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 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 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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Over the last few years, the problem of computing market equilibrium prices for exchange economies has received much attention in the theoretical computer science community. Such activity led to a flurry of polynomial time algorithms for various restricted, yet significant, settings. The most important restrictions arise either when the traders' utility functions satisfy a property known as gross substitutability or when the initial endowments are proportional (the Fisher model). In this paper, we experimentally compare the performance of some of these recent algorithms against that of the most used software packages. In particular, we evaluate the following approaches: (1) the solver PATH, available under GAMS/MPSGE, a popular tool for computing market equilibrium prices; (2) a discrete version of a simple iterative price update scheme called tâtonnement; (3) a discrete version of the welfare adjustment process; (4) convex feasibility programs that characterize the equilibrium in some special cases. We analyze the performance of these approaches on models of exchange economies where the consumers are equipped with utility functions, which are widely used in real world applications. The outcomes of our experiments consistently show that many market settings allow for an efficient computation of the equilibrium, well beyond the restrictions under which the theory provides polynomial time guarantees. For some of the approaches, we also identify models where they are are prone to failure.A preliminary version of this paper appears in [Codenotti et al. 2005a]. This paper also includes additional results from [Codenotti et al. 2005c].

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We present a polynomial time algorithm that computes an approximate equilibrium for any exchange economy with a demand correspondence satisfying gross substitutability. Such a result was previously known only for the case where the demand is a function, that is, at any price, there is only one demand vctor. The case of multi valued demands that is dealt with here arises in many settings, notably when the traders have linear utilities.We also show that exchange markets in the spending constraint model have demand correspodences satisfying gross substitutability and that they always have an equilibrium price vector with rational numbers. As a consequence, the framework considered here leads to the first exact polynomial time algorithm for this model.

It is demonstrated that for certain markets where traders have constant elasticity of substitution utility (CES) functions, the existence of a price equilibrium can be determined in polynomial time. It is also shown that for a certain range of elasticity of substitution where the CES market does not satisfy gross subsitutability that price equilibira can be computed in polynomial time. It is also shown that for markets satisfying gross substitutability, equilibria can be computed in polynomial time even if the excess demand is a correspondence. On the experimental side, equilibrium computation algorithms from computer science without running time guarantees are shown to be competitive with software packages used in applied microeconomics.Simulations also lend support to the Nash equilibrium solution concept by showing that agents employing heuristics in a restricted form of Texas Holdem converge to an approximate equilibrium. Monte Carlo simulations also indicate the long run preponderance of skill over chance in Holdem tournaments. Abstract Approved: Thesis Supervisor To my wife Jennifer, my mother Patricia, my father Daniel, and my sister Julie for their many years of love and support ii I am especially grateful to my advisor, Dr. Kasturi Varadajan. Having come to Iowa without having taken a course in computer science, it was Kasturi who first introduced me to the study of algorithms through his excellent course and sparked my first real interest in theoretical computer science. Kasturi was always generous with both time and ideas. This was critical as time after time, many days of confusion would be cleared upon his blackboard. This work would not have been possible iv without his kind advice, erudition, encouragement and, not least, his patience. And for that, I will remain grateful. v ABSTRACT It is demonstrated that for certain markets where traders have constant elasticity of substitution utility (CES) functions, the existence of a price equilibrium can be determined in polynomial time. It is also shown that for a certain range of elasticity of substitution where the CES market does not satisfy gross subsitutability that price equilibira can be computed in polynomial time. It is also shown that for markets satisfying gross substitutability, equilibria can be computed in polynomial time even if the excess demand is a correspondence. On the experimental side, equilibrium computation algorithms from computer science without running time guarantees areshown to be competitive with software packages used in applied microeconomics.Simulations also lend support to the Nash equilibrium solution concept by showing that agents employing heuristics in a restricted form of Texas Holdem converge to an approximate equilibrium. Monte Carlo simulations also indicate the long run preponderance of skill over chance in Holdem tournaments.vi Proof: Both σ and σ 0 are in B, so it must be the case that for each j, |σ j − σ 0j | < 1.Therefore, we only need to bound each component of ▽g(σ 0 ) −ᾱ by ǫ ′ 4n . One can see the formula...

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