Although the study of market equilibria has occupied center stage within Mathematical Economics for over a century, polynomial time algorithms for such questions have so far evaded researchers. We provide the first such algorithm for the linear version of a problem defined by Irving Fisher in 1891. Our algorithm is modeled after Kuhn’s primaldual algorithm for bipartite matching. 1

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 return-on-investment 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 time-dependent 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 first-price auctions and experimentally converges in the case of second-price auctions. Moreover, the point of convergence has a natural economic interpretation as the unique market equilibrium in the case of first-price mechanisms. In the case of second-price auctions, we conjecture that it converges to the “supply-aware” 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.

We present polynomial-time interior-point algorithms for solving the Fisher and Arrow-Debreu 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 ɛ-equilibrium solution. If the problem data are rational numbers and their bit-length 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 Arrow-Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method.

The traditional model of market equilibrium supports impressive existence results, including the celebrated Arrow-Debreu 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.

We give a reduction from any two-player game to a special case of the Leontief exchange economy, previously studied by Ye [29], with the property that the Nash equilibria of the game and the equilibria of the market are in one-to-one correspondence. Our reduction exposes a potential hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies is at least as hard as finding a Nash equilibrium for two-player nonzero sum games. As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [16]. In particular, among other results, we show that it is NP-hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium. Perhaps more importantly, we also prove that it is NP-hard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known PPAD-completeness result of [26], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwer’s Fixed Point Theorem. 1

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

We study efficient algorithms for computing equilibrium price in the Fisher model for a class of nonlinear concave utility functions, the logarithmic utility functions. A duality relation is derived between buyers and sellers under such utility functions. It is applied to design a polynomial time algorithm for calculating equilibrium price, for the special case when either the number of sellers or the number of buyers is bounded by a constant.

We study the algorithmic complexity of the discrete fixed point problem and develop an asymptotic matching bound for a cube in any constantly bounded finite dimension. To obtain our upper bound, we derive a new fixed point theorem, based on a novel characterization of boundary conditions for the existence of fixed points. In addition, exploring a linkage with the approximation problem of the continuous fixed point problem, we obtain asymptotic matching bounds for complexity of the approximate Brouwer fixed point problem in the continuous case for Lipschitz functions that close a previous exponential gap. It settles a fifteen years old open problem of Hirsch, Papadimitriou and Vavasis by improving both the upper and lower bounds. Our new characterization for existence of a fixed point is also applicable to functions defined on non-convex domain and makes it a potentially useful tool for design and analysis of algorithms for fixed points in general domain.

This issue’s column is written by guest columnists, Bruno Codenotti, Sriram Pemmaraju and Kasturi Varadarajan. I am delighted that they agreed to write this timely column on the topic related to the computation of market equilibria that has received much attention recently. Their column introduces the reader to several recent results and provides references for further readings.

One of the main reasons of the recent success of peer to peer (P2P) file sharing systems such as BitTorrent is its built-in tit-for-tat mechanism. In this paper, we model the bandwidth allocation in a P2P system as an exchange economy and study a tit-for-tat dynamics, namely the proportional response dynamics, in this economy. In a proportional response dynamics each player distributes its good to its neighbors proportional to the utility it received from them in the last period. We show that this dynamics not only converges but converges to a market equilibrium, a standard economic characterization of efficient exchanges in a competitive market. In addition, for some classes of utility functions we consider, it converges much faster than the classical tâtonnement process and any existing algorithms for computing market equilibria. As a part of our proof we study the double normalization of a matrix, an operation that linearly scales the rows of a matrix so that each row sums to a prescribed positive number, followed by a similar scaling of the columns. We show that the double normalization process of any non-negative matrix always converges. This complements the previous studies in matrix scaling that has focused on the convergence condition of the process when the row and column normalizations are considered as separate steps. 1