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Market Equilibrium via a Primal-Dual Algorithm for a Convex Program
"... We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primal-dual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by ..."
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Cited by 127 (27 self)
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We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primal-dual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by this setting and the manner in which our algorithm circumvents it.
Leontief Economies Encode Nonzero Sum Two-Player Games
"... We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria. We give a ..."
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Cited by 37 (4 self)
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We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria. We give a reduction from two-player games to a special family of Leontief exchange economies, which are guaranteed to have equilibria, with the property that the Nash equilibria of any game are in one-to-one correspondence with the equilibria of the corresponding economy. Our reduction exposes a potential hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies (where an equilibrium is guaranteed to exist) is at least as hard as finding a Nash equilibrium for two-player nonzero sum
Settling the complexity of Arrow-Debreu equilibria in markets with additively separable utilities
- IN: PROCEEDINGS OF THE 50TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2009
"... We prove that the problem of computing an Arrow-Debreu market equilibrium is PPAD-complete even when all traders use additively separable, piecewise-linear and concave utility functions. In fact, our proof shows that this market-equilibrium problem does not have a fully polynomial-time approximation ..."
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Cited by 31 (5 self)
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We prove that the problem of computing an Arrow-Debreu market equilibrium is PPAD-complete even when all traders use additively separable, piecewise-linear and concave utility functions. In fact, our proof shows that this market-equilibrium problem does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time.
Market equilibrium via the excess demand function
- In Proceedings STOC’05
, 2005
"... 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 polyno ..."
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Cited by 30 (2 self)
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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
On the polynomial time computation of equilibria for certain exchange economies
- IN SODA
, 2005
"... The problem of computing equilibria for exchange economies has recently started to receive a great deal of attention in the theoret ical computer science community. It has been shown that equi l ibr ia can be computed in polynomial t ime in various special cases, the most impor tant of which are whe ..."
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Cited by 30 (6 self)
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The problem of computing equilibria for exchange economies has recently started to receive a great deal of attention in the theoret ical computer science community. It has been shown that equi l ibr ia can be computed in polynomial t ime in various special cases, the most impor tant of which are when traders have l inear, Cobb-Douglas, or a range of CES ut i l i ty functions. These im-por tant special cases are instances when the market sat-isfies a proper ty called weak gTvss substitutability. Clas-sical results in economics, which theoret ical computer scientists ( including us) appear to have been hitherto unaware of, show that the equi l ibr ium prices in such markets are character ized by an infinite number of lin-ear inequalit ies and therefore form a convex set. In this paper, we show that under fairly general assumptions,
The spending constraint model for market equilibrium: Algorithmic, existence and uniqueness results
- In Proceedings of 36th Annual ACM Symposium on Theory of Computing (STOC). ACM
"... The traditional model of market equilibrium supports impressive existence results, including the celebrated Arrow-Debreu Theorem. However, in this model, polynomial time algorithms for computing (or approximating) equilibria are known only for linear utility functions. We present a new, and natural, ..."
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Cited by 30 (9 self)
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The traditional model of market equilibrium supports impressive existence results, including the celebrated Arrow-Debreu Theorem. However, in this model, polynomial time algorithms for computing (or approximating) equilibria are known only for linear utility functions. We present a new, and natural, model of market equilibrium that not only admits existence and uniqueness results paralleling those for the traditional model but is also amenable to efficient algorithms.
Cell Breathing in Wireless LANs: Algorithms and Evaluation
- IN IEEE TRANS. ON MOBILE COMPUTING (TMC)
, 2007
"... Wireless LAN administrators often have to deal with the problem of sporadic client congestion in popular locations within the network. Existing approaches that relieve congestion by balancing the traffic load are encumbered by the modifications that are required to both access points and clients. W ..."
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Cited by 30 (3 self)
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Wireless LAN administrators often have to deal with the problem of sporadic client congestion in popular locations within the network. Existing approaches that relieve congestion by balancing the traffic load are encumbered by the modifications that are required to both access points and clients. We propose Cell Breathing, a well-known concept in cellular telephony, as a load balancing mechanism to handle client congestion in a wireless LAN. We develop power management algorithms for controlling the coverage of access points to handle dynamic changes in client workloads. We further incorporate hand-off costs and manufacturer specified power level constraints into our algorithms. Our approach does not require modification to clients or to the standard. It only changes the transmission power of beacon packets and does not change the transmission power of data packets to avoid the interactions with autorating. We analyze the worst-case bounds of the algorithms and show that they are either optimal or close to optimal. In addition, we evaluate our algorithms empirically using synthetic and real wireless LAN traces. Our results show that cell breathing significantly outperforms the commonly used fixed power scheme and performs at par with sophisticated load balancing schemes that require changes to both the client and access points.
Market equilibria for homothetic, quasi-concave utilities and economies of scale in production
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Spending is not easier than trading: on the computational equivalence of Fisher and Arrow-Debreu equilibria
- Journal of the ACM
"... Abstract. It is a common belief that computing a market equilibrium in Fisher’s spending model is easier than computing a market equilibrium in Arrow-Debreu’s exchange model. This belief is built on the fact that we have more algorithmic success in Fisher equilibria than Arrow-Debreu equilibria. For ..."
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Cited by 20 (3 self)
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Abstract. It is a common belief that computing a market equilibrium in Fisher’s spending model is easier than computing a market equilibrium in Arrow-Debreu’s exchange model. This belief is built on the fact that we have more algorithmic success in Fisher equilibria than Arrow-Debreu equilibria. For example, a Fisher equilibrium in a Leontief market can be found in polynomial time, while it is PPAD-hard to compute an approximate Arrow-Debreu equilibrium in a Leontief market. In this paper, we show that even when all the utilities are additively separable, piecewise-linear, and concave functions, finding an approximate equilibrium in Fisher’s model is complete in PPAD. Our result solves a long-term open question on the complexity of market equilibria. To the best of our knowledge, this is the first PPAD-completeness result for Fisher’s model. 3 1
Fast-converging tatonnement algorithms for one-time 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 One-Time 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 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
