A complementary pivot algorithm for markets under separable, piecewise-linear concave utilities

Jugal Garg, Ruta Mehta, Milind Sohoni, Vijay V. Vazirani

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Using the powerful machinery of the linear complementarity problem and Lemke's algorithm, we give a practical algorithm for computing an equilibrium for Arrow-Debreu markets under separable, piecewise-linear concave (SPLC) utilities, despite the PPAD-completeness of this case. As a corollary, we obtain therst elementary proof of existence of equilibrium for this case, i.e., without using fixed point theorems. In 1975, Eaves [10] had given such an algorithm for the case of linear utilities and had asked for an extension to the piecewise-linear, concave utilities. Our result settles the relevant subcase of his problem as well as the problem of Vazirani and Yannakakis of obtaining a path following algorithm for SPLC markets, thereby giving a direct proof of membership of this case in PPAD. We also prove that SPLC markets have an odd number of equilibria (up to scaling), hence matching the classical result of Shapley about 2-Nash equilibria [24], which was based on the Lemke-Howson algorithm. For the linear case, Eaves had asked for a combinatorial interpretation of his algorithm. We provide this and it yields a particularly simple proof of the fact that the set of equilibrium prices is convex.

Original languageEnglish (US)
Title of host publicationSTOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing
Pages1003-1015
Number of pages13
DOIs
StatePublished - 2012
Externally publishedYes
Event44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States
Duration: May 19 2012May 22 2012

Other

Other44th Annual ACM Symposium on Theory of Computing, STOC '12
CountryUnited States
CityNew York, NY
Period5/19/125/22/12

Keywords

  • piecewise-linear concave utilities
  • separable

ASJC Scopus subject areas

  • Software

Fingerprint Dive into the research topics of 'A complementary pivot algorithm for markets under separable, piecewise-linear concave utilities'. Together they form a unique fingerprint.

Cite this