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

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 ε 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 (1/ε)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses poly( m , log (1/ε)) 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.

Optimally Deceiving a Learning Leader in Stackelberg Games

“…Best response queries are weaker than utility queries, but they arise naturally in practice, and are also expressive enough to implement fictitious play, a dynamic first proposed by Brown (1949), and proven to converge by Robinson (1951) for two-player zero-sum games to an approximate Nash equilibrium. In terms of equilibrium computation, Goldberg and Marmolejo-Cossío (2018) also provide query-efficient algorithms for computing approximate Nash equilibria for bimatrix games via best response queries provided one agent has a constant number of strategies.…”

“…Any best response correspondence BR satisfying this assumption is called polytopal as is formally defined below. Definition 3.1 (Polytopal BR correspondence (Goldberg & Marmolejo-Cossío, 2018)). A best response correspondence BR : ∆ m−1 → 2 [n] \ {∅} is polytopal if it also satisfies the following:…”