Strong comparative statics of equilibria

Some results in the monotone comparative statics literature tell us that if a parameter increases, some old equilibria are smaller than some new equilibria. We give a sufficient condition such that at a new parameter value every old equilibrium is smaller than every new equilibrium. We also adapt a standard algorithm to compute a minimal such newer parameter value and apply this algorithm to a game of network externalities. Our results are independent of a theory of equilibrium selection and are valid for games of strategic complementarities.     

Local coordination and market equilibria

We reformulate the local stability analysis of market equilibria in a competitive market as a local coordination problem in a market game, where the map associating market prices to best-responses of all traders is common knowledge and well-defined both in and out of equilibrium. Initial expectations over market variables differ from their equilibrium values and are not common knowledge. This results in a coordination problem as traders use the structure of the market game to converge back to equilibrium. We analyse a simultaneous move and a sequential move version of the market game and explore the link with local rationalizability.     

Competitive equilibria in semi-algebraic economies

This paper develops a method to compute the equilibrium correspondence for exchange economies with semi-algebraic preferences. Given a class of semi-algebraic exchange economies parameterized by individual endowments and possibly other exogenous variables such as preference parameters or asset payoffs, there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters.This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational. Therefore, the result provides theoretical foundations for a systematic analysis of multiplicity in applied general equilibrium.     

Functional rational expectations equilibria in market games

The rational expectations equilibrium (REE) has been criticized as an equilibrium concept in market game environments. Such an equilibrium may not exist generically, or it may introduce unrealistic assumptions about an economic agent’s knowledge or computational ability. We define an REE as a probability measure over uncertain states of nature which exploits all available information in a market game, and which exists for almost all economies. Furthermore, if retrading is allowed, it is possible for agents to compute such a ‘functional rational expectations equilibrium’ using straightforward numerical fixed point algorithms. The approach is demonstrated in a detailed numerical example.     

Coalition-proof Bertrand equilibria

Summary. This paper examines the coalition-proof Nash equilibria of a Bertrand model of price competition where firms supply all demand. When firms are asymmetric we prove existence and provide a sufficient condition for uniqueness. For symmetric firms, we show that an equilibrium is necessarily unique. We also examine whether this unique equilibrium outcome is implementable through a sequential move game where the firms take turns at announcing prices. Finally we examine the limiting property of such equilibria as the number of firms go to infinity.Received: 20 March 2002, Revised: 5 August 2003JEL Classification Numbers: D43, D41, L13.Correspondence to: Kunal SenguptaWe are deeply indebted to an anonymous referee for very helpful and incisive comments that led to substantial improvements in the paper. We also gratefully acknowledge the hospitality of the Department of Finance, Hong Kong University of Science and Technology where much of the work on this paper was carried out.     

Sequential location equilibria under incomplete information

Summary We consider a simple model of incomplete information in location theory. Two firms compete in a two stage framework: a sequential location stage and a price competition stage. Firm 1 knows both its own constant marginal cost technology and that of Firm 2, whereas the latter has incomplete information about firm 1's technology. The location stage turns out to be a monotonic signaling game and theunique D1 equilibrium is a pure strategy separating equilibrium if firm 1's cost advantage is below some bound, and otherwise a pooling equilibrium if the prior probability that Firm 1 is of the low cost type is high, or a semi-pooling equilibrium if it is low. This surprising result is due to the fact that the location gap between the two types of Firm 1 is bounded because of natural economic reasons, which may prevent the separation of the two types. Hence, incomplete information matters: the equilibrium locations differ quite significantly from the full information equilibrium locations.We would like to thank an anonymous referee for very helpful comments and also the participants in seminars at GREQE (Marseille), Université de Montréal, UBC, HEC (Paris), in the Location Theory session of the World Congress of the Econometric Society (Barcelona) and in the Game Theory Conference at the University of Western Ontario for their comments. We remain, of course, solely responsible for the content of the paper. Financial support from FCAR (Québec), SSHRCC (Canada) and CNRS (France) is gratefully acknowledged.     

