A simple search model of money with heterogeneous agents and partial acceptability

Summary. Simple search models have equilibria where some agents accept money and others do not. We argue such equilibria should not be taken seriously. This is unfortunate if one wants a model with partial acceptability. We introduce heterogeneous agents and show partial acceptability arises naturally and robustly. There can be multiple equilibria with different degrees of acceptability. Given the type of heterogeneity we allow, the model is simple: equilibria reduce to fixed points in [0,1]. We show that with other forms of heterogeneity equilibria are fixed points in set space, and there is no method to reduce this to a problem in R¹.Received: 4 September 2002, Revised: 23 September 2002JEL Classification Numbers: C78, E40.A. Shevchenko, R. Wright: We thank seminar participants at the Federal Reserve Bank of Cleveland, Indiana University, Purdue University, University of Toronto, the 2002 Midwest Macroeconomics Conference at Vanderbilt University, and the 2001 Conference on Economic Dynamics at the University of Essex. The National Science Foundation and the Federal Reserve Bank of Cleveland provided financial support. Braz Ministerio de Camargo and Gabriel Camera provided some helpful suggestions. Correspondence to: R. Wright     

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.     

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.     

Sequential auctions with imperfect quantity commitment

This paper studies sequential second price auctions with imperfect quantity commitment in environments involving single-unit demands, independent private values, and non-decreasing marginal costs. The paper characterizes the symmetric equilibrium strategy and demonstrates that the equilibrium price sequence is conditionally non-increasing, showing a downwards drift in cases in which the marginal cost exceeds the reserve price with positive probability. The paper also argues that unlike a strong seller who sets reserve prices strictly above marginal costs, a weak seller will typically prefer to commit to such inefficiently low reserve prices.     

Contests with a stochastic number of players

We study Tullock's (1980) n-player contest when each player has an independent probability 0 < p 1 of participating. A unique symmetric equilibrium is found for any n and p and its properties are analyzed. In particular, we show that for a fixed n > 2 individual equilibrium spending as a function of p is single-peaked and satisfies a single-crossing property for any two different numbers of potential players. However, total equilibrium spending is monotonically increasing in p and n. We also demonstrate that ex-post over-dissipation is a feature of the pure-strategy equilibrium in our model. It turns out that if the contest designer can strategically decide whether to reveal the actual number of participating players or not, then the actual number of participants is always revealed.     

A game of timing and visibility

We consider the following abstraction of competing publications. There are n players in the game. Each player i chooses a point x_i in the interval [0,1], and a player's payoff is the distance from its point x_i to the next larger point, or to 1 if x_i is the largest. For this game, we give a complete characterization of the Nash equilibrium for the two-player game, and, more important, we give an efficient approximation algorithm to compute numerically the symmetric Nash equilibrium for the n-player game. The approximation is computed via a discrete version of the game. In both cases, we show that the (symmetric) equilibrium is unique. Our algorithmic approach to the n-player game is non-standard in that it does not involve solving a system of differential equations. We believe that our techniques can be useful in the analysis of other timing games.     

A MARKET WITH A SOCIAL CONSUMPTION EXTERNALITY*

We model a spatial market in which the utility of each consumer is affected by the consumers who buy precisely the same product. The marginal contribution of consumers x's purchase on consumer y depends on |x–y|, which declines as |x–y| increases. Such modelling of preferences fits goods that signal a consumer's place in society—clothing styles, automobiles and jewellry are examples. For 2n + 1 firms we find the unique symmetric equilibrium and derive comparative statics on the optimal number of firms, the largest number of firms the market can support, and the behaviour of profits per firm as n increases.     

The refoundation of the symmetric equilibrium in Schumpeterian growth models

We provide a refoundation of the symmetric growth equilibrium characterizing the research sector of vertical R&D-driven growth models. We argue that the usual assumptions made in this class of models leave the agents indifferent as to where targeting research: hence, the problem of the allocation of R&D investment across sectors is indeterminate. By introducing an “?-contamination of confidence” in the expected distribution of R&D investment, we prove that the symmetric structure of R&D investment is the unique rational expectations equilibrium compatible with ambiguity-averse agents adopting a maxmin strategy.     

k-Price Auctions

In this paper we analyze the equilibria structure of k-price auctions, k ≥ 3, under the independent-private-value assumption. We discuss agents with an arbitrary attitude toward risk. That is, agents may be risk averse or risk seeking, or they may have an alternating attitude toward risk. We provide a characterization of a continuous symmetric equilibrium, prove that there exists at most one such equilibrium, and show that every such equilibrium is differentiable and increasing. We also show some additional general properties of the equilibrium strategies in these auctions. Journal of Economic Literature Classification Numbers: C72, C73, D83.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

Bargains,Barter, and Money

We examine a search money model in which there is a symmetric coincidence of wants in all barter matches. However, when bargaining outcomes are asymmetric across matches, the barter economy is inefficient. Then a robust monetary equilibrium exists provided that money holders enjoy adequate bargaining terms. Fiat money may be welfare improving. In contrast to the literature, it is the asymmetry in bargains across matches rather than asymmetry in demands that generates these results. Journal of Economic Literature Classification Numbers: C78, E40.     

