Electoral competition with privately-informed candidates

This paper formulates and analyzes a general model of elections in which candidates receive private signals about voters' preferences prior to committing to political platforms. We fully characterize the unique pure-strategy equilibrium: After receiving her signal, each candidate locates at the median of the distribution of the median voter's location, conditional on the other candidate receiving the same signal. Sufficient conditions for the existence of pure strategy equilibrium are provided. Though the electoral game exhibits discontinuous payoffs for the candidates, we prove that mixed strategy equilibria exist generally, that equilibrium expected payoffs are continuous in the parameters of the model, and that mixed strategy equilibria are upper hemicontinuous. This allows us to study the robustness of the median voter theorem to private information: Pure strategy equilibria may fail to exist in models “close” to the Downsian model, but mixed strategy equilibria must, and they will be “close” to the Downsian equilibrium.     

First-price auction symmetric equilibria with a general distribution

In this paper I obtain the mixed strategy symmetric equilibria of the independent private values, first-price auction for any distribution of valuations. The equilibrium strategy is a pure strategy at the continuity points of the distribution and bids a mixed strategy at the discontinuity points of the distribution.     

Asymmetric Equilibria in a Model with Costly Voting

Most of the voting models limit their analysis to the investigation of symmetric equilibria where “similar” voters make “similar” voting decisions. In this paper we examine the validity of this restriction in a model with costly plurality voting. We first show that in any pure strategy equilibrium every two individuals who have the same preferences and participate in elections, would vote for the same candidate. However, this result does not hold for mixed strategies equilibria.     

Rationing rules and Bertrand-Edgeworth equilibria in large markets

In a market with concave downward sloping demand and symmetric firms which compete in prices with constant marginal costs and capacity limits it is shown that the supports of the symmetric (mixed strategy) Nash equilibria converge to the unique competitive price provided that unsatisfied demand is allocated according to the surplus-maximizing rationing rule.     

A NONCOOPERATIVE SHIPPING COURNOT DUOPOLY WITH LINEAR‐QUADRATIC TRANSPORT COSTS AND CIRCULAR SPACE*

The mixed strategy equilibria in a location quantity duopoly model with circular markets are investigated. We find that a continuum of equilibria exists when the transport cost function is linear. However, if costs are strictly concave or convex, most strategies fail to qualify as equilibria. For any integer n, there are nondegenerate mixed strategy equilibria in which each firm locates at n possible locations with equal probability. This result explains possible outcomes in which firms’ on‐path locations are not minimally or maximally differentiated.     

Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring

We analyze repeated prisoners' dilemma games with imperfect private monitoring and construct mixed trigger strategy equilibria. Such strategies have a simple representation, where a player's action only depends upon her belief that her opponent(s) are continuing to cooperate. When monitoring is almost perfect, the symmetric efficient outcome can be approximated in any prisoners' dilemma game, while every individually rational feasible payoff can be approximated in a class of such games. The efficiency result extends when there are more than two players. It requires that monitoring be sufficiently accurate but does not require very low discounting when a public randomization device is available. Journal of Economic Literature Classification Numbers: C73, D82.     

Group efforts when performance is determined by the “best shot”

We investigate the private provision of a public good whose level is determined by the maximum effort made by a group member. Costs of effort are either commonly known or privately known. For symmetric perfect-information games, any number of players may be active and we characterize the unique (mixed-strategy) equilibrium in which active contributors use the same strategy. Increasing the number of active players leads to stochastically lower individual efforts and level of the public good. When information is private, the symmetric equilibrium is in pure strategies. Increasing the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Comparative statics with respect to costs and levels of risk aversion are derived. Finally, whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency.     

Costly Search, Capacity Constraints, and Bertrand Equilibrium Price Dispersion

This article analyzes the impact of transaction (search) costs and capacity constraints in an almost competitive market with homogeneous firms that compete on price. We characterize conditions under which Nash equilibria with price dispersion exist; in equilibrium, firms play pure strategies in prices and consumers adopt a symmetric mixed search strategy. Price dispersion is possible even though consumers all have the same search cost and valuation for the item and prices charged by all firms are common knowledge.     

