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Learning Convex Partitions and Computing Game-Theoretic Equilibria from Best Response Queries

  • Paul W. Goldberg
  • Francisco J. Marmolejo-Cossío
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

Abstract

Suppose that an m-simplex is partitioned into n convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance \(\varepsilon \) from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant m uses \(poly(n, \log \left( \frac{1}{\varepsilon } \right) )\) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses \(poly(m, \log \left( \frac{1}{\varepsilon } \right) )\) queries. We show via Kakutani’s fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies.

Keywords

Query protocol Equilibrium computation Revealed preferences 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Paul W. Goldberg
    • 1
  • Francisco J. Marmolejo-Cossío
    • 1
  1. 1.University of OxfordOxfordUK

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