Difference-form contest success functions and effort levels in contests

We examine players' equilibrium effort levels in a contest with difference-form contest success functions in which two players compete with each other to win a prize. We show the following. At the pure-strategy Nash equilibrium of a simultaneous-move game, and in the subgame-perfect equilibrium of a sequential-move game, only one of the players expends effort or neither player expends effort. If one player's composite strength is far greater than the other player's, only the player with greater composite strength expends effort whether they move simultaneously or sequentially. If the players' valuations for the prize and their marginal probabilities of winning at (0, 0) are sufficiently small, neither player expends effort whether they move simultaneously or sequentially.     

Designing an optimal contest

The paper brings a mechanism design perspective to the study of contests. We consider the problem of selecting a contest success function when the contest designer may also value the prize. We show that any equilibrium outcome that can be achieved by a concave increasing contest success function can be replicated by a linear contest success function. An expected utility maximizing designer should employ a linear homogeneous contest success function. We explicitly derive the optimal contest for a risk-neutral designer and present comparative statics results. Tullock's contest is optimal only when the designer's valuation for the prize is low.     

Asymmetric contests with general technologies

Summary. We investigate the pure-strategy Nash equilibria of asymmetric, winner-take-all, imperfectly discriminating contests, focussing on existence, uniqueness and rent dissipation. When the contest success function is determined by a production function with decreasing returns for each contestant, there is a unique pure-strategy equilibrium. If marginal product is also bounded, limiting total expenditure is equal to the value of the prize in large contests even if contestants differ. Partial dissipation occurs only when infinite marginal products are permitted. Our analysis relies heavily on the use of share functions and we discuss their theory and application.Received: 28 May 2003, Revised: 26 April 2004, JEL Classification Numbers: C72, D72. Correspondence to: Richard CornesMuch of the research in this paper was undertaken while the first author was a Visiting Scholar at the Centre for Economic Studies, University of Munich. The support of the Centre is gratefully acknowledged.     

Homothetic functions revisited

Summary. This paper examines the proposition that homotheticity is equivalent to the property that (e.g., in the context of a production function) the marginal rate of substitution is constant along any ray from the origin. This claim is made in many places, but hitherto the prerequisites have not been stated explicitly. In the present contribution it is demonstrated that an additional condition is required for the claim to hold. We present a theorem that achieves equivalence by also assuming ‘nowhere ray constancy’. It turns out that this condition is implied by assumptions often made, e.g., in production theory. Further, a complete characterization is given of the class of functions that satisfy ray constant marginal rates of substitution or, somewhat more generally, a condition of ray parallel gradients. In addition to homothetic functions this class contains functions homogeneous of degree 0 (i.e., ray constant) and functions which are homothetic in disjoint regions separated by regions of ray constancy. Received: November 5, 1999; revised version: October 31, 2000     

No-arbitrage Equilibria with Differential Information: An Existence Proof

On the example of a pure exchange financial economy with two periods incomplete nominal-asset markets and differential information of the adverse selection’s type, Cornet-De Boisdeffre (J Math Econ 38:393-410, 2002) introduced refined concepts of no-arbitrage prices and equilibria, which extended to the asymmetric information setting the classical concepts of the symmetric information literature. We now assess existence issues and extend a standard property of symmetric information models. Namely, we prove that a no-arbitrage equilibrium always exists, as long as financial markets preclude arbitrage, under the same standard conditions, whether agents have symmetric or asymmetric information.     

A Model of Commodity Differentiation with Indivisibilities and Production

This paper presents an existence theorem in a general equilibrium model of a production economy with commodity differentiation and indivisibilities. The model is motivated by the existence of markets with indivisible commodities, such as the markets for automobiles and computers. As is standard in the literature, the space of commodity characteristics is described by a compact metric space and a commodity vector is described by an integer-valued Borel measure on the space of commodity characteristics. An atomless measure space of producers and consumers is assumed to overcome the problem of non-convexity of the production and consumption sets induced by indivisibilities. This paper is based on a chapter from James Marton’s dissertation. We would like to thank Marcus Berliant, Wilhelm Neuefeind, and an anonymous referee for their valuable comments and suggestions.     

