## 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 language | English (US) |
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Title of host publication | STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing |

Pages | 1003-1015 |

Number of pages | 13 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

Event | 44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States Duration: May 19 2012 → May 22 2012 |

### Other

Other | 44th Annual ACM Symposium on Theory of Computing, STOC '12 |
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Country | United States |

City | New York, NY |

Period | 5/19/12 → 5/22/12 |

## Keywords

- piecewise-linear concave utilities
- separable

## ASJC Scopus subject areas

- Software