Abstract. The WALRAS algorithm calculates competitive equilibria via a distributed tatonnementlike process, in which agents submit single-good demand functions to market-clearing auctions. The algorithm is asynchronous and decentralized with respect to both agents and markets, making it suitable for distributed implementation. We present a formal description of this algorithm, and prove that it converges under the standard assumption of gross substitutability. We relate our results to the literature on general equilibrium stability and some more recent work on decentralized algorithms. We present some experimental results as well, particularly for cases where the assumptions required to guarantee convergence do not hold. Finally, we consider some extensions and generalizations to the WALRAS algorithm.

to perturbations in its parmneters, and why. For example, comparative analysis could be asked to explain why the period of an oscillating spring/block system would increase if the mass of the block were larger. This paper formalizes the problem of comparative analysis and presents a technique, differential qualitative (DQ) analysis, which solves the task, providing explanations suitable for use by design systems, automated diagnosis, intelligent tutoring systems, and explanation based generalization.

Number of text pages (including references, figure legends, tables, figures): 38 Number of figures: 4 Number of tables: 4 Number of words in Abstract: 208 Total number of characters in paper (including spaces): 61,419

Markets have internal dynamics leading to excess volatility and other phenomena that are difficult to explain using rational expectations models. This paper studies these using a nonequilibrium price formation rule, developed in the context of trading with market orders. Because this is so much simpler than a standard inter-temporal equilibrium model, it is possible to study multi-period markets analytically. There price dynamics have second order oscillatory terms. Value investing does not necessarily cause prices to track values. Trend following causes short term trends in prices, but also causes longer-term oscillations. When value investing and trend following are combined, even though there is little linear structure, there can be boom-bust cycles, excess and temporally correlated volatility, and fat tails in price fluctuations. The long term evolution of markets can be studied in terms of flows of money. Profits can be decomposed in terms of aggregate pairwise correlations. Under reinvestment of profits this leads to a capital allocation model that is equivalent to a standard model in population

We consider the problem of computing market equilibria and show three results. (i) For exchange economies satisfying weak gross substitutability we analyze a simple discrete version of tâtonnement, and prove that it converges to an approximate equilibrium in polynomial time. This is the first polynomialtime approximation scheme based on a simple tâtonnement process. It was only recently shown, using vastly more sophisticated techniques, that an approximate equilibrium for this class of economies is computable in polynomial time. (ii) For Fisher’s model, we extend the frontier of tractability, by developing a polynomial time algorithm that applies well beyond the homothetic case and the gross substitutability case. (iii) For production economies, we obtain the first polynomial-time algorithms for computing an approximate equilibrium when the consumers ’ side of the economy satisfies weak gross substitutability and the producers ’ side is restricted to positive production. 1

This paper develops necessary and sufficient conditions for monotone comparative statics predictions in several classes of stochastic optimization problems. The results are formulated so as to highlight the tradeoffs between assumptions about payoff functions and assumptions about probability distributions; they characterize “minimal sufficient conditions” on a pair of functions (for example, a utility function and a probability distribution) so that the expected utility satisfies necessary and sufficient conditions for comparative statics predictions. The paper considers two main classes of assumptions on primitives: single crossing properties and log-supermodularity. Single crossing properties arise naturally in portfolio investment problems and auction games. Logsupermodularity is closely related to several commonly studied economic properties, including decreasing absolute risk aversion, affiliation of random variables, and the monotone likelihood ratio property. The results are used to extend the existing literature on investment problems and games of incomplete information, including auction games and pricing games.

