Game-Theoretic Probability

• Game-theoretic probability is a framework that defines probability through sequential, adversarial games where players use capital processes and hedging to force convergence.
• It unifies classical limit laws with robust control and minimax duality, enabling risk-neutral valuation and dynamic trading strategies without relying on traditional probabilistic axioms.
• The paradigm supports testing by betting and sequential laws like the strong law of large numbers, providing practical methods for calibration and option pricing in finance.

Game-theoretic probability is a non-measure-theoretic formalism in which probability arises from the analysis of perfect-information sequential games. Rather than postulating background probability laws or σ-additive measures, game-theoretic probability derives probabilistic statements as phenomena, forced or complied with, through prescribed capital processes, hedges, and adversarial protocols among players such as Forecaster, Skeptic, Reality, and Nature. This framework unifies the pathwise emergence of classical probabilistic laws with robust control, minimax duality, and supermartingale arguments within a flexible algebra of gambles.

1. Core Protocols and Players

Game-theoretic probability protocols are generally formulated as perfect-information games played over countable rounds. Typical participants include:

• Forecaster: Announces forecasts, such as means ($m_n$), variances ($v_n$), or probability distributions, round by round.
• Skeptic: Constructs capital processes by choosing linear or nonlinear stakes (including hedges) and tests the soundness of Forecaster’s assertions.
• Reality/Nature: Selects outcomes or adversarial sequences, constrained only by protocol rules or boundedness.
• Random Number Generator (sometimes): When randomized moves are permitted.

Central to the paradigm is the capital process, updated recursively by Skeptic’s selected bets and Reality’s revealed outcomes, often via formulas such as $$K_n = K_{n-1} + M_n(x_n - m_n) + V_n\bigl((x_n - m_n)^2 - v_n\bigr)$$ in quadratic-hedge games (Vovk, 2013, Miyabe et al., 2014, Miyabe et al., 2011).

Collateral duties enforce nonnegativity of capital and restrict unbounded gains, ensuring the soundness of game-theoretic assertions. Game-theoretic probability fundamentally reinterprets probabilistic convergence, laws of large numbers, and calibration as statements about the attainability or impossibility of certain capital processes under all adversarial moves or sequences.

2. Forcing, Compliance, and Game-theoretic Probability

A key mechanism is forcing: Skeptic’s strategy forces an event $E$ if his capital diverges whenever $E$ fails, independently of Forecaster or Reality’s actions (Miyabe et al., 2014, Vovk, 2013). Conversely, compliance means Reality adopts a strategy ensuring $E$ always holds while Skeptic’s capital remains bounded.

Game-theoretic probability for an event $E$ is defined via Skeptic’s capital requirements:

• The upper game-theoretic probability $\overline{P}(E)$ is the infimum of initial capital needed for a supermartingale strategy ensuring $\sup_n K_n < 1$ on all paths in $E$.
• The lower game-theoretic probability $\underline{P}(E)$ is $1-\overline{P}(E^c)$.

In many situations, Skeptic can construct explicit supermartingales to force strong laws, convergence of series, or pathwise calibration phenomena (Miyabe et al., 2011, Miyabe et al., 2012, Sato et al., 2016). Game-theoretic probabilities characterize the attainability of events in the absence of underlying measure-theoretic structures.

3. Minimax Duality and Emergence of Risk-Neutral Measures

Recent work recasts game-theoretic probability as a robust minimax optimization over gamble spaces. The game between a Gambler and World (possibly “Nature” or “Reality”) is framed by:

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

where $\mathcal{Z}$ is the set of admissible gambles and $X$ is the target payoff (Frongillo, 24 Dec 2025). Minimax duality results show that game-theoretic probability can be understood as an inf–sup over acceptable gambles coinciding with sup–inf over probability measures under suitable regularity, generalizing Sion’s theorem and classical superhedging duality (Vovk, 2016).

In financial contexts, such as Kolokoltsov’s zero-sum investor–Nature games, risk-neutral probability measures and martingale constraints emerge automatically as solutions to robust optimization, without any probabilistic axioms (Kolokoltsov, 2011). The game-theoretic Bellman operator collapses to a finite maximization over extreme risk-neutral distributions, and option prices are computed by backward induction along these measures. This framework extends to incomplete markets, path-dependent payoffs, American options, and real options, with dynamic-programming recursions yielding nonlinear or fractional Black-Scholes PDEs in the continuous-time limit.

4. Sequential Laws: SLLN, LIL, and Convergence Theorems

Game-theoretic protocols recover classical strong laws via forcing and compliance. For example, the analogue of Kolmogorov’s SLLN is:

$$\sum_{n=1}^\infty \frac{v_n}{n^2} < \infty \implies \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n (x_i - m_i) = 0$$

forced by Skeptic and matched by Reality’s explicit strategy on the “failure” side (Vovk, 2013, Miyabe et al., 2014). Generalizations with hedges (quadratic, stronger, or weaker) underpin Marcinkiewicz–Zygmund laws, self-normalized SLLNs, and rate-of-convergence theorems, with asymptotic rates precisely controlled by the “price” sequence of hedges (Miyabe et al., 2011, Miyabe et al., 2012, Sato et al., 2016).

