Efficient Computation of Equilibrium Prices for Markets with Leontief Utilities

Codenotti, Bruno; Varadarajan, Kasturi

doi:10.1007/978-3-540-27836-8_33

Cited by 54 publications

(47 citation statements)

References 17 publications

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“…Note that in general the equilibria of Leontief exchange economies can be irrational ( [5], Section 3) so that the existential problem does not belong to NP, and we thus talk of NP-hardness as opposed to NPcompleteness.…”

Section: Hardness Resultsmentioning

confidence: 99%

“…In a very short time, polynomial-time algorithms have been developed for computing the prices for different special cases of this problem using techniques such as primal-dual [9,21], auction algorithms [15,16], and convex programming [29,20,32,5,4,3]. However, it seems that all the markets for which these polynomial-time algorithms have been derived share a common property: their equilibrium set is convex.…”

Section: Introductionmentioning

confidence: 99%

“…The set of equilibria in these markets can be disconnected [18,6]. Furthermore, no efficient algorithm is known for computing the equilibrium prices in these markets, except in the case of proportional endowments, where the set of equilibria is convex [5]. Our result shows that polynomial time algorithms handling the equilibrium problem in such a scenario where multiple disconnected equilibria can readily appear, would have an extremely important computational consequence for bimatrix games.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Leontief economies encode nonzero sum two-player games

Codenotti

Saberi

Varadarajan

et al. 2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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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 games.As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [17]. In particular, among other results, we show that it is NP-hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium.Perhaps more importantly, we also prove that it is NPhard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known PPAD-completeness result of [30], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwer's Fixed Point Theorem.On the algorithmic side, we present an algorithm for finding an approximate equilibrium for some special Leontief economies, which achieves quasi-polynomial time whenever each trader does not demand too much more of any good than some other good.

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Section: Hardness Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Leontief economies encode nonzero sum two-player games

Codenotti

Saberi

Varadarajan

et al. 2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

Self Cite

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show abstract

“…However, the latter program had limited success (most notably, an efficient algorithm for approximating equilibria for the Fisher model under Leontief utilities [8,27]), and was dealt a serious blow with the recent resolution of the long-standing open problem of finding the complexity of computing an equilibrium under additively separable, piecewise-linear, concave utilities (plc utilities). First, [6] proved PPAD-hardness for the Arrow-Debreu model under plc utilities.…”

Section: Introductionmentioning

confidence: 99%

A Perfect Price Discrimination Market Model with Production, and a (Rational) Convex Program for It

Goel

Vazirani

2010

Algorithmic Game Theory

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Recent results showing PPAD-completeness of the problem of computing an equilibrium for Fisher's market model under additively separable, piecewise-linear, concave utilities (plc utilities) have dealt a serious blow to the program of obtaining efficient algorithms for computing equilibria in "traditional" market models and has prompted a search for alternative models that are realistic as well as amenable to efficient computation. In this paper we show that introducing perfect price discrimination into the Fisher model with plc utilities renders its equilibrium polynomial time computable. Moreover, its set of equilibria are captured by a convex program that generalizes the classical Eisenberg-Gale program, and always admits a rational solution. We also give a combinatorial, polynomial time algorithm for computing an equilibrium.Next, we introduce production into our model, and again give a (rational) convex program that captures its equilibria. We use this program to obtain surprisingly simple proofs of both welfare theorems or this model. Finally, we also give an application of our price discrimination market model to online display advertising marketplaces.

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“…It was known that polynomial-time algorithms existed to compute the market equilibria of Leontief economy in Fisher setting [5]. The recent hardness results were developed through a one-to-one correspondence between the market equilibria in a special case of Leontief economy and the Nash equilibria in bi-matrix games, initially discovered by Ye [17].…”

Section: Introductionmentioning

confidence: 99%