Equilibrium opacity in ultimatum‐offer bargaining

We consider ultimatum bargaining between a seller and a buyer of an asset. They know each other's valuation of the asset. Both can defer their decisions to delegates. These delegates have opaque preferences. Seller and buyer choose the opacity of their delegate. For the seller's delegate this choice is restricted to a random reservation price drawn from the set of symmetric two‐point distributions around the seller's true reservation price. The opacity choice of the buyer's delegate is restricted to a random willingness‐to‐pay drawn from the set of symmetric two‐point distribution around the buyer's true willingness‐to‐pay. We characterize the set of pure‐strategy equilibria in their delegation choices. Multiple equilibria arise. Except for two corner solutions, both players will exploit the strategy of opacity. A large set of efficient equilibria exist. For these, opacity choices do not reduce the probability of transacting, but benefit the buyer compared with the no‐delegation equilibrium. We also study the robustness of the results with respect to the player's ability to also resort to a tougher delegate in addition to the opacity choice.     

Elections,Information, and State‐Dependent Candidate Quality

The quality of political candidates often depends on the current state of the world, for example because their personal characteristics are more valuable in some situations than in others. We explore the implications of state‐dependent candidate quality in a model of electoral competition where voters are uncertain about the state. Candidates are fully informed and completely office‐motivated. With a reasonable restriction on voters' beliefs, an equilibrium where candidates' positions reveal the true state does not exist. Nonrevealing equilibria always exist. Some main findings are that candidates' positions can diverge more in equilibrium when they differ more in state‐dependent quality and when the electorate is less well informed.     

An Evolutionary Interpretation of Mixed-Strategy Equilibria

A convincing interpretation of mixed-strategy equilibria describes them as steady states in a large population in which players use pure strategies but the population as a whole mimics a mixed strategy. I study the conditions under which an evolutionary, stochastic learning process converges to the appropriate distribution over pure strategies in the population. I find that not all mixed equilibria can be justified as the result of an evolutionary process even if the equilibrium is unique. For symmetric 2 × 2 and 3 × 3 games I give necessary and sufficient conditions for convergence, which are related to the concept of an ESS, and forn × ngames I give a sufficient condition.Journal of Economic LiteratureClassification Numbers: C73, D83.     

Diligence and laziness in the Chinese countryside revisited

A game-theoretic model is used to asses the impact of changing disutility of labour and distribution schemes on labour supply in Chinese agricultural product teams. It is argued that a previous analysis of the problem by chinn (1980) overlooks some inherent complications. It is shown that application of the symmetric, pure strategy Nash equilibrium concept yields no equilibria over a significant range of values for relevant variables. Asymmetric, pure strategy equilibria are shown to exist over this range. Allowing production teams continuous choice of labour inputs restores symmetric equilibrium.     

Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections

We prove that the support of mixed strategy equilibria of two-player, symmetric, zero-sum games lies in the uncovered set, a concept originating in the theory of tournaments, and the spatial theory of politics. We allow for uncountably infinite strategy spaces, and as a special case, we obtain a long-standing claim to the same effect, due to R. McKelvey (Amer. J. Polit. Sci.30 (1986), 283-314), in the political science literature. Further, we prove the nonemptiness of the uncovered set under quite general assumptions, and we establish, under various assumptions, the coanalyticity and measurability of this set. In the concluding section, we indicate how the inclusion result may be extended to multiplayer, non-zero-sum games. Journal of Economic Literature Classification Numbers: C72, D72.     

On the Relationship between Risk-Dominance and Stochastic Stability

In a 2 × 2 symmetric game with two symmetric equilibria, one risk-dominates another if and only if the equilibrium strategy is a unique best response to any mixture that gives itself at least a probability of one-half. In a two-person strategic form game, we call a Nash equilibriumglobally risk-dominantif it consists of strategies such that each one of them is a unique best response to any mixture that gives the other at least a probability of one-half. We show that if a weakly acyclic two-person game has a globally risk-dominant equilibrium, then this is the one that is selected by the stochastic equilibrium selection process of Young.Journal of Economic LiteratureClassification Numbers: C72, C73.     

