Minimax Duality in Game-Theoretic Probability

TL;DR

This paper establishes a minimax duality framework for game-theoretic probability, linking it to classical measure-theoretic results and enabling the derivation of new and existing theorems within a unified game-theoretic context.

Contribution

It introduces a new minimax theorem for finite-time game-theoretic probability, unifying and extending prior results through a novel gamble space framework.

Findings

Proves a finite-time minimax theorem for game-theoretic probability.

Derives several existing and new game-theoretic probability results.

Suggests a broader generalization akin to Ville's theorem.

Abstract

Game-theoretic probability uses the structure of gambles to define a concept like probability, but which is more flexible and robust. We show that results in game-theoretic probability can be thought of as minimax theorems for specific zero-sum games between two players, Gambler and World. The traditional measure-theoretic versions arise when World must play first. This perspective suggests the possibility of a more general minimax theorem from which a wide array of game-theoretic results would follow. After developing a new framing of game-theoretic probability via gamble spaces, we prove such a theorem for finite time. Applying this minimax theorem to games derived from existing measure-theoretic statements, we prove several existing and novel game-theoretic statements. This general minimax theorem can be thought of as a composite Ville's theorem, as we discuss along with future directions.