Binary-Choice Kelly Criterion

• Binary-choice Kelly criterion is a log-optimal rule that defines the fraction of wealth to wager in repeated binary bets to maximize expected logarithmic growth.
• It features a closed-form solution (f* = (p(b+1)-1)/b) under no-shorting and no-leverage conditions, ensuring a unique, concave optimum for bet sizing.
• Practical insights include the use of fractional Kelly and variance-controls to mitigate estimation errors and asymptotic volatility, linking the theory to risk management in betting and finance.

The binary-choice Kelly criterion is the log-optimal staking rule for repeated binary gambles with multiplicative wealth dynamics. In its classical form, each trial has two outcomes—win with probability $p$ and loss with probability $q=1-p$—and a bettor stakes a fraction of current wealth so as to maximize expected logarithmic growth. For net odds $b>0$, staking a fraction $f$ multiplies wealth by $1+bf$ on a win and by $1-f$ on a loss, yielding the objective

$g(f)=p\ln(1+bf)+q\ln(1-f).$

Under the usual no-shorting/no-leverage restriction $0\le f\le 1$, this problem has a unique concave optimum, the Kelly fraction, which is the canonical solution for long-run geometric growth in binary betting and the discrete prototype for broader log-optimal portfolio theory (Hakobyan et al., 23 Mar 2025).

1. Formal model and closed-form solution

In the binary-choice setup, the one-period return $R$ takes values $b$ with probability $q=1-p$0 and $q=1-p$1 with probability $q=1-p$2. Wealth after $q=1-p$3 rounds is

$q=1-p$4

and the per-trial log-return is

$q=1-p$5

The expected log-growth is therefore

$q=1-p$6

The objective is strictly concave because

$q=1-p$7

The first-order condition

$q=1-p$8

has the unique interior solution

$q=1-p$9

Under no shorting and no leverage, the admissible Kelly stake is the truncation of this expression to $b>0$0: $b>0$1 The fair-game boundary is $b>0$2, for which $b>0$3; if $b>0$4, the optimal action under the no-shorting/no-leverage constraint is not to bet. At even net odds, $b>0$5, the formula reduces to the familiar

$b>0$6

(Hakobyan et al., 23 Mar 2025).

Several equivalent parameterizations recur in the literature. With decimal odds $b>0$7,

$b>0$8

In prediction-market notation, if the market-implied probability is $b>0$9, then

$f$0

These are algebraic restatements of the same binary Kelly rule (Uhrín et al., 2021).

The same closed form appears in even-money Bernoulli games written with outcome variable $f$1 and fixed stake fraction $f$2. There the per-trial log-utility

$f$3

is globally concave on $f$4, and its maximizer is

$f$5

(Miller, 24 Feb 2025).

2. Optimal growth, edge, and information-theoretic structure

Substituting the optimal fraction into the growth objective gives a closed-form maximum. Using

$f$6

one obtains

$f$7

This expression makes the edge threshold explicit: positive long-run growth requires favorable odds in the sense $f$8, equivalently $f$9 (Hakobyan et al., 23 Mar 2025).

In the even-money case, the optimal growth simplifies to an entropy identity. Evaluated at $1+bf$0,

$1+bf$1

where

$1+bf$2

is the binary Shannon entropy. Thus the fair case $1+bf$3 yields zero expected geometric growth, while a biased game with $1+bf$4 yields $1+bf$5 (Miller, 24 Feb 2025).

A related information-theoretic formulation appears in the horse-race framework. For the binary special case, the optimal expected doubling rate can be written as

$1+bf$6

and the use of side information raises the optimal doubling rate by the pragmatic information of the messages. In Weinberger’s formulation, the increase in doubling rate equals the mutual-information term associated with conditioning winning probabilities on messages, tying binary Kelly growth directly to information usage in sequential decision-making (0903.2243).

This information-theoretic viewpoint reappears in prediction markets. With Kelly traders on a binary event, the market-clearing price is a wealth-weighted average of beliefs, and wealth updates proportionally to likelihood ratios after each outcome. In that setting, the market’s cumulative log loss is within $1+bf$7 of the best participant’s, so the binary Kelly mechanism acts as both a staking rule and an aggregation rule for probabilistic beliefs (Beygelzimer et al., 2012).

