Win-Martingales: Models and Fluctuation Analysis

• Win-Martingales are martingale processes that represent evolving win probabilities converging to 0 or 1, capturing the information flow in binary outcomes.
• They satisfy universal first-moment identities like E[N_b]=1/b and E[D_{a,b}]=(1-b)/(b-a), while their higher-order behaviors depend on specific model constructions.
• Win-Martingales underpin diverse frameworks—from Wright–Fisher diffusions to prediction-market models—and are pivotal in variational entropy and divergence minimization problems.

Searching arXiv for relevant papers on win-martingales and closely related work. Win-martingales are continuous or discrete-time martingale models for evolving win probabilities that start from an initial value in $(0,1)$ and terminate in the degenerate outcome set $\{0,1\}$. In the canonical two-outcome formulation, the process represents the conditional probability of eventual victory given the currently available information, so the martingale property is a direct consequence of conditional expectation. The notion appears both in contest models with many competitors, where one tracks a family of conditional winning-probability martingales, and in continuous-path optimal-transport and information-theoretic problems, where one studies the class of laws on $C([0,1];\mathbb R)$ satisfying $X_0=x_0\in(0,1)$, $X_1\in\{0,1\}$, and martingale dynamics with absolutely continuous quadratic variation [(Aldous et al., 2012); (Backhoff-Veraguas et al., 2023); (Backhoff et al., 16 Feb 2026)]. The topic connects fluctuation theory, prediction-market modeling, Wright–Fisher diffusions, martingale optimal transport, and entropy- or divergence-based variational principles.

1. Formal definition and probabilistic interpretation

In the binary setting, a win-martingale is a process of the form

$M_t=P(Y=1\mid\mathcal F_t),$

where $Y\in\{0,1\}$ is the terminal win indicator and $(\mathcal F_t)$ is the information filtration. By the tower property, $(M_t)$ is a martingale; one has $M_0=P(Y=1)\in(0,1)$ and $\{0,1\}$0 (Pipping-Gamón et al., 26 Jan 2026). In continuous-time formulations used in the recent optimal-transport and entropy literature, the process is idealized to have continuous sample paths on $\{0,1\}$1, with prescribed start $\{0,1\}$2 and terminal Bernoulli law (Backhoff-Veraguas et al., 2023, Backhoff-Veraguas et al., 2024).

A more general many-contestant version considers a collection $\{0,1\}$3 adapted to a common filtration, with $\{0,1\}$4 interpreted as the conditional probability that contestant $\{0,1\}$5 ultimately wins. In the formulation of a $\{0,1\}$6-feasible process, these coordinates satisfy $\{0,1\}$7, $\{0,1\}$8, $\{0,1\}$9, each $C([0,1];\mathbb R)$0 is a continuous-path martingale, and there is an almost surely finite stopping time $C([0,1];\mathbb R)$1 and a random index $C([0,1];\mathbb R)$2 such that $C([0,1];\mathbb R)$3 while all other coordinates equal $C([0,1];\mathbb R)$4 (Aldous et al., 2012). This realizes the same principle in a simplex-valued rather than scalar setting.

The common interpretation is that win-martingales encode information flow rather than physical scores. In sports analytics, prediction markets, or contest models, the process is the endogenous evolution of conditional winning probabilities under a filtration. In population-genetic models, the same structure appears through allele frequencies under pure drift, and in particular through Wright–Fisher diffusions, whose coordinates are martingales summing to one and eventually collapse to fixation (Aldous et al., 2012).

2. Canonical fluctuation quantities and distribution-free identities

For a family of contestant win-probabilities $C([0,1];\mathbb R)$5, two natural fluctuation counts are emphasized in the contest-model literature. For thresholds $C([0,1];\mathbb R)$6,

$C([0,1];\mathbb R)$7

counts how many contestants ever reach level $C([0,1];\mathbb R)$8, and

$C([0,1];\mathbb R)$9

counts total downcrossings of $X_0=x_0\in(0,1)$0 by all coordinate martingales (Aldous et al., 2012). These quantities are model-dependent in distribution but admit model-independent first moments.

Specifically, in any $X_0=x_0\in(0,1)$1-feasible process with $X_0=x_0\in(0,1)$2, the exact identities

$X_0=x_0\in(0,1)$3

hold by optional-sampling and classical downcrossing arguments (Aldous et al., 2012). The derivation reduces first to a single absorbed continuous martingale $X_0=x_0\in(0,1)$4 with initial value $X_0=x_0\in(0,1)$5, for which

$X_0=x_0\in(0,1)$6

and then sums over contestants (Aldous et al., 2012). These formulas are notable because they depend only on the martingale structure and boundary conditions, not on detailed path dynamics.

