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65
Market Equilibrium via a PrimalDual Algorithm for a Convex Program
"... 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 primaldual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by ..."
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Cited by 115 (24 self)
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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 primaldual 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.
Dynamics of bid optimization in online advertisement auctions
 In Proceedings of the 16th International World Wide Web Conference
, 2007
"... We consider the problem of online keyword advertising auctions among multiple bidders with limited budgets, and study a natural bidding heuristic in which advertisers attempt to optimize their utility by equalizing their returnoninvestment across all keywords. We show that existing auction mechani ..."
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Cited by 57 (2 self)
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We consider the problem of online keyword advertising auctions among multiple bidders with limited budgets, and study a natural bidding heuristic in which advertisers attempt to optimize their utility by equalizing their returnoninvestment across all keywords. We show that existing auction mechanisms combined with this heuristic can experience cycling (as has been observed in many current systems), and therefore propose a modified class of mechanisms with small random perturbations. This perturbation is reminiscent of the small timedependent perturbations employed in the dynamical systems literature to convert many types of chaos into attracting motions. We show that the perturbed mechanism provably converges in the case of firstprice auctions and experimentally converges in the case of secondprice auctions. Moreover, the point of convergence has a natural economic interpretation as the unique market equilibrium in the case of firstprice mechanisms. In the case of secondprice auctions, we conjecture that it converges to the “supplyaware” market equilibrium. Thus, our results can be alternatively described as a tâtonnement process for convergence to market equilibrium in which prices are adjusted on the side of the buyers rather than the sellers. We also observe that perturbation in mechanism design is useful in a broader context: In general, it can allow bidders to “share ” a particular item, leading to stable allocations and pricing for the bidders, and improved revenue for the auctioneer.
A Path to the ArrowDebreu Competitive Market Equilibrium
 MATH. PROGRAMMING
, 2004
"... We present polynomialtime interiorpoint algorithms for solving the Fisher and ArrowDebreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛequilibrium solution. If the p ..."
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Cited by 40 (7 self)
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We present polynomialtime interiorpoint algorithms for solving the Fisher and ArrowDebreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛequilibrium solution. If the problem data are rational numbers and their bitlength is L, then the bound to generate an exact solution is O(n 4 L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound O(n 8 log(1/ɛ)) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the ArrowDebreu equilibrium, similar to the central path developed for linear programming interiorpoint methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixedpoint theorem. The defining equations are bilinear and possess some primaldual structure for the application of the Newtonbased pathfollowing method.
Leontief Economies Encode Nonzero Sum TwoPlayer Games
"... We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria. We give a ..."
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Cited by 34 (4 self)
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We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria. We give a reduction from twoplayer games to a special family of Leontief exchange economies, which are guaranteed to have equilibria, with the property that the Nash equilibria of any game are in onetoone correspondence with the equilibria of the corresponding economy. Our reduction exposes a potential hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies (where an equilibrium is guaranteed to exist) is at least as hard as finding a Nash equilibrium for twoplayer nonzero sum
Settling the complexity of ArrowDebreu equilibria in markets with additively separable utilities
 IN: PROCEEDINGS OF THE 50TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2009
"... We prove that the problem of computing an ArrowDebreu market equilibrium is PPADcomplete even when all traders use additively separable, piecewiselinear and concave utility functions. In fact, our proof shows that this marketequilibrium problem does not have a fully polynomialtime approximation ..."
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Cited by 30 (5 self)
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We prove that the problem of computing an ArrowDebreu market equilibrium is PPADcomplete even when all traders use additively separable, piecewiselinear and concave utility functions. In fact, our proof shows that this marketequilibrium problem does not have a fully polynomialtime approximation scheme unless every problem in PPAD is solvable in polynomial time.
The spending constraint model for market equilibrium: Algorithmic, existence and uniqueness results
 In Proceedings of 36th Annual ACM Symposium on Theory of Computing (STOC). ACM
"... The traditional model of market equilibrium supports impressive existence results, including the celebrated ArrowDebreu Theorem. However, in this model, polynomial time algorithms for computing (or approximating) equilibria are known only for linear utility functions. We present a new, and natural, ..."
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Cited by 28 (8 self)
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The traditional model of market equilibrium supports impressive existence results, including the celebrated ArrowDebreu Theorem. However, in this model, polynomial time algorithms for computing (or approximating) equilibria are known only for linear utility functions. We present a new, and natural, model of market equilibrium that not only admits existence and uniqueness results paralleling those for the traditional model but is also amenable to efficient algorithms.
Market equilibrium via the excess demand function
 In Proceedings STOC’05
, 2005
"... 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 polyno ..."
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Cited by 28 (2 self)
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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 polynomialtime 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
On the polynomial time computation of equilibria for certain exchange economies
 IN SODA
, 2005
"... The problem of computing equilibria for exchange economies has recently started to receive a great deal of attention in the theoret ical computer science community. It has been shown that equi l ibr ia can be computed in polynomial t ime in various special cases, the most impor tant of which are whe ..."
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Cited by 26 (6 self)
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The problem of computing equilibria for exchange economies has recently started to receive a great deal of attention in the theoret ical computer science community. It has been shown that equi l ibr ia can be computed in polynomial t ime in various special cases, the most impor tant of which are when traders have l inear, CobbDouglas, or a range of CES ut i l i ty functions. These impor tant special cases are instances when the market satisfies a proper ty called weak gTvss substitutability. Classical results in economics, which theoret ical computer scientists ( including us) appear to have been hitherto unaware of, show that the equi l ibr ium prices in such markets are character ized by an infinite number of linear inequalit ies and therefore form a convex set. In this paper, we show that under fairly general assumptions,
Market equilibria in polynomial time for fixed number of goods or agents
 In FOCS
, 2008
"... We consider markets in the classical ArrowDebreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he/she buys) and an initial bundle. At an “equilibrium ” set of prices for goods, if each individual buyer separately exchanges the initial bundl ..."
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Cited by 24 (3 self)
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We consider markets in the classical ArrowDebreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he/she buys) and an initial bundle. At an “equilibrium ” set of prices for goods, if each individual buyer separately exchanges the initial bundle for an optimal bundle at the set prices, the market clears, i.e., all goods are exactly consumed. Classical theorems guarantee the existence of equilibria, but computing them has been the subject of much recent research. In the related area of MultiAgent Games, much attention has been paid to the complexity as well as algorithms. While most general problems are hard, polynomial time algorithms have been developed for restricted classes of games, when one assumes the number of strategies is constant [20, 11]. For the Market Equilibrium problem, several important special cases of utility functions have been tackled. Here we begin a program for this problem similar to that for multiagent games, where general utilities are considered. We begin by showing that if the utilities are separable piecewise linear concave (PLC) functions, and the number of goods (or alternatively the number of buyers) is constant, then we can compute an exact equilibrium in polynomial time. Our technique for the constant number of goods is to decompose the space of price vectors into cells using certain hyperplanes, so that in each cell, each buyer’s threshold marginal utility is known. Still, one needs to solve a linear optimization problem in each cell. We then show the main result that for general (nonseparable) PLC utilities, an exact equilibrium can be found in polynomial time provided the number of goods is constant. The starting point of the algorithm is a “celldecomposition ” of the space of price vectors using polynomial surfaces (instead of hyperplanes). We use results from computational algebraic geometry to bound the number of such cells. For solving the problem inside each cell, we introduce and use a novel LPduality