The 30th anniversary of The Journal of Portfolio Management is a milestone in the rich intellectual history of modern finance, firmly establishing the relevance of quantitative models and scientific inquiry in the practice of financial management. One of the most enduring ideas from this intellectual history is the Efficient Markets Hypothesis (EMH), a deceptively simple notion that has become a lightning rod for its disciples and the proponents of behavioral economics and finance. In its purest form, the EMH obviates active portfolio management, calling into question the very motivation for portfolio research. It is only fitting that we revisit this groundbreaking idea after three very successful decades of this Journal. In this article, I review the current state of the controversy surrounding the EMH and propose a new perspective that reconciles the two opposing schools of thought. The proposed reconciliation, which I call the Adaptive Markets Hypothesis (AMH), is based on an evolutionary approach to economic interactions, as well as some recent research in the cognitive neurosciences that has been transforming and revitalizing the intersection of psychology and economics. Although some of these ideas have not yet been fully articulated within a rigorous quantitative framework, long time students of the EMH and seasoned practitioners will no doubt recognize immediately the possibilities generated by this new perspective. Only time will tell whether its potential will be fulfilled. I begin with a brief review of the classic version of the EMH, and then summarize the most significant criticisms leveled against it by psychologists and behavioral economists. I argue that the sources of this controversy can

This paper formalizes the relation between comparative statics and the out-of-equilibrium explanation for how a system evolves after a change in parameters. The paper has two main results. First, an increase in an exogenous parameter sets o# learning dynamics that involve larger values of the endogenous variables. Second, equilibrium selections that are not monotone increasing in the exogenous variables must be predicting unstable equilibria. Moreover, under some conditions monotone comparative statics and stability are equivalent. JEL Classification: C61, C62, C72, C73 Keywords: Monotone comparative statics, supermodularity, strategic complements, learning, correspondence principle. # Discussions with Ilya Segal and Chris Shannon were very important for this work, I am very grateful for all their help. For comments and advice, I also thank Robert Anderson, Juan Dubra, Nestor Gandelman, Ernesto Lopez Cordova, Marcelo Moreira, Charles Pugh, Matthew Rabin, Tarun Sabarwal and Miguel Villa...

A satisfactory complete normative criterion for individualistic ethical decision-making under uncertainty such as Harsanyi’s (Journal of Political Economy, 1955) requires a single fundamental utility function for all individuals which is fully in-terpersonally comparable. The paper discusses reasons why interpersonal com-parisons of utility (ICU’s) have been eschewed in the past and argues that most existing approaches, both empirical and ethical, to ICU’s are flawed. Either they confound facts with values, or they are based on unrealistic hypothetical decisions in an “original position”. Instead ICU’s need to be recognized for what they really are — preferences for different kinds of people. INTERPERSONAL COMPARISONS OF UTILITY...I still believe that it is helpful to speak as if inter-personal comparisons of utility rest upon scientiÆcfoundations – that is, upon observation or introspection....I still think, when I make interpersonal comparisons... that my judgments are more like judgments of value than judgments of veriÆablefact. Nevertheless, to those of my friends who think differently, I would urge that, in practice, our difference is not very important. They think that propositions based upon the assumption of equality are essentially part of economic science. I think that the assumption of equality comes from outside, and that its justiÆcation is more ethical than scientiÆc.But we all agree that it is Ætting that such assumptions should be made and their implications explored with the aid of the economist’s technique.

Current risk-management practices are based on probabilities of extreme dollar losses (e.g., measures like Value at Risk), but these measures capture only part of the story. Any complete risk-management system must address two other important factors—prices and preferences. Together with probabilities, these compose the three P’s of “Total Risk Management. ” This article describes how the three P’s interact to determine sensible risk profiles for corporations and for individuals—guidelines for how much risk to bear and how much to hedge. By synthesizing existing research in economics, psychology, and decision sciences and through an ambitious research agenda to extend this synthesis into other disciplines, a complete and systematic approach to rational decision making in an uncertain world is within reach. lthough rational decision making in the face of uncertainty is by no means a new aspect of the human condition, 1 A recent events have helped to renew and deepen interest in risk management. Two forces in particular have shaped this trend: advances in financial technology (models for pricing derivative instruments and computationally efficient means for implementing them) and an ever-increasing demand for new and exotic financial engineering products (perhaps because of increased market volatility or simply because of the growing complexity of the global financial system). These forces, coupled with such recent calamities as those of Orange County, Gibson Greetings, Metallgesellschaft, Procter & Gamble, and Barings Securities, provide more than sufficient motivation for a thriving risk-management industry. Current risk-management practices focus almost exclusively on the statistical aspects of risk. For example, one of the most popular riskmanagement