Nash Equilibria in Pure Strategies

We consider an n ‐person non‐zero‐sum non‐cooperative game in normal form, where the strategy sets are some closed intervals of the real line. It is shown that if the pay‐off functions are continuous on the whole space and if for each pay‐off function the smallest local maximum in the strategy variable is a global maximum, then the game possesses a pure strategy Nash equilibrium.     

The Neglect of Monotone Comparative Statics Methods

Monotone methods enable comparative static analysis without the restrictive assumptions of the implicit-function theorem. Ease of use and flexibility in solving comparative static and game-theory problems have made monotone methods popular in the economics literature and in graduate courses, but they are still absent from undergraduate mathematical economics courses and textbooks. In this article, the authors illustrate the generality of monotone comparative statics relative to the implicit function approach. For example, to sign the effect of a discrete policy shift on a choice variable, the marginal returns will increase with the policy parameter. They also apply monotone methods in game theory settings. As mathematical economics courses and majors gain popularity, incorporating monotone methods into curriculum and textbooks would provide a modern treatment of comparative static analysis.     

Demonstrating the Possibility of Pareto Inferior Nash Equilibria

This study reviews the status of student outcomes assessment in departments of economics at colleges and universities across the country.     

Nash Equilibria of Repeated Games with Observable Payoff Vectors

We study a model of repeated games with imperfect monitoring where the payoff vector is observable. In this situation, any profitable deviation is detectable by all the players but the identity of the deviator may be unknown. We design collective punishments directed against the set of potential deviators. A particular class of signals is studied for which a characterization of the set of equilibrium payoffs is obtained. Journal of Economic Literature Classification Numbers: C73.     

On the Number of Pure Strategy Nash Equilibria in Random Games

How many pure Nash equilibria can we expect to have in a finite game chosen at random? Solutions to the above problem have been proposed in some special cases. In this paper we assume independence among the profiles, but we allow either positive or negative dependence among the players' payoffs in a same profile. We provide asymptotic results for the distribution of the number of Nash equilibria when either the number of players or the number of strategies increases. We will show that different dependence assumptions lead to different asymptotic results. Journal of Economic Literature Classification Number C72.     

A Property of Nash Equilibria in Repeated Games with Incomplete Information

We consider two-person non-zero-sum infinitely repeated games with lack of information on one side. The characterization of Nash equilibrium payoffs obtained by Hart allows for complex strategies, which are actually required by some equilibrium payoffs in some games. We show that appropriate one-shot public communication mechanisms make Nash equilibrium payoffs achievable by means of simple strategies. Furthermore, these mechanisms satisfy a notion of self-fulfillment.Journal of Economic LiteratureClassification Numbers: D82, C72.     

Comparative Statics for the Consumer Problem

This paper studies comparative statics in the consumer utlity maximization problem with two goods. The analysis can be global, and it allows for indivisibilities, discrete changes, and non-binding budget constraints. It does not rely either on topological or convexity assumptions. Instead, it uses lattice programming techniques. The conditions derived are superextremal variant properties on the utility function, in appropriately constructed lattices on the consumption set. Sufficient conditions are given for one or both goods to be normal, for one to be inferior, and for some cases of price effects, using what are called value orders and lattices.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

Comparative Statics of Money-goods Specifications of the Utility Function

The introduction of real-cash balances into the neoclassical model of the consumer wrecks havoc, in general, on the empirically observable refutable comparative statics properties of the model. We provide the most general solution of this problem to date by deriving a symmetric and negative semidefinite generalized Slutsky matrix that is empirically observable and which contains all other such comparative statics results as a special case. In addition, we clarify and correct two aspects of Samuelson and Sato's (1984) treatment of this problem. Received March 9, 2001; revised version received September 3, 2001     

Stable Provision vs. Cournot–Nash Equilibria in Pure Public Good Economies

Shitovitz and Spiegel (1998, Journal of Economic Theory 83 , pp. 1–18) demonstrated that in pure public good economies, for some consumers the Lindahl consumption bundles can be inferior (utility-wise) to their Cournot–Nash allocation. In this paper, we prove that in any finite pure public good economy there exists a core allocation that is unanimously preferred, utility-wise, by all consumers over their Cournot–Nash consumption bundles.     

Factor-Market Distortions, Dynamic Stability, and Paradoxical Comparative Statics

In the late 1960s and early 1970s there was a strong revival of interest in the role of factor-market distortions in those branches of our subject which rely on the Lerner–Samuelson two-by-two model of production. Among the legacies from that period were several comparative statical propositions which seemed counterintuitive or paradoxical. However, the revival lost momentum when confronted with Neary's observation that the major paradoxes are associated with unstable equilibria and therefore rarely observed. On the other hand, there are now seen to be several errors in Neary's analysis. For example, in the important case of a closed economy (and, by interpretation, in the case of a large open economy) he overlooked the fact that paradoxes can be associated with stable equilibria.     