In the binary one-dimensional setting, the running maximum

$X_0=x_0\in(0,1)$7

is the central fluctuation observable. For discrete-time binary Doob martingales, optional stopping at the first-passage time $X_0=x_0\in(0,1)$8 yields

$X_0=x_0\in(0,1)$9

equivalently

$X_1\in\{0,1\}$0

with sharpness governed by the absence of overshoot and of terminal-time first-hit effects (Pipping-Gamón et al., 26 Jan 2026). Conditioning on eventual loss gives the bound

$X_1\in\{0,1\}$1

for $X_1\in\{0,1\}$2 (Pipping-Gamón et al., 26 Jan 2026).

Under continuous-path regularity, the discrete correction terms disappear, and exact identities replace inequalities. If $X_1\in\{0,1\}$3 has continuous sample paths on $X_1\in\{0,1\}$4, then for $X_1\in\{0,1\}$5,

$X_1\in\{0,1\}$6

while

$X_1\in\{0,1\}$7

In particular, $X_1\in\{0,1\}$8 (Pipping-Gamón et al., 26 Jan 2026). These formulas provide benchmark laws for maxima of correctly specified binary forecast martingales.

3. Extremal constructions and variance phenomena

Although the first moments of $X_1\in\{0,1\}$9 and $M_t=P(Y=1\mid\mathcal F_t),$0 are universal in the many-contestant setting, their distributions can vary sharply across feasible models. Two extremal constructions organize this variability (Aldous et al., 2012).

The first is the progressive-elimination or “Survivor” model. One starts with all active contestants, evolves a suitable Wright–Fisher diffusion on active fractions, and whenever some $M_t=P(Y=1\mid\mathcal F_t),$1 hits $M_t=P(Y=1\mid\mathcal F_t),$2, that coordinate is frozen at $M_t=P(Y=1\mid\mathcal F_t),$3 while the remainder continue evolving. Iteration yields a model in which $M_t=P(Y=1\mid\mathcal F_t),$4 takes either $M_t=P(Y=1\mid\mathcal F_t),$5 or $M_t=P(Y=1\mid\mathcal F_t),$6, so $M_t=P(Y=1\mid\mathcal F_t),$7 is as concentrated as possible around its mean $M_t=P(Y=1\mid\mathcal F_t),$8 (Aldous et al., 2012).

The second is the sequential-examination or “Millionaire” model. Contestants are ordered, and one reveals $M_t=P(Y=1\mid\mathcal F_t),$9 until it either hits $Y\in\{0,1\}$0 with chance $Y\in\{0,1\}$1 or $Y\in\{0,1\}$2 with chance $Y\in\{0,1\}$3; if it loses, one examines $Y\in\{0,1\}$4, and so forth. In the limit $Y\in\{0,1\}$5, this produces

$Y\in\{0,1\}$6

with variance $Y\in\{0,1\}$7, which is the largest possible among feasible processes (Aldous et al., 2012). Correspondingly, in any $Y\in\{0,1\}$8- or $Y\in\{0,1\}$9-feasible process,

$(\mathcal F_t)$0

and equality is attained by the Geometric$(\mathcal F_t)$1 law in the maximal-spread construction (Aldous et al., 2012).

A related maximal-spread construction exists for downcrossings. In the sequential-examination limit,

$(\mathcal F_t)$2

and this yields the largest possible asymptotic order $(\mathcal F_t)$3 for the variance, at fixed $(\mathcal F_t)$4 (Aldous et al., 2012). At the same time, a reflection-coupling construction shows that there are $(\mathcal F_t)$5-feasible processes where $(\mathcal F_t)$6 remains $(\mathcal F_t)$7 as $(\mathcal F_t)$8 with $(\mathcal F_t)$9 bounded away from $(M_t)$0 (Aldous et al., 2012). This makes clear that universal fluctuation identities for win-martingales are primarily first-moment statements; higher-order behavior depends strongly on mechanism.