3. Volatility, asymptotic variance, and fractional Kelly

The standard Kelly solution maximizes asymptotic log-growth, but the associated stake is often regarded as aggressive. A recent variance-based treatment introduces the asymptotic variance of per-trial log-growth as a unified risk descriptor in the binary setting: $1+bf$8 Because

$1+bf$9

$1-f$0 is strictly increasing on $1-f$1. Thus larger Kelly fractions increase asymptotic volatility monotonically (Hakobyan et al., 23 Mar 2025).

For i.i.d. trials, the law of large numbers and central limit theorem imply

$1-f$2

and

$1-f$3

Equivalently,

$1-f$4

This approximation quantifies fluctuations around mean log-growth and underlies asymptotic risk measures (Hakobyan et al., 23 Mar 2025).

Two such measures have been proposed. The asymptotic Sharpe ratio is

$1-f$5

which measures mean log-growth per unit asymptotic volatility of log-growth. The asymptotic ridge coefficient is

$1-f$6

which penalizes aggressive sizing via asymptotic variance. In the binary case, the corresponding first-order condition is

$1-f$7

or explicitly,

$1-f$8

Under no shorting and no leverage, this admits a unique solution in $1-f$9, producing a disciplined fractional Kelly fraction that decreases as $g(f)=p\ln(1+bf)+q\ln(1-f).$0 increases (Hakobyan et al., 23 Mar 2025).

The widely used ad hoc variant is fractional Kelly: $g(f)=p\ln(1+bf)+q\ln(1-f).$1 Then

$g(f)=p\ln(1+bf)+q\ln(1-f).$2

is strictly increasing and strictly concave in $g(f)=p\ln(1+bf)+q\ln(1-f).$3, with maximum at $g(f)=p\ln(1+bf)+q\ln(1-f).$4, while

$g(f)=p\ln(1+bf)+q\ln(1-f).$5

is strictly increasing in $g(f)=p\ln(1+bf)+q\ln(1-f).$6. The data further note that reducing $g(f)=p\ln(1+bf)+q\ln(1-f).$7 decreases growth linearly but reduces asymptotic volatility more than linearly. This motivates variance targeting by solving $g(f)=p\ln(1+bf)+q\ln(1-f).$8 or ridge sizing by solving the penalized first-order condition (Hakobyan et al., 23 Mar 2025).

A simpler quadratic approximation also appears in the control-theoretic literature. Expanding $g(f)=p\ln(1+bf)+q\ln(1-f).$9 yields the approximate optimizer

$0\le f\le 1$0

At even odds, $0\le f\le 1$1, this approximation coincides exactly with the true Kelly fraction; for general $0\le f\le 1$2 it differs, often shrinking the stake in favorable cases with $0\le f\le 1$3 (Hsieh, 2020).

4. Finite-horizon moments, martingale regimes, and drawdown approximations

The binary-choice Kelly criterion is asymptotic in its primary objective, but several finite-horizon quantities are available in closed form. In the even-money Bernoulli model with fixed fraction $0\le f\le 1$4,

$0\le f\le 1$5

and, equivalently, if $0\le f\le 1$6 wins and $0\le f\le 1$7 losses occur with $0\le f\le 1$8,

$0\le f\le 1$9

The expected wealth factor is

$R$0

so

$R$1

At $R$2,

$R$3

which grows exponentially for $R$4 (Miller, 24 Feb 2025).

The second moment is

$R$5

and the variance is

$R$6

For small $R$7 and large $R$8,

$R$9

This provides a finite-horizon volatility proxy distinct from asymptotic log-growth variance (Miller, 24 Feb 2025).

The same paper partitions the stake domain using the sign of the per-trial log-utility. Define $b$0 by

$b$1

Then, for $b$2, the wealth process is a submartingale for $b$3, a martingale at $b$4, and a supermartingale for $b$5. For small edges, the threshold obeys the approximation

$b$6

This regime split clarifies that positive expected log-growth and positive expected wealth drift are not identical notions, but here the sign of $b$7 determines the martingale classification stated in the paper (Miller, 24 Feb 2025).

At a fixed finite horizon $b$8, the central-limit approximation for log-wealth yields an approximate shortfall probability. For a log drawdown threshold $b$9 relative to mean,

$q=1-p$00

The same source explicitly notes the limitations: this is asymptotic, ignores path dependence and absorbing constraints, and exact finite-$q=1-p$01 behavior is a binomial mixture over two log-values rather than Gaussian (Hakobyan et al., 23 Mar 2025).

5. