A benchmark "package auction" is introduced in which bidders may determine their own packages on which to bid. If all bidders bid straightforwardly, then the outcome is a point in the core of the exchange economy that minimizes the seller's revenue. When goods are substitutes, straightforward bidding strategies comprise an ex post Nash equilibrium. Compared to the Vickrey auction, the benchmark ascending package auction has cheaper information processing, better handling of budget constraints, and less vulnerability to joint bidding strategies among bidders who would otherwise be losers. Improvements are suggested that speed the auction and limit opportunities for collusion.

In this paper we consider the classic problem of finding the market equilibrium prices under linear utility functions. A notion of approximate market equilibrium was proposed by Deng, Papadimitriou and Safra [5]. Using this notion, we present the first fully polynomial-time approximation scheme for finding a market equilibrium price vector. The main tool in our algorithm is the polynomial-time algorithm of Devanur et al. [6] for a variant of the problem in which there is a clear demarcation between buyers and sellers. Their algorithm is used as a subroutine in our algorithm.

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

Abstract — We formulate a peer-to-peer filesharing system as an exchange economy: a price is associated with each file, and users exchange files only when they can afford it. This formulation solves the free-riding problem, since uploading files is a necessary condition for being able to download. However, we do not explicitly introduce a currency; users must upload files in order to earn a budget for downloading. We discuss existence, uniqueness, and dynamic stability of the competitive equilibrium, which is always guaranteed to be Pareto efficient. In addition, a novel aspect of our approach is an allocation mechanism for clearing the market out of equilibrium. We analyze this mechanism when users can anticipate how their actions affect the allocation mechanism (price anticipating behavior). For this regime we characterize the Nash equilibria that will occur, and show that as the number of users increases, the Nash equilibrium rates become approximately Pareto efficient. I.

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

With the proliferation of the Internet comes the possibility of aggregating vast collections of computers into largescale computational platforms. Research driven by this realization has yielded a new software architecture, known as The Computational Grid [16], for building high-performance distributed applications and systems. The vision outlined by its architects is for applications to draw computational “power ” from a distributed pool of resources in

The Walrasian general equilibrium model is the centrepiece of modern economic theory, but progress in understanding its dynamical properties has been meagre. This article shows that the instability of WalrasÕ t^atonnement process is due to the public nature of prices, which leads to excessive correlation in the behaviour of economic agents. When prices are private information, a dynamic with a globally stable stationary state obtains in economies that are unstable in the t^atonnment process. We provide an agent-based model of a multi-sector Walrasian economy with production and exchange, in which prices are private information. This economy is dynamically well behaved. In WalrasÕ original description of general equilibrium (Walras, 1954 [1874]), market clearing was effected by a central authority. This authority, which has come to be known as the ÔauctioneerÕ, remains today because no one has succeeded in producing a plausible decentralised dynamic model of producers and consumers engaged in market interaction in which prices and quantities move towards market-clearing levels. Only under implausible assumptions can the continuous ÔauctioneerÕ dynamic be shown to be stable (Fisher, 1983), and in a discrete model, even these assumptions (gross substitutability, for instance) do not preclude instability and chaos in price

This paper compares the performance of four different trading institutions in laboratory markets. Two institutions, the continuous double auction and the single call market, are commonly employed on organized exchanges. Two other “hybrid ” institutions, the uniform price double auction and multiple call market, link the other institutions in different dimensions. The laboratory environment features four buyers and four sellers who receive random values and costs in each period and who have a one-unit trading capacity. Therefore, each period provides an observation of price formation and exchange in a thin market environment. We find that trading efficiency is lowest in the institutions that permit only one transaction opportunity each period, primarily due to insufficient trading volume. However, the institutions that permit a single trading opportunity force all traders to transact at a uniform price, which tends to generate prices that more accurately reflect underlying market conditions. *This is a revision of a draft presented at “Money, Markets and Methods: A Conference in Honor

Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For