are perfectly correlated across individuals for the first investment but independent across individuals for the second. We consider an environment in which men and women match, with wealthier individuals more successful in matching. We show how individuals' concern about relative wealth can affect their investment decisions, and we provide conditions under which individuals bias their investments either toward or away from the investment with correlated returns. A modification of the model is used to explain why agents investments might exhibit a home country bias.
We examine a market in which long-lived firms face a short-term incentive to exert low effort, but could earn higher profits if it were possible to commit to high effort. There are two types of firms, "inept" firms who can only exert low effort, and "competent" firms who have a choice between high and low effort. There is occasional exit, and competent and inept potential entrants compete for the right to inherit the departing firm's reputation. Consumers receive noisy signals of effort choice, and so competent firms choose high effort in an attempt to distinguish themselves from inept firms. A competent firm is most likely to enter the market by purchasing an average reputation, in the hopes of building it into a good reputation, than either a very low reputation or a very high reputation. Inept firms, in contrast, find it more profitable to either buy high reputations and deplete them or buy low reputations.
The statement of Proposition 1.2 is incorrect and the proof of Proposition 3 is incorrect. The correct statement of Proposition 1.2 and a proof of Proposition 3 is available here.
characters in the proofs inexplicably did not print in the published version. A
corrected version can be found in the July 2001 issue.
The correct electronic version of the paper is also downloadable from the Review of Economic Studies website at http://www.restud.com/.
We examine an evolutionary model with "local interactions," so that agents are more likely to interact with some agents than with others. We first review the result that equilibrium strategy choices with given local interactions correspond to correlated equilibria of the underlying game. We then allow the pattern of interactions itself to be shaped by evolutionary pressures. If agents do not have the ability to avoid unwanted interactions, then heterogeneous outcomes can appear, including outcomes in which different groups play different Pareto ranked equilibria. If agents do have the ability to avoid undesired interactions, then we derive conditions under which outcomes must be not only homogeneous but efficient.
Investors making complementary investments typically do not have incentives to invest efficiently when they cannot contract prior to their decisions. When they bargain over the surplus generated by their investments, they will usually not obtain the full fruits of the investment. Intuitively, this hold-up problem should be ameliorated if, in the bargaining stage, each agent has alternatives to the partner he is bargaining with. We characterize the matching and division of surplus in finite economies for any initial investment decisions. We provide conditions on those decisions that guarantee that each agent will capture the change in the aggregate social surplus that results from any investment change he makes. We further show that for any given problem, there exists a bargaining rule by which pairs split their surplus that will support efficient investment choices in equilibrium. We also show, however, that overinvestment or underinvestment can occur for natural bargaining rules.
There is a companion paper dealing with large (continuum population) economies, Efficient Non-Contractible Investments in Large Economies.
We consider a market in which there are two types of workers, "red" and "green," where these labels have no direct payoff implications. Workers can choose to acquire costly skills. Skilled workers must search for firms with a job vacancy, while firms with vacancies also search for unemployed workers. A unique symmetric equilibrium exists in which firms ignore workers' colors. There may also exist an asymmetric equilibrium in which firms only search for green workers, more green than red workers acquire skills, skilled green workers receive higher wage rates than skilled red workers, and the unemployment rate is higher among skilled red than green workers, though there are more unemployed skilled green than red workers. Discrimination between ex ante identical individuals thus arises as an equilibrium phenomenon. Our analysis differs from previous models of discrimination in assuming that firms have perfect information about workers with whom they are matched, and strictly prefer to hire minority workers (contingent on meeting a worker), and in generating predictions concerning unemployment as well as wage rates.
For a PDF file of the Omitted Calculations, click here.
This paper continues the study of the model introduced in Social Norms, Savings Behavior, and Growth. In this economy, there is socially inefficient competition among people. Self-enforcing social norms can potentially control the inefficient competition. However, the inefficient behavior often cannot be suppressed in equilibrium among those with the lowest income due to the ineffectiveness of sanctions against those in the society with the least to lose. We demonstrate that in such cases, it may be possible for society to be divided into distinct classes, with inefficient behavior suppressed in the upper classes but not in the lower.
Evolutionary game theory provides an answer to two of the central questions in economic modeling: When is it reasonable to assume that people are rational? And, when is it reasonable to assume that behavior is part of a Nash equilibrium (and if it is reasonable, which equilibrium)? The traditional answers are not compelling, and much of evolutionary modeling is motivated by the need for a better answer. Evolutionary game theory suggests that, in a range of settings, agents do (eventually) play a Nash equilibrium. Moreover, evolutionary modeling has shed light on the relative plausibility of different Nash equilibria.
There are some typos in the references of the printed version (and the version distributed on the September 1998 JEL CD-ROM). As far as I know, the version on later CD's is correct. For a PDF file of the correct References, click here.
