Computation of Market Equilibria via the Excess Demand Function

We consider the computation of equilibria for exchange economies. The general problem is unlikely to admit efficient algorithms. We develop and adapt a number of tools which allow us to take advantage of the structure of equilibria, when the market satisfies a property, called weak gross substitutability, which guarantees that the equilibria form a convex set. Using these tools we derive two polynomial time algorithms: the first one is a simple and efficient discrete version of the t?atonnement process, while the second one is based on the Ellipsoid method, and achieves a better dependence on the approximation parameter. Our approach does not make use of the specific form of the utility functions of the individual traders, and it is thus more general than previous work

On the Existence of Low-Rank Explanations for Mixed Strategy Behavior

Nash equilibrium is used as a model to explain the observed behavior of players in strategic settings. For example, in many empirical applications we observe player behavior, and the problem is to determine if there exist payoffs for the players for which the equilibrium corresponds to observed player behavior. Computational complexity of Nash equilibria is an important consideration in this framework. If the instance of the model that explains observed player behavior requires players to have solved a computationally hard problem, then the explanation provided is questionable. In this paper we provide conditions under which Nash equilibrium is a reasonable explanation for strategic behavior, i.e., conditions under which observed behavior of players can be explained by games in which Nash equilibria are easy to compute. We identify three structural conditions and show that if the data set of observed behavior satisfies any of these conditions, then it is consistent with payoff matrices for which the observed Nash equilibria could have been computed efficiently. Our conditions admit large and structurally complex data sets of observed behavior, showing that even with complexity considerations, Nash equilibrium is often a reasonable model.Comment: Updated writeup. 19 page

Proportional Dynamics in Exchange Economies

We study the Proportional Response dynamic in exchange economies, where each player starts with some amount of money and a good. Every day, the players bring one unit of their good and submit bids on goods they like, each good gets allocated in proportion to the bid amounts, and each seller collects the bids received. Then every player updates the bids proportionally to the contribution of each good in their utility. This dynamic models a process of learning how to bid and has been studied in a series of papers on Fisher and production markets, but not in exchange economies. Our main results are as follows: - For linear utilities, the dynamic converges to market equilibrium utilities and allocations, while the bids and prices may cycle. We give a combinatorial characterization of limit cycles for prices and bids. - We introduce a lazy version of the dynamic, where players may save money for later, and show this converges in everything: utilities, allocations, and prices. - For CES utilities in the substitute range $[0,1)$, the dynamic converges for all parameters. This answers an open question about exchange economies with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known. We also note that proportional response is a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.Comment: 25 pages, 6 figure

A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market

We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities.Comment: Preliminary version in ICALP 201

Tracing Equilibrium in Dynamic Markets via Distributed Adaptation

Competitive equilibrium is a central concept in economics with numerous applications beyond markets, such as scheduling, fair allocation of goods, or bandwidth distribution in networks. Computation of competitive equilibria has received a significant amount of interest in algorithmic game theory, mainly for the prominent case of Fisher markets. Natural and decentralized processes like tatonnement and proportional response dynamics (PRD) converge quickly towards equilibrium in large classes of Fisher markets. Almost all of the literature assumes that the market is a static environment and that the parameters of agents and goods do not change over time. In contrast, many large real-world markets are subject to frequent and dynamic changes. In this paper, we provide the first provable performance guarantees of discrete-time tatonnement and PRD in markets that are subject to perturbation over time. We analyze the prominent class of Fisher markets with CES utilities and quantify the impact of changes in supplies of goods, budgets of agents, and utility functions of agents on the convergence of tatonnement to market equilibrium. Since the equilibrium becomes a dynamic object and will rarely be reached, our analysis provides bounds expressing the distance to equilibrium that will be maintained via tatonnement and PRD updates. Our results indicate that in many cases, tatonnement and PRD follow the equilibrium rather closely and quickly recover conditions of approximate market clearing. Our approach can be generalized to analyzing a general class of Lyapunov dynamical systems with changing system parameters, which might be of independent interest

The Complexity of Nash Equilibria as Revealed by Data

In this paper we initiate the study of the computational complexity of Nash equilibria in bimatrix games that are specified via data. This direction is motivated by an attempt to connect the emerging work on the computational complexity of Nash equilibria with the perspective of revealed preference theory, where inputs are data about observed behavior, rather than explicit payoffs. Our results draw such connections for large classes of data sets, and provide a formal basis for studying these connections more generally. In particular, we derive three structural conditions that are sufficient to ensure that a data set is both consistent with Nash equilibria and that the observed equilibria could have been computed effciently: (i) small dimensionality of the observed strategies, (ii) small support size of the observed strategies, and (iii) small chromatic number of the data set. Key to these results is a connection between data sets and the player rank of a game, defined to be the minimum rank of the payoff matrices of the players. We complement our results by constructing data sets that require rationalizing games to have high player rank, which suggests that computational constraints may be important empirically as well

Competitive Equilibrium for Chores: from Dual Eisenberg-Gale to a Fast, Greedy, LP-based Algorithm

We study the computation of competitive equilibrium for Fisher markets with $n$ agents and $m$ divisible chores. Prior work showed that competitive equilibria correspond to the nonzero KKT points of a non-convex analogue of the Eisenberg-Gale convex program. We introduce an analogue of the Eisenberg-Gale dual for chores: we show that all KKT points of this dual correspond to competitive equilibria, and while it is not a dual of the non-convex primal program in a formal sense, the objectives touch at all KKT points. Similar to the primal, the dual has problems from an optimization perspective: there are many feasible directions where the objective tends to positive infinity. We then derive a new constraint for the dual, which restricts optimization to a hyperplane that avoids all these directions. We show that restriction to this hyperplane retains all KKT points, and surprisingly, does not introduce any new ones. This allows, for the first time ever, application of iterative optimization methods over a convex region for computing competitive equilibria for chores. We next introduce a greedy Frank-Wolfe algorithm for optimization over our program and show a state-of-the-art convergence rate to competitive equilibrium. In the case of equal incomes, we show a $\mathcal{\tilde O}(n/\epsilon^2)$ rate of convergence, which improves over the two prior state-of-the-art rates of $\mathcal{\tilde O}(n^3/\epsilon^2)$ for an exterior-point method and $\mathcal{\tilde O}(nm/\epsilon^2)$ for a combinatorial method. Moreover, our method is significantly simpler: each iteration of our method only requires solving a simple linear program. We show through numerical experiments on simulated data and a paper review bidding dataset that our method is extremely practical. This is the first highly practical method for solving competitive equilibrium for Fisher markets with chores.Comment: 25 pages, 17 figure