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17
Distributed algorithms via gradient descent for fisher markets.
 In Proc. 12th ACM Conference on Electronic Commerce,
, 2011
"... ABSTRACT Designing distributed algorithms that converge quickly to an equilibrium is one of the foremost research goals in algorithmic game theory, and convex programs have played a crucial role in the design of algorithms for Fisher markets. In this paper we shed new light on both aspects for Fish ..."
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ABSTRACT Designing distributed algorithms that converge quickly to an equilibrium is one of the foremost research goals in algorithmic game theory, and convex programs have played a crucial role in the design of algorithms for Fisher markets. In this paper we shed new light on both aspects for Fisher markets with linear and spending constraint utilities. We show fast convergence of the Proportional Response dynamics recently introduced by Wu and Zhang [WZ07]. The convergence is obtained from a new perspective: we show that the Proportional Response dynamics is equivalent to a gradient descent algorithm (with respect to a Bregman divergence instead of euclidean distance) on a convex program that captures the equilibria for linear utilities. We further show that the convex program program easily extends to the case of spending constraint utilities, thus resolving an open question raised by
Proportional response dynamics in the Fisher market
"... Abstract. In this paper, we show that the proportional response dynamics, a utility based distributed dynamics, converges to the market equilibrium in the Fisher market with constant elasticity of substitution (CES) utility functions. By the proportional response dynamics, each buyer allocates his b ..."
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Abstract. In this paper, we show that the proportional response dynamics, a utility based distributed dynamics, converges to the market equilibrium in the Fisher market with constant elasticity of substitution (CES) utility functions. By the proportional response dynamics, each buyer allocates his budget proportional to the utility he receives from each good in the previous time period. Unlike the tâtonnement process and its variants, the proportional response dynamics is a large step discrete dynamics, and the buyers do not solve any optimization problem at each step. In addition, the goods are always cleared and assigned to the buyers proportional to their bids at each step. Despite its simplicity, the dynamics converges fast for strictly concave CES utility functions, matching the best upperbound of computing the market equilibrium via solving a global convex optimization problem. 1
New Convex Programs and Distributed Algorithms for Fisher Markets with Linear and Spending Constraint Utilities
"... In this paper we shed new light on convex programs and distributed algorithms for Fisher markets with linear and spending constraint utilities. • We give a new convex program for the linear utilities case of Fisher markets. This program easily extends to the case of spending constraint utilities as ..."
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In this paper we shed new light on convex programs and distributed algorithms for Fisher markets with linear and spending constraint utilities. • We give a new convex program for the linear utilities case of Fisher markets. This program easily extends to the case of spending constraint utilities as well, thus resolving an open question raised by [Vaz10]. • We show that the gradient descent algorithm with respect to a Bregman divergence converges with rate O(1/t) under a condition that is weaker than having Lipschitz continuous gradient (which is the usual assumption in the optimization literature for obtaining the same rate). • We show that the Proportional Response dynamics recently introduced by Zhang [Zha09] is equivalent to a gradient descent algorithm for solving the new convex program. This insight also gives us better convergence rates, and helps us generalize it to spending constraint utilities. 1
Tatonnement Beyond Gross Substitutes? Gradient Descent to the Rescue
, 2013
"... Tatonnement is a simple and natural rule for updating prices in Exchange (ArrowDebreu) markets. In this paper we define a class of markets for which tatonnement is equivalent to gradient descent. This is the class of markets for which there is a convex potential function whose gradient is always eq ..."
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Tatonnement is a simple and natural rule for updating prices in Exchange (ArrowDebreu) markets. In this paper we define a class of markets for which tatonnement is equivalent to gradient descent. This is the class of markets for which there is a convex potential function whose gradient is always equal to the negative of the excess demand and we call it Convex Potential Function (CPF) markets. We show the following results. • CPF markets contain the class of Eisenberg Gale (EG) markets, defined previously by Jain and Vazirani. • The subclass of CPF markets for which the demand is a differentiable function contains exactly those markets whose demand function has a symmetric negative semidefinite Jacobian. • We define a family of continuous versions of tatonnement based on gradient descent using a Bregman divergence. As we show, all processes in this family converge to an equilibrium for any CPF market. This is analogous to the classic result for markets satisfying the Weak Gross Substitutes property. • A discrete version of tatonnement converges toward the equilibrium for the following markets of complementary goods; its convergence rate for these settings is analyzed using a common potential function. – Fisher markets in which all buyers have Leontief utilities. The tatonnement process reduces the distance to the equilibrium, as measured by the potential function, to an ɛ fraction of its initial value in O(1/ɛ) rounds of price updates. – Fisher markets in which all buyers have complementary CES utilities. Here, the distance to the
Theory and Algorithms for Modern Problems in Machine Learning and an Analysis of Markets
, 2008
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Discrete price updates yield fast convergence in ongoing markets with finite warehouses
, 2010
"... This paper shows that in suitable markets, even with outofequilibrium trade allowed, a simple price update rule leads to rapid convergence toward the equilibrium. In particular, this paper considers a Fisher market repeated over an unbounded number of time steps, with the addition of finite sized ..."