4. Wright–Fisher structures and the infinite-contestant limit

The Wright–Fisher diffusion plays a canonical role in the theory of win-martingales. In the contest setting, the infinitely-many-alleles Wright–Fisher diffusion with no mutation and no selection yields a natural $(M_t)$1-feasible process after a labeling-and-consistency construction (Aldous et al., 2012). In finite dimensions, the $(M_t)$2-allele Wright–Fisher diffusion on the simplex has generator

$(M_t)$3

and each coordinate is a martingale with variance rate $(M_t)$4 (Aldous et al., 2012). In the limit $(M_t)$5, one obtains a diffusion on the ranked infinite simplex that can be lifted to a $(M_t)$6-feasible win-martingale system (Aldous et al., 2012).

This canonical process preserves the universal identities

$(M_t)$7

but the exact laws of $(M_t)$8 and $(M_t)$9 remain open (Aldous et al., 2012). The associated open problem is equivalent to determining joint threshold-hitting probabilities such as

$M_0=P(Y=1)\in(0,1)$0

which solves an elliptic PDE on $M_0=P(Y=1)\in(0,1)$1 but has no known closed-form solution (Aldous et al., 2012).

The Wright–Fisher diffusion also reappears as an optimizer in more recent variational formulations. In the setting of reciprocal specific relative entropy between continuous martingales, the optimization is carried out over the class $M_0=P(Y=1)\in(0,1)$2 of continuous martingale laws $M_0=P(Y=1)\in(0,1)$3 such that $M_0=P(Y=1)\in(0,1)$4, $M_0=P(Y=1)\in(0,1)$5 almost surely, and quadratic variation is absolutely continuous with density $M_0=P(Y=1)\in(0,1)$6 (Backhoff et al., 16 Feb 2026). The unique minimizer is a time-changed neutral Wright–Fisher diffusion with volatility density

$M_0=P(Y=1)\in(0,1)$7

that is,

$M_0=P(Y=1)\in(0,1)$8

(Backhoff et al., 16 Feb 2026). This establishes the neutral Wright–Fisher process as a canonical win-martingale in an information-theoretic sense.

5. Entropy, divergence minimization, and extremal win-martingales

A major recent direction treats win-martingales as admissible laws in variational problems relative to Brownian motion. In the maximal-entropy formulation, one considers the class $M_0=P(Y=1)\in(0,1)$9 of continuous-path martingale laws with $\{0,1\}$00 and $\{0,1\}$01, and minimizes the specific relative entropy with respect to Wiener measure (Backhoff-Veraguas et al., 2023). If under a law $\{0,1\}$02 the canonical process has absolutely continuous quadratic variation with density $\{0,1\}$03, then

$\{0,1\}$04

(Backhoff-Veraguas et al., 2023). The unique minimizer is therefore called the max-entropy win-martingale (Backhoff-Veraguas et al., 2023).

The optimizer is characterized by the stochastic differential equation

$\{0,1\}$05

with $\{0,1\}$06 and $\{0,1\}$07 almost surely (Backhoff-Veraguas et al., 2023). The derivation uses a martingale-transport first-order condition and a scaling argument, leading to the bounded solution

$\{0,1\}$08

of the ODE

$\{0,1\}$09

on $\{0,1\}$10 (Backhoff-Veraguas et al., 2023). The minimal specific entropy has the closed form

$\{0,1\}$11

(Backhoff-Veraguas et al., 2023).

A related but distinct divergence is the reciprocal specific relative entropy of a continuous martingale law $\{0,1\}$12 relative to Wiener measure: $\{0,1\}$13 This functional penalizes deviations of the instantaneous variance $\{0,1\}$14 from $\{0,1\}$15, and over the class of win-martingales it is uniquely minimized by the time-changed neutral Wright–Fisher diffusion described above (Backhoff et al., 16 Feb 2026). The corresponding HJB equation has optimizer

$\{0,1\}$16

and the associated value function admits an explicit representation via a separation-of-variables ansatz (Backhoff et al., 16 Feb 2026).

These two variational problems do not select the same optimizer. One selects the sine-volatility martingale of Backhoff-Veraguas and Beiglböck (Backhoff-Veraguas et al., 2023); the other selects the time-changed neutral Wright–Fisher diffusion (Backhoff et al., 16 Feb 2026). This suggests that “canonical” win-martingales depend sensitively on the chosen divergence.

