We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primal--dual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by this setting and the manner in which our algorithm circumvents it.

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Market equilibrium via a primal--dual algorithm for a convex program

We provide the first polynomial time exact algorithm for computing an Arrow-Debreu market equilibrium for the case of linear utilities. Our algorithm is based on solving a convex program using the ellipsoid algorithm and simultaneous diophantine approximation. As a side result, we prove that the set of assignments at equilibria is convex and the equilibria prices themselves are log-convex. Our convex program is explicit and intuitive, which allows maximizing a concave function over the set of equilibria. On the practical side, Ye developed an interior point algorithm (Ye, 2004) to find an equilibrium based on our convex program. We also derive separate combinatorial characterizations of equilibrium for Arrow-Debreu and Fisher cases. Our convex program can be extended for many non-linear utilities (Codenotti and Varadarajan, 2004; Jain and Ye) and production models (Jain). Our paper also makes a powerful theorem even more powerful in the area of geometric algorithms and combinatorial optimization. The main idea in this generalization is to allow ellipsoids not to contain the whole convex region but a part of it. This theorem is of independent interest.

A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities

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 polynomial-time approximation scheme based on a simple atonnement 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 substitutes 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.

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Market equilibrium via the excess demand function

In this paper we study algorithms for computing market equilibrium in markets with linear utility functions. The buyers in the market have an initial endowment given by a portfolio of items. The market equilibrium problem is to compute a price vector which ensures market clearing, i. e. the demand of a good equals its supply, and given the prices, each buyer maximizes its utility. The problem is of considerable interest in Economics. This paper presents a formulation of the market equilibrium problem as a parameterized linear program. We construct the dual of these parametrized linear programs. We show that finding the market equilibrium is the same as finding a linear-program from the family of programs where the optimal dual solution satisfies certain properties. The market clearing conditions arise naturally from complementary slackness conditions.We then define an auction mechanism which computes prices such that approximate market clearing is achieved. The algorithm we obtain outperforms previously known methods.

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Auction algorithms for market equilibrium

The problem of computing equilibria for exchange economies has recently started to receive a great deal of attention in the theoretical computer science community. It has been shown that equilibria can be computed in polynomial time in various special cases, the most important of which are when traders have linear, Cobb-Douglas, or a range of CES utility functions. These important special cases are instances when the market satisfies a property called weak gross substitutability. Classical results in economics, which theoretical computer scientists (including us) appear to have been hitherto unaware of, show that the equilibrium prices in such markets are characterized by an infinite number of linear inequalities and therefore form a convex set. In this paper, we show that under fairly general assumptions, there are polynomial-time algorithms to compute equilibria in such markets. To the best of our knowledge, these are the first polynomial-time algorithms for exchange markets under the general setting of weak gross substitutability. To show this result, we need to build on the proofs that characterize the equilibria as a convex set.As a consequence, we obtain alternative polynomial-time algorithms for computing equilibria with linear, Cobb-Douglas, a range of CES, as well as certain other non-homogeneous utility functions that satisfy weak gross substitutability. Unlike previous polynomial-time algorithms, our approach does not make use of the specific form of these utility functions and is in this sense more general. We expect our framework to work or be readily adaptable to handle other exchange markets, provided that the utility functions satisfy weak gross substitutability.

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D. J. Newman

Papers

Complexity of circumscribed and inscribed ellipsoid methods for solving equilibrium economical models