Minimax Duality in Game-Theoretic Probability

TL;DR

This work develops a general finite-time minimax framework for game-theoretic probability based on gamble spaces, enabling a broad class of results to be viewed as zero-sum minimax theorems between Gambler and World. By formalizing upper and lower game-theoretic expectations and consistency notions, the authors derive a chain of price inequalities, establish a finite-time sequential minimax theorem, and connect game-theoretic results with measure-theoretic probability via minimax duality. They also illuminate composite Ville-type statements, extend minimax duality to sequential and finite horizons, and discuss connections to online learning, finance, and statistics, including finitely additive measures and e-variables. The framework provides a unifying lens to lift measure-theoretic results to worst-case, pathwise game-theoretic analogues, while clarifying when and why such dualities fail and what additional conditions are required. Overall, the paper advances a robust, extensible bridge between traditional probability theory and game-theoretic, pathwise perspectives with broad implications for theory and applications.

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.

Figures (6)

• Figure 1: Example derivations from the famous conversation between Pascal and Fermat. Depicted is a best-of-3 contest between Alexa and Calen, with a prize of $100, where Alexa already lost the first match. Thus the only way for Alexa to win the prize is to win both remaining games ($WW$). Pascal reasoned that were Alexa to win the next game, she would now be on even footing with Calen, and it would be natural for them to earn an even split of the prize. Thus the amount owed to Alexa in state $W$ should be $50. As Alexa cannot win the prize after losing the next match (and is thus owed $0 in state $L$), similar reasoning suggests that the amount owed Alexa now is $25. Fermat instead proposed dividing the money according to the possible ways for Alexa to win the prize, in this case 1 out of 4, also concluding that Alexa is owed $25 now. These two derivations can be seen as the two optimal strategies in a zero-sum game---not the one between Alexa and Calen, but between two external players, Gambler and World, which respectively bet on and choose the match outcomes. Gambler chooses a gambling strategy $\psi$, which places a bet (under even odds) on the next match based on the history so far. World chooses a probability measure $P$ on the sequence of outcomes $\{W,L\}^2$ from the initial situation. Let $X:\{W,L\}^2\to\mathbb{R}$ encode the value of the final outcome to Alexa, namely $X(WW) = 100$, $X(WL) = X(LW) = X(LL) = 0$. The payoff to World in the game, $u(P,\psi)$, is the expected difference between $X$ and the winnings of Gambler. Specifically, identifying $\{W,L\}$ with $\{-1,1\}$ for convenience, we may write $u(P,\psi) = \mathbb{E}_P[X(Y_1Y_2) - \psi({\boldsymbol{\varepsilon}})Y_1 - \psi(Y_1)Y_2]$, where $Y_1,Y_2 \in \{-1,1\}$ represent the outcomes. When Gambler must play first, the optimal strategy $\psi^*$, depicted in blue, bets $25 on $W$ in the initial situation, and either refrains from betting if Alexa loses or bets another $50 on $W$ if she wins. This strategy is exactly what Pascal derived by backward induction. (More precisely, it is the amount staked in each state, which in this case is equal to the value owed at that state.) On the other hand, when World must play first, the optimal strategy is the uniform distribution $P$ on $\{W,L\}^2$, which is exactly Fermat's approach. Both optimal strategies are unique, and as minimax duality holds in this game, both give the same payoff of $25 to World. The conclusion in both cases is again that Alexa's standing $X$ is "worth" $25. As a final point, imagine that the contest was best-of-4, and Alexa and Calen had each already won a game. If we split the prize upon a tie, we have $X = (\$0,\$50,\$50,\$100)$ top to bottom. Interestingly, Fermat's weights do not change: we still have $P = (1/4,1/4,1/4,1/4)$. But Pascal's derivation would change, starting with $50 but placing no bet, and regardless of the first match outcome, placing a $25 bet on $W$ in the next round. In this sense, the game-theoretic approach is a refinement of the measure-theoretic (§ \ref{['sec:intro-composite-ville']}).
• Figure 2: The replication cost to sell or buy $X$, respectively, relative to existing gambles $\mathcal{Z}$. This cost is exactly $\alpha$ if the depicted gamble $Z$ achieves the infimum in eq. \ref{['eq:prop-upper-ex-rep-cost']}.
• Figure 3: A visualization of Example \ref{['ex:single-interval']} showing a nontrivial price gap $\underline{\textsf{E}} X' = -\tfrac 1 4 < \tfrac 1 4 = \overline{\textsf{E}} X'$. We can conveniently plot both the upper and lower expectations by reflecting the latter; rather that plotting the smallest affine function dominating $-X'$, we plot the largest affine lower bound.
• Figure 4: The chain of prices of some variable $X$ implied by eq. \ref{['eq:intro-price-defs-chain']}. Observe that negations of these statements can be inferred too: if it is always profitable to buy $X$ at a price under the assumption listed, it is never profitable to sell at that price. Hence, these prices are from the perspective of one agent only; if both agents have the same assumption about World, no trade would occur unless both agents are indifferent. Taken in pairs, one can think of these quantities as the bid-ask spreads offered by market makers with assumptions (i-iii) about World.
• Figure 5: Visualizations of $\mathrm{dcl}(\mathcal{Z})$ in red and $-\mathrm{dcl}(-\mathcal{Z})$ in blue for the two gambles spaces in Examples \ref{['ex:egl-greater-egu']} and \ref{['ex:egl-leq-egu-but-no-consistent']}. Overlap between these regions gives a variable $X$ for which $\underline{\textsf{E}} X > \overline{\textsf{E}} X$. Consistent measures could be visualized as the normal cone to $\mathrm{dcl}(\mathcal{Z})$, which is empty in both examples.

Theorems & Definitions (176)

• Definition 2.1: Gamble space
• Definition 2.2: Game-theoretic upper expectation
• Proposition 2.3
• proof
• Remark 2.4
• Remark 2.5
• Example 2.6: Multiplicative LLN strategy
• Example 2.7: Fair coin
• Example 2.8: Outcome interval
• Example 2.9: Variance