Blackwell Approachability and Minimax Theory

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 (Blackwell [1]) and its generalization due to Hou [6] are still valid. Second, minimax structure grants a result in the spirit of Blackwell’s weak-approachability conjecture (Blackwell [1]), later resolved by Vieille [11], that any set is either approachable by one player, or avoidable by the opponent. This analysis also reveals a strategy for the opponent.