6. Specific Wasserstein divergence, special exponents, and calibration laws

The divergence framework has been generalized by replacing specific relative entropy with specific $\{0,1\}$17-Wasserstein divergence. For continuous-martingale laws $\{0,1\}$18 with volatility densities $\{0,1\}$19, the specific $\{0,1\}$20-Wasserstein divergence satisfies

$\{0,1\}$21

(Backhoff-Veraguas et al., 2024). When the reference law is the constant martingale $\{0,1\}$22, optimization over the class of win-martingales becomes equivalent to minimizing

$\{0,1\}$23

subject to the terminal win constraint (Backhoff-Veraguas et al., 2024).

For $\{0,1\}$24, the unique extremal win-martingale is obtained from a separation-of-variables ansatz

$\{0,1\}$25

where $\{0,1\}$26 solves a boundary-value problem derived from a first-order martingale-optimal-transport condition (Backhoff-Veraguas et al., 2024). The case $\{0,1\}$27 is especially explicit: $\{0,1\}$28 so the optimal win-martingale solves

$\{0,1\}$29

(Backhoff-Veraguas et al., 2024). This process remains in $\{0,1\}$30 on $\{0,1\}$31 and hits $\{0,1\}$32 only at time $\{0,1\}$33 (Backhoff-Veraguas et al., 2024). Under the reparameterization $\{0,1\}$34, one has

$\{0,1\}$35

and for the log-odds $\{0,1\}$36,

$\{0,1\}$37

(Backhoff-Veraguas et al., 2024). The paper further identifies this law with a Schrödinger-problem/entropic-bridge limit (Backhoff-Veraguas et al., 2024).

Win-martingales also furnish model-agnostic calibration diagnostics for sequential probability forecasts. For binary Doob martingales with continuous paths, the exact law of the path maximum gives the benchmark

$\{0,1\}$38

for the peak win probability attained on trajectories that eventually lose (Pipping-Gamón et al., 26 Jan 2026). This supports expectation-calibration checks, empirical CDF comparisons, and threshold-based exceedance tests. For example, under correct specification,

$\{0,1\}$39

(Pipping-Gamón et al., 26 Jan 2026). The practical significance is that large apparent “collapses” can be evaluated against a martingale reference law rather than anecdotal intuition.

7. Relations to randomness, misconceptions, and open problems

The term “win-martingale” occurs in distinct but related literatures. In prediction and contest models, it denotes conditional winning-probability martingales ending at $\{0,1\}$40 or $\{0,1\}$41 [(Aldous et al., 2012); (Backhoff-Veraguas et al., 2023); (Pipping-Gamón et al., 26 Jan 2026)]. In algorithmic randomness, by contrast, martingales are betting strategies on bit sequences, and integer-valued martingales are used to define weakened randomness notions (Bienvenu et al., 2010). That literature studies when a martingale “wins” by making unbounded capital, rather than a process representing a win probability. The overlap is conceptual rather than terminological: both settings exploit martingale fairness, stopping-time arguments, and threshold-hitting structure, but the objects and objectives differ (Bienvenu et al., 2010).

A common misconception is that the law of a win-martingale is essentially determined by its start and endpoint. The results above show otherwise. First moments such as $\{0,1\}$42 or exact maximum identities such as $\{0,1\}$43 under continuity are universal [(Aldous et al., 2012); (Pipping-Gamón et al., 26 Jan 2026)], but higher-order fluctuation laws can vary dramatically across models (Aldous et al., 2012). Likewise, different divergence principles select different “canonical” win-martingales: the maximal-entropy criterion yields the sine-volatility SDE (Backhoff-Veraguas et al., 2023), whereas reciprocal specific relative entropy selects the time-changed neutral Wright–Fisher diffusion (Backhoff et al., 16 Feb 2026), and specific $\{0,1\}$44-Wasserstein optimization yields a family depending on $\{0,1\}$45, with an explicit form at $\{0,1\}$46 (Backhoff-Veraguas et al., 2024).

Several open problems remain explicit in the literature. In the $\{0,1\}$47-Wright–Fisher contest model, the laws of $\{0,1\}$48 and $\{0,1\}$49 are not known (Aldous et al., 2012). In algorithmic randomness, open questions include whether allowing all integer multiples of $\{0,1\}$50 changes integer-valued randomness and whether there is a pure Kolmogorov-complexity or effective-test characterization of integer-valued or finitely-valued randomness (Bienvenu et al., 2010). A plausible implication is that the win-martingale framework remains technically fertile precisely because universal martingale constraints coexist with strong model dependence at the level of pathwise distribution, divergence geometry, and extremal structure.