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

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Part of the Lecture Notes in Computer Science book series (LNISA,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

Full Online Version of Paper: https://arxiv.org/abs/1807.06170.

F. J. Marmolejo Cossío—Supported by the Mexican National Council of Science and Technology (CONACyT).

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Fig. 2.

Notes

  1. 1.

    CD-GBS runs in polynomial time for constant m. The time-intensive operation consists of identifying uncovered intervals, but since the dimension of the ambient simplex is constant, each empirical polytope \({\widehat{P}}_i\) has at most a constant number of bounding hyperplanes. These hyperplanes can each be extruded by \(\varepsilon \), and checking whether there exists a point outside all these extrusions can be done in time polynomial in n via brute force. In fact, all other algorithms in this paper have efficient runtimes (in their relevant parameters) due to similar reasoning.

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Goldberg, P.W., Marmolejo-Cossío, F.J. (2018). Learning Convex Partitions and Computing Game-Theoretic Equilibria from Best Response Queries. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_12

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  • DOI: https://doi.org/10.1007/978-3-030-04612-5_12

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