Blackwell Approachability and Minimax Theory

TL;DR

This paper explores the connection between Blackwell Approachability in vector-valued repeated games and classical minimax theory, establishing new theoretical results and strategies in this context.

Contribution

It demonstrates that Blackwell's Approachability Theorem holds without minimax assumptions and links approachability with minimax theory, providing new insights and strategies.

Findings

Blackwell's Approachability Theorem is valid without minimax assumptions.

Any set in the game is either approachable or avoidable, extending previous conjectures.

The paper reveals a strategy for the opponent based on minimax structure.

Abstract

This manuscript investigates the relationship between Blackwell Approachability, a stochastic vector-valued repeated game, and minimax theory, a single-play scalar-valued scenario. First, it is established in a general setting --- one not permitting invocation of minimax theory --- that Blackwell's Approachability Theorem and its generalization due to Hou are still valid. Second, minimax structure grants a result in the spirit of Blackwell's weak-approachability conjecture, later resolved by Vieille, that any set is either approachable by one player, or avoidable by the opponent. This analysis also reveals a strategy for the opponent.