Subgame-perfect equilibria of finite- and infinite-horizon games

We show that subgame-perfect equilibria of infinite-horizon games arise as limits, as the horizon grows long and epsilon small, of subgame-perfect epsilon-equilibria of games which are truncated after a finite horizon. A number of applications show that this result provides a useful technique for analyzing the existence and uniqueness of infinite-horizon equilibria. We extend our result to the sequential equilibrium concept.     

Variational convergence: Approximation and existence of equilibria in discontinuous games

We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.     

Competitive equilibria of economies with a continuum of consumers and aggregate shocks

This paper studies competitive equilibria of a production economy with aggregate productivity shocks. There is a continuum of consumers who face borrowing constraints and individual labor endowment shocks. The dynamic economy is described in terms of sequences of aggregate distributions. The existence of competitive equilibrium is proven and a recursive characterization is established. In particular, it is shown that for any competitive equilibrium, there is a payoff equivalent competitive equilibrium that is generated by a suitably defined recursive equilibrium.     

On the belief (in-)dependence of sequential equilibria

In this paper, we demonstrate that the existence of sequential equilibria may depend on the players holding beliefs which are tailored to the respective equilibrium but otherwise difficult to justify. We also discuss potential remedies.     

Stationary equilibria in stochastic games: structure, selection, and computation

This paper introduces an algorithm to compute stationary equilibria in stochastic games that is guaranteed to converge for almost all such games. Since in general the number of stationary equilibria is overwhelming, we pay attention to the issue of equilibrium selection. We do this by extending the linear tracing procedure to the class of stochastic games, called the stochastic tracing procedure. As a by-product of our results, we extend a recent result on the generic finiteness of stationary equilibria in stochastic games to oddness of equilibria.     

Sequential equilibria of asymmetric ascending auctions: The case of log-normal distributions

Summary. The sequential equilibrium of an ascending-price auction of a single item is derived explicitly for the case of log-normal distributions and a multiplicative valuation model comprising both common and private factors, and allowing asymmetries. If the prior distribution on the common factors is diffuse, or of the form obtained by Bayesian updating from a diffuse prior distribution, then the equilibrium strategies are log-linear with coefficients obtained by solving a set of linear equations. A similar construction applies to normal distributions and additive terms in the valuation model. An example illustrates the predictions derived from the model. Received: December 11, 1996; revised version: July 15, 1997     

Adaptive learning in extensive form games and sequential equilibrium

Summary. This paper studies adaptive learning in extensive form games and provides conditions for convergence points of adaptive learning to be sequential equilibria. Precisely, we present a set of conditions on learning sequences such that an assessment is a sequential equilibrium if and only if there is a learning sequence fulfilling the conditions, which leads to the assessment. Received: November 5, 1996; revised version: May 28, 1997     

Computing a quasi-perfect equilibrium of a two-player game

Refining an algorithm due to Koller, Megiddo and von Stengel, we show how to apply Lemke’s algorithm for solving linear complementarity programs to compute a quasi-perfect equilibrium in behavior strategies of a given two-player extensive-form game of perfect recall. A quasi-perfect equilibrium is known to be sequential, and our algorithm thus resolves a conjecture of McKelvey and McLennan in the positive. A quasi-perfect equilibrium is also known to be normal-form perfect and our algorithm thus provides an alternative to an algorithm by von Stengel, van den Elzen and Talman. For the case of a zero-sum game, we devise variants of the algorithm that rely on linear programming rather than linear complementarity programming and use the simplex algorithm or other algorithms for linear programming rather than Lemke’s algorithm. We argue that these latter algorithms are relevant for recent applications of equilibrium computation to artificial intelligence.     

