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

TL;DR

This paper introduces algorithms for learning convex partitions of a simplex with label queries and applies these to bound the query complexity of computing approximate game-theoretic equilibria, especially in games with few strategies.

Contribution

The paper presents two new algorithms for convex partition learning and connects them to bounds on query complexity for approximate equilibria in bimatrix and multi-player games.

Findings

Algorithms achieve polylogarithmic query complexity in the number of regions or dimensions.

Bounds are established for approximate well-supported equilibria in bimatrix games with constant strategies.

Partial extensions to multi-player games are provided.

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 $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ 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. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.