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 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.

💡 Research Summary

The paper investigates the relationship between Blackwell’s approachability theory—originally formulated for stochastic vector‑valued repeated games—and classical minimax theory, which concerns single‑play scalar‑valued games. The authors work in a very general setting where the usual assumptions that guarantee the applicability of von Neumann’s minimax theorem (linear payoffs, compact convex action sets) are deliberately omitted. Their first contribution is to show that Blackwell’s original approachability theorem and its later generalization by Hou remain valid even when minimax theory cannot be invoked. This is achieved by carefully distinguishing between two quantifier‑order notions: “1‑force” (the player moving first) and “2‑force” (the player moving second). They define forcing and avoidance in terms of these quantifiers and develop a geometric tool called a half‑space‑forcing candidate, which captures the local relationship between the current average payoff vector and the target set.

A central technical result (Proposition 3.1) proves that for any half‑space H, if the X‑player can 2‑force H (i.e., guarantee the payoff lies in H when moving second) and the scalarized game ⟨f(·,·),λ⟩ satisfies the minimax property for every direction λ, then X can also 1‑force H. The proof relies on the existence of a saddle point for the scalarized game, showing that minimax structure collapses the distinction between the two quantifier orders for half‑spaces. The authors also exhibit counter‑examples where the minimax property fails, demonstrating that the equivalence between 1‑force and 2‑force can break down for non‑half‑space sets.

In the repeated‑game setting (Section 4), the authors introduce the notion of a half‑space‑forcing candidate (a triple (φ,ψ,H) where φ is the current average payoff, ψ∈S is a nearest point in the target set, and H is the half‑space orthogonal to the segment