Market Equilibria with Hybrid Linear-Leontief Utilities

2009 50th Annual IEEE Symposium on Foundations of Computer Science

Dai

et al. 2009

Self Cite

103

117

Section: Our Contributionmentioning

confidence: 99%

Settling the Complexity of Arrow-Debreu Equilibria in Markets with Additively Separable Utilities

2009 50th Annual IEEE Symposium on Foundations of Computer Science

Dai

et al. 2009

Self Cite

103

117

“…-For Fisher's model, polynomial-time algorithms are given not only for linear markets but also for Leontief and many other markets, e.g., the hybrid linear-Leontief markets [20]. 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].…”

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

confidence: 99%

“…Assume (14) holds for some k ∈ [2n]. Then by (5) we have a 2k [T * ] − 1 ≤ O(1/n 21 ) and thus, a 2k [T * ] < 1 + 1/n20 . This implies that a k,2k < 1 + 1/n20 .…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Algorithms and Computation

Teng

2009

Self Cite

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.

Settling the complexity of computing two-player Nash equilibria

Deng

Teng

2009

J. ACM

Self Cite

498

610

We prove that BIMATRIX, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991.Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems:-BIMATRIX does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. -The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for BIMATRIX is not polynomial unless every problem in PPAD is solvable in randomized polynomial time.Our results also have a complexity implication in mathematical economics:-Arrow-Debreu market equilibria are PPAD-hard to compute.