Continuous Bayesian mixtures of fixed-proportion betting strategies yield a one-to-one correspondence between prior density concentration rates and the rate of approach to classical limits such as the law of the iterated logarithm (LIL) or Erdős–Feller–Kolmogorov–Petrowsky variants (Sato et al., 2016). Explicit formulas are derived for upper-class functions governing eventual bounds on normalized sums, extending Cover’s universal portfolios to game-theoretic self-normalization schemes.

5. Testing by Betting, Calibration, and Descriptive Analysis

The central principle for statistical testing in game-theoretic probability is testing by betting: If Skeptic’s capital $K_n$ grows large, Forecaster’s predictions are discredited to the extent that Skeptic’s gain exceeds what would be expected under valid models (Shafer, 2023). Unlike Cournot’s principle, game-theoretic methods authorize optional continuation—Skeptic may “wing it,” adapt tests, or stop at arbitrary times, with Ville’s inequality guaranteeing validity of capital thresholds.

The Kelly criterion, and its competitive generalization, allows for purely descriptive analyses: competing probability distributions bet against each other, with confidence sets replaced by sets of distributions doing well in relative capital competitions (often Fisher’s likelihood intervals) (Shafer, 2023). In these analyses, description intervals do not depend on frequentist or Bayesian coverage, but directly reflect forecasting performance.

Game-theoretic probability also clarifies and calibrates running-evidence statistics, such as insuring against loss of evidence by replacing the running maximum of capital $K^*_n$ with admissible calibrator functions $F(K^*_n)$, characterized by precise integral inequalities (Dawid et al., 2010).

6. Derandomization, Compliance, and Randomness

The paradigm encompasses derandomization techniques: Any randomized Reality strategy (e.g., coin-tossing) that achieves an “almost sure” event can be converted into a deterministic strategy that strongly complies with the event (Miyabe et al., 2014). This construction is explicit, using forcing strategies of Skeptic and a “meta-game,” and generalizes algorithmic randomness constructions. Similarly, perfect-information protocols admit Reality’s matching strategies, guaranteeing divergence and non-convergence when certain tail-sums (e.g., variances) diverge (Vovk, 2013, Miyabe et al., 2011).

Randomized forecasting is universally powerful against countable statistical tests only when infinite precision is allowed; discretization introduces calibration errors and exposes vulnerabilities, generalizing Oakes’ example in the game-theoretic benchmarking of forecasters (0808.3746).

7. Duality, Extensions, and Connections to Measure-Theoretic Probability

Game-theoretic probability admits a rigorous duality with classical measure-theoretic probability. For broad classes of positive lower semicontinuous functionals on path spaces (e.g., price processes), game-theoretic and measure-theoretic expectations coincide (Vovk, 2016). Superhedging duality and minimax theorems extend to wider classes of paths and protocols, accommodating infinite-horizon sequential games and composite versions of Ville’s theorem (Frongillo, 24 Dec 2025). Connections to martingale inequalities, online learning, regret minimization, and robust optimization are systematically elucidated via the structure of gamble spaces and minimax duality.

Open directions include the extension of minimax arguments to infinite horizons, constructive conversions of measure-theoretic supermartingales to game-theoretic analogues, and the development of robust practical algorithms for nonparametric, high-dimensional, or learning-based forecasting settings (Frongillo, 24 Dec 2025, Shafer, 2023).

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Selected Reference Table: Core Papers and Their Contributions

arXiv id	Main Contribution	Key Concepts/Results
(Kolokoltsov, 2011)	Option pricing via zero-sum game; risk-neutral measures emergent	Bellman minimax, robust control, nonlinear/fractional Black-Scholes PDEs
(Miyabe et al., 2014)	Constructive derandomization, conversion of randomized to deterministic Reality strategies	Forcing, compliance, three-step method
(Vovk, 2013)	Reality’s matching strategy for SLLN	Explicit compliance, capital bounds
(Sato et al., 2016)	Rates of SLLN via Bayesian mixtures, EFKP-LIL	Prior concentration, upper-class functions
(Miyabe et al., 2012)	Validity/sharpness of LIL with hedges	Quadratic/stronger hedges, pathwise bounds
(0808.3746)	Limits of randomized forecasting, game-theoretic Oakes’ example	Calibration error, forecast discretization
(Frongillo, 24 Dec 2025)	General minimax duality for gamble spaces	Inf–sup vs sup–inf, composite Ville, price chain
(Shafer, 2023)	Testing by betting, descriptive analysis	Kelly competition, confidence sets, optional continuation
(Dawid et al., 2010)	Insurance against loss of evidence	Admissible capital calibrators, integral characterization
(Vovk, 2016)	Duality between game-theoretic and measure-theoretic expectation	Superhedging, liminf closures, continuous path spaces
(Miyabe et al., 2011)	Series convergence, rate SLLN, compliance	Quadratic/weaker hedges, deterministic Reality strategies

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Game-theoretic probability provides a rigorous and flexible foundation for probabilistic reasoning, data analysis, and robust financial evaluation, supporting both classical limit laws and new forms of statistical inference by reducing random variability to structured adversarial pathways and capital processes.