A path to the Arrow–Debreu competitive market equilibrium

Ye, Yinyu

doi:10.1007/s10107-006-0065-5

Cited by 89 publications

(95 citation statements)

References 34 publications

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“…We know that an (approximate) market equilibrium in any Fisher's economy with CES utilities can be found in polynomial time [4,15,12,21,7,22]. In fact, Ye [21] proved that if every utility function is the minimum of a collection of homogeneous linear functions, then one can find a Fisher equilibrium in polynomial time.…”

Section: Market Equilibria: Fisher's Model Vs Arrow-debreu's Modelmentioning

confidence: 99%

Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

Chen

Teng

2009

Algorithms and Computation

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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 ArrowDebreu 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.

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Section: Market Equilibria: Fisher's Model Vs Arrow-debreu's Modelmentioning

confidence: 99%

Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

Chen

Teng

2009

Algorithms and Computation

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“…In fact, Ye shows that an ǫ-equilibrium of such an market with n traders and n goods can be found in O(n 4 log(1/ǫ)) time [27]. If data is given as rational numbers of L-bits, then an exact equilibrium can be found in O(n 4 L) time.…”

Section: General Market Exchange Problems and Algorithmic Resultsmentioning

confidence: 99%

“…Even more remarkablely, an (approximate) equilibrium in a Fisher's economy with any CES utilities can be found in polynomial time [11,27,28,10,14].…”

Section: General Market Exchange Problems and Algorithmic Resultsmentioning

confidence: 99%

On the Approximation and Smoothed Complexity of Leontief Market Equilibria

Huang¹,

Teng²

Frontiers in Algorithmics

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We show that the problem of finding an ǫ-approximate Nash equilibrium of an n × n two-person games can be reduced to the computation of an (ǫ/n) 2 -approximate market equilibrium of a Leontief economy. Together with a recent result of Chen, Deng and Teng, this polynomial reduction implies that the Leontief market exchange problem does not have a fully polynomial-time approximation scheme, that is, there is no algorithm that can compute an ǫ-approximate market equilibrium in time polynomial in m, n, and 1/ǫ, unless PPAD ⊆ P, We also extend the analysis of our reduction to show, unless PPAD ⊆ RP, that the smoothed complexity of the Scarf's general fixed-point approximation algorithm (when applying to solve the approximate Leontief market exchange problem) or of any algorithm for computing an approximate market equilibrium of Leontief economies is not polynomial in n and 1/σ, under Gaussian or uniform perturbations with magnitude σ.

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“…It is well-known that one can use interior point methods to solve the linearly constrained convex program (10) to yield both primal and dual optimal solutions in polynomial time; see [15].…”

Section: Convex Optimization For Computing An Equilibriummentioning

confidence: 99%

“…It is shown in [15] that this inequality admits an efficient barrier function for interior-point algorithms. Thus, Theorem 3.…”

Section: The Markets With Concave and Non-homogeneous Utilitiesmentioning

confidence: 99%

On Equilibrium Pricing as Convex Optimization

Chen¹,

Ye²,

Zhang³

2010

JCM

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We study competitive economy equilibrium computation. We show that, for the first time, the equilibrium sets of the following two markets:1. A mixed Fisher and Arrow-Debreu market with homogeneous and log-concave utility functions;2. The Fisher and Arrow-Debreu markets with several classes of concave non-homogeneous utility functions; are convex or log-convex. Furthermore, an equilibrium can be computed as convex optimization by an interior-point algorithm in polynomial time.

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A path to the Arrow–Debreu competitive market equilibrium

Cited by 89 publications

References 34 publications

Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

On the Approximation and Smoothed Complexity of Leontief Market Equilibria

On Equilibrium Pricing as Convex Optimization

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