Stochastic strategy adjustment in coordination games

Summary. The paper explores a model of equilibrium selection in coordination games, where agents from an infinite population stochastically adjust their strategies to changes in their local environment. Instead of playing perturbed best-response, it is assumed that agents follow a rule of ‘switching to better strategies with higher probability’. This behavioral rule is related to bounded-rationality models of Rosenthal (1989) and Schlag (1998). Moreover, agents stay with their strategy in case they successfully coordinate with their local neighbors. Our main results show that both strict Nash equilibria of the coordination game correspond to invariant distributions of the process, hence evolution of play is not ergodic but instead depends on initial conditions. However, coordination on the risk-dominant equilibrium occurs with probability one whenever the initial fraction contains infinitely many agents, independent of the spatial distribution of these agents. Received: March 14, 2000; revised version: June 21, 2001     

The structure of the set of sunspot equilibria in economies with incomplete financial markets

Summary This paper considers the set of equilibria of two-period, sunspot economies withS purely extrinsic states of nature in the second period andI assets with linearly independent nominal payoffs. The span of the payoff matrix contains the vector [1, ... , 1] (i.e., inside money). The set of economies is described in terms of (sunspot-invariant) utility functions. IfS>I> 0, there is an open, dense set of economies such that, given a vector of no arbitrage asset prices, the set of equilibrium allocations contains a smooth manifold of dimensionS—I. Such a manifold contains at least one nonsunspot equilibrium (and at most a finite number of such equilibria).The paper was written while I was a visitor at C.O.R.E., Universitè Catholique de Louvain, with the financial support of a S.P.E.S. fellowship. I would like to thank D. Cass, H. Polemarchakis and P. Siconolfi for their helpful comments.     

Private money and bank runs

Abstract This paper studies bank runs in a model with private money. We show that allowing claims on demand deposits to circulate as a medium of exchange can help prevent bank runs. In our model, there exists a unique banking equilibrium where no one demands early withdrawals of real goods and agents in need of liquidity use private money to finance consumption. With private money, the unique equilibrium not only eliminates bank runs but also improves banking efficiency. The implications of our model are consistent with the evidence from the banking history of the United States.     

Chores

We analyze situations where the provision of each of c public goods must be voluntarily assumed by exactly one of n private agents in the absence of transfer schemes or binding contracts. We model this problem as a complete information, potentially infinite horizon game where n agents simultaneously wage c wars of attrition. Providing a public good commits an agent not to take on the provision of another public good for a fixed period. We explore the strategic trade-offs that this commitment ability and the multiplicity of tasks provide. Subgame perfect equilibria (SPEs) are characterized completely for games with two agents and two public goods. For games with two identical agents and c > 1 identical public goods, we establish that an equilibrium that yields a surplus-maximizing outcome always exists and we provide sufficient conditions under which it is the unique equilibrium outcome. We show that under mild conditions, the surplus-maximizing SPE is the unique symmetric SPE. Journal of Economic Literature Classification Number: H41, C72, D13.     

General equilibrium in economies with uncertain delivery

We consider a general equilibrium model of trade ex ante with differential information in which agents choose plans of state-contingent lists of bundles. Being unable to verify that the state of nature is s and not t, an agent has to accept the delivery of any bundle in the list for delivery in state s or in the list for delivery in state t. Under the assumption that each state of nature can be verified by at least one agent, we establish existence of equilibrium and we show that the equilibrium allocation satisfies a notion of coalitional incentive compatibility.     

Unfolding Social Hierarchies

Consider a large population of finitely-lived agents organized into n different hierarchical levels. Every period, all those placed at each level are randomly matched to play a given symmetric game. Based on the resulting outcome, a ρ-fraction of agents who (within their own level) attain the highest payoffs are promoted upwards. On the other hand, newcomers replacing those who die every period enter at the lowest level and choose irreversibly the strategy to be played for the rest of their life. This choice is made, with some noise, by imitating one of the strategies adopted at the highest level.  In this setup, the unique long-run behavior of the system is fully characterized for the whole class of 2×2 coordination games and two alternative variations of the model. The results crucially depend on the key “institutional” parameters ρ and n. In particular, it is shown that inefficient behavior prevails in the long run (even when risk-dominated) if promotion is only mildly selective—high ρ—and the social system is quite hierarchical—large n. In a stylized manner, these parameter conditions may be viewed as reflecting a sort of institutional deficiency that impairs economic performance. Journal of Economic Literature Classification Numbers: C70, C72, C73, D72.     

Measuring Beliefs in an Experimental Lost Wallet Game

We measure beliefs in an experimental game. Player 1 may take x < 20 Dutch guilders, or leave it and let player 2 split 20 guilders between the players. We find that the higher is x (our treatment variable), the more likely is player 1 to take the x. Out of those who leave the x, many expect to get back less than x. There is no positive correlation between x and the amount y that 2 allocates to 1. However, there is positive correlation between y and 2's expectation of 1's expectation of y. Journal of Economic Literature Classification Numbers: C72, C92.     

A Note on the Probability of k Pure Nash Equilibria in Matrix Games

A formula is derived for the probability that a "randomly selected" n-person matrix game has exactly k pure strategy equilibria. It is shown that for all n ≥ 2, this probability converges to e⁻¹/k! as the sizes of the strategy sets of at least two players increase without bound. Thus the number of pure strategy equilibria in large random n-person matrix games is approximately Poisson distributed with mean one. The latter is a known result obtained by a new proof in this note. Journal of Economic Literature Classification Number: C72.     

Message spaces for perfect correlated equilibria

Summary. We show that a perfect correlated equilibrium distribution of an N-person game, as defined by Dhillon and Mertens (1996) can be achieved using a finite number of copies of the strategy space as the message space.Received: 31 August 2004, Revised: 1 April 2005, JEL Classification Numbers: C72.Srihari Govindan: Correspondence toGovindan would like to thank the Social Sciences and Humanities Research Council of Canada for research support.