This paper shows that Nash equilibria of local-interaction games are equivalent to correlated equilibria of the underlying game. (c) Springer Verlag
A strategy profile of a normal form game is proper if and only if it is quasi-perfect in every extensive form (with that normal form). Thus, properness requires optimality along a sequence of supporting trembles, while sequentiality only requires optimality in the limit. Building on Extensive Form Reasoning in Normal Form Games, a decision-theoretic implementation of sequential rationality, strategic independence respecting equilibrium (SIRE), is defined and compared to proper equilibrium, using lexicographic probability systems. Finally, we give tremble-based characterizations, which do not involve structural features of the game, of the rankings of strategies that underlie proper equilibrium and SIRE. (c) Academic Press, Inc.
Some typographic errors are corrected in the Erratum, Games and Economic Behavior, 19 (May 1997), 249.
We develop a simple model that captures a concern for relative standing. The concern for relative standing is instrumental in the sense that individuals do not get utility directly from their relative standing, but rather, the concern is induced because relative standing affects consumption of standard commodities. This is a simplified version of the model introduced in Social Norms, Savings Behavior, and Growth. We investigate the consequence of a concern for relative wealth in models in which individuals are making labor-leisure choice decisions. We show how individuals' decisions are affected by the aggregate income distribution and how the concern for relative wealth can generate behavior that can be interpreted as conspicuous consumption when wealth is not directly observable.
Normal-form information sets (introduced in Extensive Form Reasoning in Normal Form Games) capture situations in which players can make certain decisions as if they knew their opponents had chosen from a particular subset of their strategies. In this paper, we say that an extensive-form game represents a normal-form game if, for each such situation, the corresponding choice in the extensive form is made with the player knowing that the opponents have chosen from the relevant subset. We develop an algorithm that generates a representation whenever one exists and present a necessary and sufficient condition for a normal-form game to be representable. JEL Classification Numbers: C70, C72. (c) 1994 Academic Press, Inc.
This paper investigates the incentives of a regulator to close depository institutions, recognizing that an institution's risk taking will be influenced by the regulator's policy regarding bank closure and that there are opportunity costs in closing banks arising from their intermediation function. The regulator focuses not on the current portfolio of the bank, but on the bank's future portfolio. Even if the regulator seeks to maximize welfare, the first best is not obtainable because the regulator is unable to credibly commit to certain policies regarding closure. (c) 1994 Academic Press, Inc.
Projects with negative expected value cannot obtain financing in competitive capital markets if all potential investors are risk neutral and have identical beliefs about the distribution of the project's net revenue. We present a series of examples with heterogeneous beliefs in which it is possible for a project to obtain financing even though all investors in the project believe, conditional on the project being undertaken, that the project has negative expected value. An important feature of the examples is that the differences in beliefs are due only to differences in information, and are not simply arbitrary unexplained differences in opinions. (c) 1994 Academic Press, Inc.
Text of a talk in the session "Recent Advances in Evolutionary Economics" at the American Economic Association meetings, Anaheim, CA, January 1993. The talk describes the impact of perpetual randomness on models of evolution, using the model of Learning, Mutation, and Long Run Equilibria in Games as an illustration.
There are many economic problems which, when modeled as games of incomplete information, give rise to many sequential equilibria, severely limiting the usefulness of the model. There has recently been a large literature devoted to 'refining' the set of equilibria in order to reduce this multiplicity by restricting the set of admissible disequilibrium beliefs. This paper argues that the logical foundations of some refinements and the equilibria they focus on are problematic and, further, proposes an alternative refinement that avoids the difficulties. We also provide an existence theorem covering a broad class of signaling games often studied in economics. (c) 1993 Academic Press, Inc.
There is a tension between a belief in the strategic relevance of information sets and subgames and a belief in the sufficiency of the reduced normal form. We identify a property of extensive form information sets and subgames termed strategic independence. Strategic independence is captured by the reduced normal form and can be used to define normal form information sets and subgames. We prove a close relationship between these normal form structures and their extensive form namesakes. We then motivate and implement solution concepts corresponding to subgame perfection, sequential equilibrium, and forward induction entirely in the reduced normal form.
A companion paper, Normal Form Structures in Extensive Form Games, uses normal form information sets to study which collections of normal form structures can be represented in the one extensive form. Strategic independence is explored further in How Proper is Sequential Equilibrium?
When firms are symmetrically informed, a Stackelberg leader prefers to be leader rather than a Cournot duopolist. However, when the leader has superior information about demand, the leader may earn lower ex ante profits than it would earn if it was choosing quantities simultaneously with the follower. In this paper, I give the firm with superior information the option of delaying its quantity decision until the decision period of the less-informed firm (so that decisions are made simultaneously). Surprisingly, in the unique stable outcome, the informed firm moves first regardless of its private information. (c) 1993 Academic Press, Inc.
An evolutionary model with a finite number of players and with stochastic mutations is analyzed. The expansion and contraction of strategies is linked to their current relative success, but mutuation, perturbing the system from its deterministic evolution, are present as well. The focus is on the long run implications of ongoing mutations, which drastically reduce the set of equilibria. For 2 by 2 symmetric games with two symmetric strict Nash equilibria the risk dominant equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium.
