Minimax Duality in Game-Theoretic Probability (2512.21026v1)

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.

• The paper proves a minimax duality theorem that establishes equality between game-theoretic and measure-theoretic expectations for bounded, semi-analytic variables under finitely-generated gamble spaces.
• The methodology employs induction and a tower property for conditional expectations, rigorously deriving a chain of inequalities among different probabilistic pricing schemes.
• The results unify classical probabilistic proofs with constructive game-theoretic strategies, offering robust implications for finance, risk management, and online learning applications.

Minimax Duality and Game-Theoretic Probability: An Expert Synthesis

Overview

"Minimax Duality in Game-Theoretic Probability" (2512.21026) presents a formal bridge between game-theoretic and measure-theoretic probability, unifying the two approaches through a general minimax duality framework articulated with precise mathematical formalism. The author introduces the notion of gamble spaces—collections of contracts (gambles) that a gambler can select over an arbitrary outcome space—in which probability is not an a priori measure but emerges from the no-arbitrage (or supermartingale) constraints imposed by the allowable set of gambles.

The core thesis is that many results in game-theoretic probability, especially those giving worst-case pathwise guarantees, can be seen as instances of a minimax theorem for zero-sum games between a "Gambler" and "World," with the classical measure-theoretic results arising as limiting special cases in which World must commit to a probabilistic strategy first.

Formalization: Gamble Spaces and Duality

The article rigorously defines gamble spaces $(\Omega, \mathcal{Z})$ as the mathematical substrate of game-theoretic probability, where $\Omega$ is an abstract outcome space and $\mathcal{Z}$ the set of available gambles $Z:\Omega \to \overline{\mathbb{R}}$. The upper game-theoretic expectation of a variable $X:\Omega\to \overline{\mathbb{R}}$ is formalized as

$$[X] = \inf_{Z\in \mathcal{Z}} \sup_{\omega\in \Omega} [X(\omega) - Z(\omega)],$$

expressing the minimal starting capital required to super-replicate $X$ via available gambles.

The paper distinguishes between three "prices" or upper expectations for $X$, corresponding to different strengths of adversarial actions by World:

• (i) Consistent probabilistic World: World must commit to a probability measure $P$ such that no gamble has positive expected value;
• (ii) Arbitrary probabilistic World: World picks $P$ but cannot condition on the Gambler's action;
• (iii) Fully adversarial World: World waits to see the Gambler's choice and responds with an outcome.

The main technical results are chains of inequalities between these prices, with equality corresponding to minimax duality.

Main Results

Minimax Theorem for Game-Theoretic Probability

The central technical contribution is a new minimax theorem applicable to finite-horizon games with possibly infinite outcome and action spaces, provided the per-round gamble spaces are finitely generated and satisfy a boundedness condition. Specifically, the author proves:

• Minimax Duality (Finite-Horizon Case): For the class of sequential gamble spaces where in each round the gambles are finitely generated, bounded below, Borel measurable, and have full support, the game-theoretic and measure-theoretic expectations agree on all bounded upper semi-analytic variables: $[X] = [X]_\text{MT}$ for all such $X$.

This is established via a nontrivial induction argument using a tower property for conditional upper expectations, extending backward-induction logic familiar from online learning and Markov decision processes, but here applied in the space of universally measurable functions.

Consequences and Recoveries

• The minimax theorem recovers and generalizes several existing results, including composite versions of Ville's Theorem and pathwise martingale inequalities (LLN, Azuma-Hoeffding, matrix concentration, etc.).
• The framework unifies classical proofs (e.g., using measure-theoretic martingale differences) and game-theoretic versions (constructing explicit nonnegative capital processes / betting strategies) under a single duality principle.
• The author constructs precise conditions under which the distinction between game-theoretic and measure-theoretic results collapses, and when a proper gap arises (e.g., infinite-horizon or discontinuous variable cases).

Impossibility and Counterexamples

The paper also examines the boundaries of minimax duality:

• It provides counterexamples (on, e.g., $[0,1]$ with linear outcome gambles) where the tower property and hence minimax duality fails for discontinuous or unbounded variables, connecting these failures to results about the limits of countable additivity and the necessity (but insufficiency) of continuity axioms.
• The connection to finitely additive probability and coherent risk measures is formalized: E1-E4 (subadditivity, positive homogeneity, monotonicity, normalization) are satisfied by both game- and measure-theoretic upper expectation functionals, but countable additivity (E5) is crucial for full minimax duality.

Technical Strengths and Claims

• General Minimax Theorem: The primary advance is the extension of minimax duality to infinite outcome spaces and nontrivial gamble spaces for finite time—a significant generalization over prior results, which were restricted to either finite outcome alphabets or single (noncomposite) probability scenarios.
• Composite Ville's Theorem: The result is interpreted as a composite version of Ville’s theorem—there exists a single (possibly nonconstructive) gambling strategy that, for all sequentially consistent probability measures, becomes arbitrarily rich whenever the event (e.g., SLLN-failure) occurs.
• Sharp Chain of Inequalities: The author establishes a sharp chain of inequalities among several natural "prices," pinning down when game-theoretic probability is strictly more robust than its measure-theoretic counterpart.

Implications and Outlook

Theoretical Implications

The work brings clarity and precision to the relationship between measure-theoretic and game-theoretic probability. When minimax duality holds:

• Every measure-theoretic result can be "lifted" to a pathwise, adversarial guarantee.
• All probabilistic assertions (laws of large numbers, concentration inequalities, central limit theorems) can be seen as statements about the impossibility of constructing profitable replicating strategies against a World obeying specific bet-consistency constraints.
• The minimax theorem demonstrates that for a broad class of games, the game-theoretic notion is, in a formal sense, not "stronger," but constructively refines the measure-theoretic one (the betting strategy’s explicit form depending on the event in question).

Practical Implications

For robust statistics, risk-averse finance, and online machine learning, these results:

• Justify the use of game-theoretic frameworks (e.g., e-processes, safe testing) as not being ad hoc, but as exact duals of classic probabilistic statements under martingale-type constraints.
• Provide a formal guarantee that replication-based “worst-case” pricing coincides with supremum expected value when the underlying gamble spaces meet the required regularity/closure properties.
• Suggest that pathwise, adversarial learning-theoretic arguments can be transferred into the classical stochastic regime, and vice versa, via the minimax correspondence.

Directions for Future Research

The author highlights key challenges for further extension:

• Infinite-Horizon Duality: Extending minimax duality to the countably infinite setting remains open, especially for events/variables involving tail $\sigma$-algebras or discontinuous payoff functions $X$.
• Game-Theoretic e-Processes: The structure and conversion between measure-theoretic and genuinely game-theoretic e-process supermartingales (across a composite family of measures) is not yet fully characterized.
• Risk and Statistical Robustness: The interface between game-theoretic probability, coherent risk measurement, and robust statistics remains mathematically compelling, particularly in relating various notions of consistency and additivity.

Conclusion

This paper provides a systematic and fully formal framework for understanding the alignment and distinction between game-theoretic and measure-theoretic probability via minimax duality. By establishing general minimax theorems for finite-horizon gamble spaces, the author demonstrates that virtually all classical measure-theoretic results of probability and statistics (within the specified regularity conditions) admit an exactly equivalent, constructive game-theoretic reformulation. This duality is not only mathematically elegant but also practically important for robust methods in inference and learning.

The nonconstructive nature of the minimax result means that, while existence of replication/betting strategies is guaranteed, explicit construction often remains challenging—and remains a fertile ground for further research, especially in the quest for general infinite-horizon theorems and the operational synthesis of adversarial and stochastic paradigms.