Market Equilibrium in Exchange Economies with Some Families of Concave Utility Functions

We present explicit convex programs which characterize the equilibrium for certain additively separable utility functions and CES functions. These include some CES utility functions that do not satisfy weak gross substitutability.Exchange economy, computation of equilibria, convex feasibility problem

Nash equilibria in fisher market

Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the Fisher market game in which buyers strategize by posing different utility functions. We show that existence of a conflict-free allocation is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs

Improved Balanced Flow Computation Using Parametric Flow

We present a new algorithm for computing balanced flows in equality networks arising in market equilibrium computations. The current best time bound for computing balanced flows in such networks requires $O(n)$ maxflow computations, where $n$ is the number of nodes in the network [Devanur et al. 2008]. Our algorithm requires only a single parametric flow computation. The best algorithm for computing parametric flows [Gallo et al. 1989] is only by a logarithmic factor slower than the best algorithms for computing maxflows. Hence, the running time of the algorithms in [Devanur et al. 2008] and [Duan and Mehlhorn 2015] for computing market equilibria in linear Fisher and Arrow-Debreu markets improve by almost a factor of $n$

Computing Equilibria in Markets with Budget-Additive Utilities

We present the first analysis of Fisher markets with buyers that have budget-additive utility functions. Budget-additive utilities are elementary concave functions with numerous applications in online adword markets and revenue optimization problems. They extend the standard case of linear utilities and have been studied in a variety of other market models. In contrast to the frequently studied CES utilities, they have a global satiation point which can imply multiple market equilibria with quite different characteristics. Our main result is an efficient combinatorial algorithm to compute a market equilibrium with a Pareto-optimal allocation of goods. It relies on a new descending-price approach and, as a special case, also implies a novel combinatorial algorithm for computing a market equilibrium in linear Fisher markets. We complement these positive results with a number of hardness results for related computational questions. We prove that it is NP-hard to compute a market equilibrium that maximizes social welfare, and it is PPAD-hard to find any market equilibrium with utility functions with separate satiation points for each buyer and each good.Comment: 21 page

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

A rational convex program for linear Arrow-Debreu markets

We present a new flow-type convex program describing equilibrium solutions to linear Arrow-Debreu markets. Whereas convex formulations were previously known ([Nenakov and Primak 1983; Jain 2007; Cornet 1989]), our program exhibits several new features. It provides a simple necessary and sufficient condition and a concise proof of the existence and rationality of equilibria, settling an open question raised by Vazirani [2012]. As a consequence, we also obtain a simple new proof of the result in Mertens [2003] that the equilibrium prices form a convex polyhedral set

On the Complexity of Market Equilibria and Revenue Maximization

This thesis consists of two parts. In the first part, we concentrate on the computation of Market Equilibria and settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utilities. We prove that the problem is PPAD-complete when the Constant Elasticity of Substitution parameter $\rho$ is any constant less than -1. Building on this result, we introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory, and prove that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. In the second part, we study Revenue Maximization. We begin by resolving the complexity of the revenue-optimal Bayesian Unit-demand Item Pricing problem when the buyer's values for the items are independent. We show that the problem can be solved in polynomial time for distributions of support size 2; but its decision version is NP-complete for distributions of support size 3. Next, we study the optimal mechanism design problem for a single unit-demand buyer with item values drawn from independent distributions. We show that, for distributions of support-size 2 and the same high value, Item Pricing can achieve the same revenue as any menu of lotteries. On the other hand, we provide simple examples where randomization improves revenue. Finally, we show that unless the polynomial-time hierarchy collapses, namely P^{NP}=P^{#P}, there is no universal efficient randomized algorithm that implements an optimal mechanism even when distributions have support size 3

Fast-Converging Tatonnement Algorithms for the Market Problem

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 defines and analyzes two simple tatonnement algorithms that differ 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 relative demand for money at equilibrium prices. We give two protocols. The first assumes global knowledge of only the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters. Our second protocol assumes no global knowledge whatsoever