Left-side Relatively Weak Increases in Risk and Their Comparative Statics

This paper proposes a new concept, a left-side relatively weak increase in risk (L-RWIR) order, that extends the definition of a relatively weak increase in risk (RWIR) order. We show that, for the class of linear payoffs, one can obtain an appealing comparative statics result for L-RWIR shifts imposing additional restrictions on risk preferences of a risk-averse decision maker.JEL classification: D81.revised version received October 10, 2003Acknowledgements The authors would like to thank an anonymous referee for insightful comments and useful suggestions.     

On the Frequency of Interior Cournot–Nash Equilibria in a Public Good Economy

In a public good economy the distribution of initial income is an important determinant of how many individuals contribute to the public good in Cournot–Nash equilibrium. In this paper, first a simple formula is derived that provides a measure for the size of the set of income distributions leading to an interior Cournot–Nash equilibrium in which all individuals contribute to the public good. Furthermore, we give an estimate for the frequency that all members of a certain subgroup of the population are contributors.     

Existence of a Universal Comparative Statics Matrix for Differentiable Optimization Problems

A universal existence theorem is established that yields exhaustive and constraint-free comparative statics information for a general, differentiable optimization problem in the preferred form of a semidefinite matrix. It subsumes all comparative statics formulations of differentiable optimization problems. Its relationship to comparative statics methods extant is establishedWe are grateful to an anonymous referee for a particularly helpful comment on the contents of this paper.     

The structure of Nash equilibrium tariffs

This paper examines theoretically the structure of optimal (Nash equilibrium) tariff rates in a two-country economy with more than two traded goods. We provide a condition under which the equilibrium tariff rates are uniform in both countries, and explore the relative size of the equilibrium tariff rates in each country when the uniform tariff condition is not satisfied. The elasticities of compensated excess demand for goods play an important role in characterizing the structure of the equilibrium tariff rates. This paper undertakes the analysis using a dual approach.     

Comparative Statics of Optimal Nonlinear Income Taxation with a Publicly Provided Input and a Nonlinear Production Technology

Comparative static properties of the solution to an optimal nonlinear income tax problem are provided for a model in which the government both designs a redistributive income tax schedule and provides a public input for a nonlinear production technology. These assumptions imply that wage rates are endogenous. The endogeneity of the wages necessitates taking account of general equilibrium effects of changes in the parameters of the model that are not present when the technology is linear.     

Double implementation in Nash and undominated Nash equilibria in social choice environments

Summary Jackson [1] and Yamato [12] constructed game forms which implement social choice correspondences in Nash and undominated Nash equilibria simultaneously in exchange economies. In this paper, I deal with social choice environments and construct a game form which implements social choice correspondences satisfying monotonicity, no veto power and having at least three agents in Nash and undominated Nash equilibria under the existence of an alternative called a holocaust. The game form constructed in this paper includes an integer game but satisfies the boundedness condition introduced by Jackson [1].I am grateful to Professors Tatsuyoshi Saijo and Stephen Turnbull, and to an anonymous referee for their helpful comments and suggestions. However all remaining errors are mine. An earlier version of this paper was presented at the 1991 Annual Meeting of the Japan Association of Economics and Econometrics at Hokkaido University.After this paper was submitted toEconomic Theory I became aware of the work of Jackson, Palfrey, and Srivastava [2] who have obtained a similar result to mine as a by-product of their main result.     

The Structure of Sunspot Equilibria: The Role of Multiplicity

This paper examines the structure of sunspot equilibria in a standard two period exchange economy with real assets. We show that for a generic choice of utility functions and endowments, there exists an open set of real asset structures whose payoffs are independent of sunspots such that the economy with this asset structure has a regular sunspot equilibrium. An important implication of our result is that the multiplicity of non-sunspot equilibria is not necessary for the existence of sunspot equilibria. Our technique is general and can be applied to show the existence of sunspot equilibria in other frameworks.     

Dynamic Stability, Paradoxical Comparative Statics, and Factor–Market Distortions in an Economy with Three Production Sectors

In an earlier paper the authors clarified the relationship between the stability of long–run equilibrium and the possibility of paradoxical comparative statics in the Lerner–Samuelson two–by–two model of production with factor–market distortions (see Review of International Economics 9 (1901) :383–400). The present paper extends the analysis to an economy with three production sectors. It is found that almost all properties of long–run equilibrium in the two–by–two model with factor–market distortions continue to hold, while some new properties, such as plurality of equilibrium, appear. Specifically, the instability of the adjustment process is not ruled out; a paradox cannot coexist with stable equilibrium in a small open economy, but may do so in a closed economy.