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

  • Authors:

  • Jugal Garg;Ruta Mehta;Milind Sohoni;Vijay V. Vazirani

  • Affiliations:

  • IIT-Bombay, Mumbai, India;IIT-Bombay, Mumbai, India;IIT-Bombay, Mumbai, India;Georgia Institute of Technology, Atlanta, GA, USA

  • Venue:
  • STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
  • Year:
  • 2012

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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 the first 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.