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16
Market equilibria for homothetic, quasiconcave utilities and economies of scale in production
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Fastconverging tatonnement algorithms for onetime and ongoing market problems
 In Symposium on Theory of Computing (STOC 2008
, 2008
"... 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 OneTime Markets. The Ongoing Market allows ..."
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Cited by 18 (2 self)
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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 OneTime Markets. The Ongoing Market allows trade at nonequilibrium 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
A primaldual algorithm for computing Fisher equilibrium in absence of gross substitutability property
 In Proceedings of wine
, 2005
"... Abstract. We provide the first strongly polynomial time exact combinatorial algorithm to compute Fisher equilibrium for the case when utility functions do not satisfy the Gross substitutability property. The motivation for this comes from the work of Kelly, Maulloo, and Tan [15] and Kelly and Vazira ..."
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Cited by 9 (4 self)
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Abstract. We provide the first strongly polynomial time exact combinatorial algorithm to compute Fisher equilibrium for the case when utility functions do not satisfy the Gross substitutability property. The motivation for this comes from the work of Kelly, Maulloo, and Tan [15] and Kelly and Vazirani [16] on rate control in communication networks. We consider a tree like network in which root is the source and all the leaf nodes are the sinks. Each sink has got a fixed amount of money which it can use to buy the capacities of the edges in the network. The edges of the network sell their capacities at certain prices. The objective of each edge is to fix a price which can fetch the maximum money for it and the objective of each sink is to buy capacities on edges in such a way that it can facilitate the sink to pull maximum flow from the source. In this problem, the edges and the sinks play precisely the role of sellers and buyers, respectively, in the Fisher’s market model. The utility of a buyer (or sink) takes the form of Leontief function which is
A Fast and Simple Algorithm for Computing Market Equilibria
"... We give a new mathematical formulation of market equilibria using an indirect utility function: the function of prices and income that gives the maximum utility achievable. The formulation is a convex program and can be solved when the indirect utility function is convex in prices. We illustrate th ..."
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Cited by 8 (1 self)
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We give a new mathematical formulation of market equilibria using an indirect utility function: the function of prices and income that gives the maximum utility achievable. The formulation is a convex program and can be solved when the indirect utility function is convex in prices. We illustrate that many economies including – Homogeneous utilities of degree α ∈ [0, 1] in Fisher economies — this includes Linear, Leontief, CobbDouglas – Resource allocation utilities like multicommodity flows satisfy this condition and can be efficiently solved. Further, we give a natural and decentralized priceadjusting algorithm in these economies. Our algorithm, mimics the natural tâtonnement dynamics for the markets as suggested by Walras: it iteratively adjusts a good’s price upward when the demand for that good under current prices exceeds its supply; and downward when its supply exceeds its demand. The algorithm computes an approximate equilibrium in a number of iterations that is independent of the number of traders and is almost linear in the number of goods. Interestingly, our algorithm applies to certain classes of utility functions that are not weak gross substitutes.
On the complexity of market equilibria with maximum social welfare
 Information Processing Letters
"... Abstract. We consider the computational complexity of the Market equilibrium problem by exploring the structural properties of the Leontief exchange economy. We prove that, for economies guaranteed to have a market equilibrium, nding one with maximum social welfare or maximum individual welfare ..."
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Cited by 5 (1 self)
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Abstract. We consider the computational complexity of the Market equilibrium problem by exploring the structural properties of the Leontief exchange economy. We prove that, for economies guaranteed to have a market equilibrium, nding one with maximum social welfare or maximum individual welfare is NPhard. In addition, we prove that counting the number of equilibrium prices is #Phard.
The complexity of nonmonotone markets
 In Proceedings of ACM STOC
, 2013
"... We introduce the notion of nonmonotone utilities, which covers a wide variety of utility functions in economic theory. We show that it is PPADhard to compute an approximate ArrowDebreu market equilibrium in markets with linear and nonmonotone utilities. Building on this result, we settle the ..."
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Cited by 5 (0 self)
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We introduce the notion of nonmonotone utilities, which covers a wide variety of utility functions in economic theory. We show that it is PPADhard to compute an approximate ArrowDebreu market equilibrium in markets with linear and nonmonotone utilities. Building on this result, we settle the longstanding open problem regarding the computation of an approximate ArrowDebreu market equilibrium in markets with CES utilities, by proving that it is PPADcomplete when the Constant Elasticity of Substitution parameter, ρ, is any constant less than −1. Categories and Subject Descriptors F.2 [Analysis of algorithms and problem complexity]
Computing equilibria in a fisher market with linear singleconstraint production units
, 2005
"... We study the problem of computing equilibrium prices in a Fisher market with linear utilities and linear singleconstraint production units. This setting naturally appears in ad pricing where the sum of the lengths of the displayed ads is constrained not to exceed the available ad space. There are ..."
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We study the problem of computing equilibrium prices in a Fisher market with linear utilities and linear singleconstraint production units. This setting naturally appears in ad pricing where the sum of the lengths of the displayed ads is constrained not to exceed the available ad space. There are three approaches to solve market equilibrium problems: convex programming, auctionbased algorithms, and primaldual. Jain, Vazirani, and Ye recently proposed a solution using convex programming for the problem with an arbitrary number of production constraints. A recent paper by Kapoor, Mehta, and Vazirani proposes an auctionbased solution. No primaldual algorithm is proposed for this problem. In this paper we propose a simple reduction from this problem to the classical Fisher setting with linear utilities and without any production units. Our reduction not only imports the primaldual algorithm of Devanur et al. to the singleconstraint production setting, but also: i) imports other simple algorithms, like the auctionbased algorithm of Garg and Kapoor, thereby providing a simple insight behind the recent sophisticated algorithm of Kapoor, Mehta, and Vazirani, and ii) imports all the nice properties of the Fisher setting, for example, the existence of an equilibrium in rational numbers, and the uniqueness of the utilities of the agents at the equilibrium.
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.
FastConverging Tatonnement Algorithms for the Market Problem ∗
, 2007
"... 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 ..."
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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.