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This paper shows that in suitable markets, even with outofequilibrium trade allowed, a simple price update rule leads to rapid convergence toward the equilibrium. In particular, this paper considers a Fisher market repeated over an unbounded number of time steps, with the addition of finite sized warehouses to enable nonequilibrium trade. The main result is that suitable tatonnement style price updates lead to convergence in a significant subset of markets satisfying the Weak Gross Substitutes property. Throughout this process the warehouse are always able to store or meet demand imbalances (the needed capacity depends on the initial imbalances). Our price update rule is robust in a variety of regards: • The updates for each good depend only on information about that good (its current price, its excess demand since its last update) and occur asynchronously from updates to other prices. • The process is resilient to error in the excess demand data. • Likewise, the process is resilient to discreteness, i.e. a limit to divisibility, both of goods and money.
A dynamic axiomatic approach to firstprice auctions
 In Proceedings of the 14th ACM Conference on Electronic Commerce
, 2013
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X Tatonnement in Ongoing Markets of Complementary Goods
"... This paper continues the study, initiated by Cole and Fleischer in [Cole and Fleischer 2008], of the behavior of a tatonnement price update rule in Ongoing Fisher Markets. The prior work showed fast convergence toward an equilibrium when the goods satisfied the weak gross substitutes property and ha ..."
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This paper continues the study, initiated by Cole and Fleischer in [Cole and Fleischer 2008], of the behavior of a tatonnement price update rule in Ongoing Fisher Markets. The prior work showed fast convergence toward an equilibrium when the goods satisfied the weak gross substitutes property and had bounded demand and income elasticities. The current work shows that fast convergence also occurs for the following types of markets: — All pairs of goods are complements to each other, and — the demand and income elasticities are suitably bounded. In particular, these conditions hold when all buyers in the market are equipped with CES utilities, where all the parameters ρ, one per buyer, satisfy −1 < ρ ≤ 0. In addition, we extend the above result to markets in which a mixture of complements and substitutes occur. This includes characterizing a class of nested CES utilities for which fast convergence holds. An interesting technical contribution, which may be of independent interest, is an amortized analysis for handling asynchronous events in settings in which there are a mix of continuous changes and discrete events.
N.K.: Towards polynomial simplexlike algorithms for market equilibria
, 2013
"... In this paper we consider the problem of computing market equilibria in the Fisher setting for utility models such as spending constraint and perfect, pricediscrimination. These models were inspired from modern ecommerce settings and attempt to bridge the gap between the computationally hard but ..."
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In this paper we consider the problem of computing market equilibria in the Fisher setting for utility models such as spending constraint and perfect, pricediscrimination. These models were inspired from modern ecommerce settings and attempt to bridge the gap between the computationally hard but realistic separable, piecewiselinear and concave utility model and, the tractable but less relevant linear utility case. While there are polynomial time algorithms known for these problems, the question of whether there exist polynomial time Simplexlike algorithms has remained elusive, even for linear markets. Such algorithms are desirable due to their conceptual simplicity, ease of implementation and practicality. This paper takes a significant step towards this goal by presenting the first Simplexlike algorithms for these markets assuming a positive resolution of an algebraic problem of Cucker, Koiran and Smale. Unconditionally, our algorithms are FPTASs; they compute prices and allocations such that each buyer derives at least a 1 1+εfraction of the utility at a true market equilibrium, and their running times are polynomial in the input length and 1/ε. We start with convex programs which capture market equilibria in each setting and, in a systematic way, convert them into linear complementarity problem (LCP) formulations. Then, departing from previous approaches which try to pivot on a single polyhedron associated to the LCP obtained, we carefully construct a polynomiallength sequence of polyhedra, one containing the other, such that starting from an optimal solution to one allows us to obtain an optimal solution to the next in the sequence in a polynomial number of complementary pivot steps. Our framework to convert a convex program into an LCP and then come up with a Simplexlike algorithm that moves on a sequence of connected polyhedra may be of independent interest. 1
Market communication in production economies
 In: Proceedings of the 6th international conference on Internet and network economics. WINE’10
, 2010
"... Abstract. We study the information content of equilibrium prices using the market communication model of Deng, Papadimitriou, and Safra [4]. We show that, in the worst case, communicating an exact equilibrium in a production economy requires a number of bits that is a quadratic polynomial in the num ..."
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Abstract. We study the information content of equilibrium prices using the market communication model of Deng, Papadimitriou, and Safra [4]. We show that, in the worst case, communicating an exact equilibrium in a production economy requires a number of bits that is a quadratic polynomial in the number of goods, the number of agents, the number of firms, and the number of bits used to represent an endowment. 1