Existence of equilibria in countable games: An algebraic approach

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Waldʼs pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group, the payoffs are functions of the group operation, and mixed strategies are not requested to be σ-additive. As a byproduct we show that if finitely additive mixed strategies are allowed, then Waldʼs game admits an equilibrium. Finally we extend the main results to uncountable games.     

Judgment aggregation in search for the truth

We analyze the problem of aggregating judgments over multiple issues from the perspective of whether aggregate judgments manage to efficiently use all voters' private information. While new in judgment aggregation theory, this perspective is familiar in a different body of literature about voting between two alternatives where voters' disagreements stem from conflicts of information rather than of interest. Combining the two bodies of literature, we consider a simple judgment aggregation problem and model the private information underlying voters' judgments. Assuming that voters share a preference for true collective judgments, we analyze the resulting strategic incentives and determine which voting rules efficiently use all private information. We find that in certain, but not all cases a quota rule should be used, which decides on each issue according to whether the proportion of ‘yes’ votes exceeds a particular quota.     

A note on asymmetric and mixed strategy equilibria in the search-theoretic model of fiat money

Summary. The simple search-theoretic model of fiat money has three symmetric Nash equilibria: all agents accept money with probability 1; all agents accept money with probability 0; and all agents accept money with probability y in (0,1). Here I construct an asymmetric pure strategy equilibrium, payoff-equivalent to the symmetric mixed strategy equilibrium, where a fraction N in (0,1) of agents always accept money and 1-N never accept money. Counter to what has been conjectured previously, I find N > y. I also introduce evolutionary dynamics, and show that the economy converges to monetary exchange iff the initial proportion of agents accepting money exceeds N. Received: September 10, 1997; revised version: April 24, 1998     

Threshold uncertainty in the private-information subscription game

We introduce threshold uncertainty, à la Nitzan and Romano (1990), into a private-values model of voluntary provision of a discrete public good. Players are allowed to make any level of contribution toward funding the good, which is provided if the cost threshold is reached. Otherwise, contributions are refunded. Conditions ensuring existence and uniqueness of a Bayesian equilibrium are established. Further restricting the threshold uncertainty to a uniform distribution, we show the equilibrium strategies are very simple, even allowing for any number of players with asymmetric distributions of values. Comparative statics with respect to changes in players' distributions are derived, allowing changes in both the intensity and the dispersion of values. For example, increased uncertainty, in the sense of mean-preserving spreads of players' distributions of values, increases equilibrium contributions. Finally, we show the equilibrium is interim incentive inefficient. The sharpness of our results greatly contrasts with the more qualified insights of earlier private-values models with known cost threshold, which relied on there being two symmetric players and generally exhibited multiple equilibria.     

Private provision of a discrete public good: efficient equilibria in the private-information contribution game

We study the voluntary provision of a discrete public good via the contribution game. Players independently and simultaneously make nonrefundable contributions to fund a discrete public good, which is provided if and only if contributions cover the cost of production. We characterize nonconstant continuous symmetric equilibria, giving sufficient conditions for their existence. We show the common normalization by which players’ values are distributed over [0, 1] is not without loss of generality: if the distribution over this interval has continuous density f with f(0) >  0, then no (nonconstant) continuous symmetric equilibrium exists. We study in detail the case in which players’ private values are uniformly distributed, showing that, generically, when one continuous equilibrium exists, a continuum of continuous equilibria exists. For any given cost of the good, multiple continuous equilibria cannot be Pareto ranked. Nevertheless, not all continuous equilibria are interim incentive efficient. The set of interim incentive efficient equilibria is exactly determined. The authors thank Manfred Dix, George Mailath, Andrew Postlewaite, and an anonymous referee for their comments.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.