Learning buyers' valuation distribution in posted-price selling

Summary. For his proof of the existence of a general competitive equilibrium Abraham Wald assumed a strictly pseudomonotone inverse market demand function or, equivalently, that market demand satisfies the Weak Axiom of Revealed Preference. It is well known that more recent existence theorems do not need this assumption. In order to clarify its role in Wald's proof, the question of existence of an equilibrium for a modified version of the Walras-Cassel model is reduced to the solvability of a related variational inequality problem. In general, the existence of a solution to such a problem can only be proved by advanced mathematical methods. We provide an elementary induction proof which demonstrates the essence of Abraham Wald's famous contribution. Received: July 22, 1997; revised version: December 11, 1997     

Existence of Nash equilibria in finite extensive form games with imperfect recall: A counterexample

This paper provides an example showing that for finite extensive form games without perfect recall existence of Nash equilibria in behaviour strategies is not guaranteed in general.     

Stackelberg equilibrium with many leaders and followers. The case of zero fixed costs

I study a version of the Stackelberg game with many identical firms in which leaders and followers use a continuous cost function with no fixed cost. Using lattice theoretical methods I provide a set of conditions that guarantee that the game has an equilibrium in pure strategies. With convex costs the model shows the same properties as a quasi-competitive Cournot model. The same happens with concave costs, but only when the number of followers is small. When this number is large the leaders preempt entry. I study the comparative statics and the limit behavior of the equilibrium and I show how the main determinants of market structure interact. More competition between the leaders always displaces the followers. Instead, how a stronger threat of entry affects the equilibrium depends on the technology. With strictly convex costs it is the followers that eventually displace the leaders.     

Existence and efficiency of a stationary subgame-perfect equilibrium in coalitional bargaining models with nonsuperadditive payoffs

We provide the existence theorem of stationary subgame-perfect equilibrium (SSPE) in a noncooperative coalitional bargaining game model with random proposers. Our model contains a bargaining situation where the coalitional game is nonsuperadditive. We also provide a necessary and sufficient condition for the existence of a pure-strategy SSPE satisfying the efficiency property when the discount factor is close to one. Furthermore, we provide examples where the delay in agreement occurs, even in a random-proposers model, when the game is nonsuperadditive. I am grateful to Akira Okada and an anonymous referee for their useful comments and helpful suggestions.     

Teaching Nash Equilibrium and Dominance: A Classroom Experiment on the Beauty Contest

Abstract: The authors' aim in this article was to show how the use of classroom experiments may be a good pedagogical tool to teach the Nash equilibrium (NE) concept. The basic game is a version of the beauty contest game (BCG), a simple guessing game in which repetition lets students react to other players' choices and converge iteratively to the equilibrium solution. The authors perform this experiment with undergraduate students with no previous training in game theory. After four rounds, they observe a clear decreasing tendency in the average submitted number in all groups. Thus, the findings show that by playing a repeated BCG, students quickly learn how to reach the NE solution.     

地方政府、企业行为与污水排放——基于西部各省的面板数据分析

通过建立的寻租博弈模型从微观视角对西部地区地方政府和企业行为与污水排放量的关系展开探讨可知,由于地方政府和企业之间有一定的利益共融性,寻租现象可能存在。对西部各省面板数据进行实证分析可知,寻租程度和西部污水排放量存在着正相关关系。     

The marginal pricing rule revisited

The purpose of the paper is to introduce a tighter definition for the marginal (cost) pricing rule. By means of an example, we illustrate the improvements that one gets with the new definition with respect to the former one using Clarke’s normal cone, and we discuss its consequences in terms of the existence of equilibria.      

Existence,uniqueness, and stability of equilibrium in an OLG economy

Summary. This paper presents sufficient conditions for the existence of a unique and globally stable steady state equilibrium for OLG economies with production. The conditions impose separate requirements on the utility and production functions. Moreover, the conditions do not require assumptions concerning the third order derivatives of the production and utility functions.Received: 12 August 2002, Revised: 7 January 2003, JEL Classification Numbers: D50, D91, E13, O41.I would like to thank Nick Baigent, Laurie Conway, Karl Farmer, Christian Gehrke, and Hideo Konishi for helpful comments. I am grateful to an anonymous referee for his or her comments.