Implementing Arrow-Debreu equilibria by trading infinitely-lived securities

Summary. We show that Arrow-Debreu equilibria with countably additive prices in infinite-time economy under uncertainty can be implemented by trading infinitely-lived securities in complete sequential markets under two different portfolio feasibility constraints: wealth constraint, and essentially bounded portfolios. Sequential equilibria with no price bubbles implement Arrow-Debreu equilibria, while those with price bubbles implement Arrow-Debreu equilibria with transfers. Transfers are equal to price bubbles on initial portfolio holdings. Price bubbles arise in sequential equilibrium under the wealth constraint if some securities are in zero supply or negative prices are permitted, but cannot arise with essentially bounded portfolios.Received: 19 November 2003, Revised: 24 February 2004, JEL Classification Numbers: D50, G12, E44.Correspondence to: Jan WernerWe acknowledge helpful discussions with Roko Aliprantis, Subir Chattopaydhyay, Steve LeRoy, Manuel Santos, and seminar participants at Brown University, University of Pennsylvania, NBER Workshop in General Equilibrium Theory, SITE 2000, the 2000 World Congress of the Econometric Society, and Federal Reserve Bank of Kansas City. The views expressed herein are those of the authors and do not necessarily reflect the views of Federal Reserve Bank of Kansas City or the Federal Reserve System.     

Computing optimal tax equilibria

This paper proposes a technique for computing optimal taxes in a full general equilibrium model. It is based upon a fixed point algorithm of the type that is widely used to solve Walrasian general equilibrium models. Computing an optimal tax equilibrium is more difficult than solving a general equilibrium model only to the extent that the derivatives of the social welfare function and of the consumer demand functions must be calculated in the former case. Solutions to several sample problems are provided to demonstrate the technique. They suggest that optimal tax rates are exceedingly sensitive to the specification of the model used to derive them.     

Homotopy methods to compute equilibria in game theory

This paper presents a survey of the use of homotopy methods in game theory. Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by various selection theories. We present the relevant techniques underlying homotopy algorithms. We give detailed expositions of the Lemke–Howson algorithm and the van den Elzen–Talman algorithm to compute Nash equilibria in 2-person games, and the Herings–van den Elzen, Herings–Peeters, and McKelvey–Palfrey algorithms to compute Nash equilibria in general n-person games. We explain how the main ideas can be extended to compute equilibria in extensive form and dynamic games, and how homotopies can be used to compute all Nash equilibria.     

Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies

We examine the connection between Walrasian equilibria of a limit economy (with infinitesimal firms) and noncooperative (Cournot) equilibria of approximating finite economies (with significant firms). Nonconvex production sets, decreasing returns in the aggregate, and endogenous determination of the number of active firms are allowed. A Walrasian equilibrium is a limit of pure strategy noncooperative equilibria only if a condition (loosely analogous to downward sloping demand in the partial equilibrium constant returns to scale case) holds. The condition is also sufficient to guarantee the existence of a robust sequence of pure strategy noncooperative equilibria which converges to the Walrasian equilibrium.     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

Excess supply equilibria

This paper addresses a fundamental problem in economic theory: How can there be equilibria of the economic system where some commodity is in excess supply, yet that commodity's relative price shows no tendency to fall? Of course, the principal example of such a phenomenon is an economy experiencing a prolonged period of involuntary unemployment of the labor force during which there is no significant change in the real wage.In the following pages, I shall describe a two-commodity, general equilibrium model that has a continuum of unemployment equilibria, one for any given unemployment rate. The important feature of this model is that workers establish their wage rates in an attempt to maximize expected utility. The information upon which these wage setting decisions are based is provided by actual labor market transactions.Despite the voluntary nature of the wage setting decision, I shall argue that each equilibrium of this economy exhibits involuntary unemployment in the Keynesian sense. For there will always be another equilibrium with a lower real wage, a higher level of employment, and at which (at least when workers are risk neutral) each worker achieves a higher level of expected utility.