We argue that many goods and decisions are not allocated or made through markets. We interpret an agent's status as a ranking device that determines how well he or she fares in the nonmarket sector. The existence of a nonmarket sector can endogenously generate a concern for relative position in, for example, the income distribution so that higher income implies higher status. Moreover, it can naturally yield multiple equilibria. It is thus possible to explain differences in growth rates across countries without recourse to differences in underlying preferences, technologies, or endowments. Different social organizations lead to different reduced-form preferences, which lead to different growth rates.
The proof of existence of the aristocratic equilibrium is incorrect. The erratum shows by example that aristocratic equilibria do exist. Other equilibria are discussed in detail in Class Systems and the Enforcement of Social Norms.
Interest in evolutionary game theory has recently increased dramatically. I present a brief overview of the subject and the symposium. An evolutionary model consists of a large population of myopic and unsophisticated players playing some game repeatedly through time. Strategies that are "good" replies to the distribution of actions chosen by the current population will be played by a larger fraction of the population in the next period. Thus, players learn from the experience of the population. Results on dynamics, convergence, and equilibrium concepts and their interpretation are discussed.
We show that efficient collusion by any subset of bidders in second price private value auctions is possible, even when the bidders are heterogeneous. An important property of efficient collusion is that a bidder's net payoff from participating in collusion is independent of her valuation. We show that the cooperative game whose characteristic function (evaluated at a coalition) is the ex ante collusive surplus (of that coalition) has a nonempty core. A feature of more technical interest is the mechanism characterization when the private information of agents does not enter in a linear (or even piecewise linear) manner.
A simple non-pathological example demonstrates that, when an incumbent firm has private information about the state of demand, an entrant's profits and the likelihood of entry may be decreasing in the state of demand.
We introduce a distinction between a firm and its network of workers. In a competitive world, if networks are easily lured away, the workers must receive the entire value of their contribution to the firm. How then can service firms have equity value? A model is analyzed in which workers are paid less as a group than their value, even in a competitive world. The workers are assumed to have a nonwage benefit for working at the current firm; this benefit is privately known. These privately known benefits make it impossible for the workers to agree on a division of their value should they leave the existing firm for a new enterprise. The result is that the workers may receive a total compensation that is less than their contribution to the firm.
A yes or no decision must be made about some issue. All agents must agree. The "Coase theorem" asserts that the efficient outcome can be acheived. Suppose the value (positive or negative) that an individual attaches to an affirmative decision is privately known to that individual. It is proved, under very mild conditions, that with independent types, as the number of agents increases, the probability of an affirmative efficient decision necessarily goes to zero. An example in which it is common knowledge that an affirmative decision is efficient and yet the probability of such a decision goes to zero is given.
A dynamic model of differentiated oligopoly in which all firms have private information about their costs and simultaneously signal their information by their pricing decisions is studied.
As in one-sided signaling models, each privately informed firm has an incentive to misrepresent costs. However, simultaneous signaling generates a substantially different comparison of signaling behavior with behavior in the nonsignaling benchmark. Call the action that maximizes a firm's expected profits-assuming that the other firms do not condition their beliefs on the firm's actions-the myopic response. In one-sided signaling, behavior in the nonsignaling benchmark is given by the myopic response. With simultaneous signaling, behavior in the nonsignaling benchmark is not the myopic response, since the myopic response is a response to the other agents'signaling choices, not their nonsignaling choices. As a result, comparisons with nonsignaling benchmarks become more delicate and more interesting.
Consider the following game: There are n agents, each with private information, simultaneously choosing actions in each of two periods. Agents revise their beliefs after observing first period actions and before choosing second period actions. In a separating equilibrium, all private information is revealed in the first period. Existence of separating equilibria is demonstrated under (almost) the same set of assumptions that guarantee it in each of the n one-sided signaling games (where only one agent has private information payoff relevant in the second period). The simultaneity of decisions leads to a substantial technical difficulty that must be overcome.
While signaling models with a continuum of types have well-determined (often unique) separating equilibria, models with a finite set of types do not. For signaling games, the following is proved: Suppose there is a sequence of finite sets of types, each set refining its predecessor and the limit set dense in an interval. Then every sequence of separating equilibrium paths has a convergent subsequence, the limit of any convergent sequence is a separating equilibrium path of the limit game, and the convergence is uniform. If the limit game has a unique equilibrium path then every sequence converges to it.
This paper provides two results that are useful in proving the existence of and characterizing separating equilibria in signaling games. A key element in the analysis of separating equilibria is the examination of the implied incentive compatibility constraints. It is shown that these constraints imply differentiability of strategies. In addition, a monotonicity condition (which is similar to the single crossing condition) is analyzed that is necessary and sufficient for there to be a strategy satisfying the incentive compatibility constraints. As a direct consequence of these two results, the analysis of Paul Milgrom and John Roberts (1982) is considerably strengthened.
Some extensions are provided in Mailath and von Thadden (2013).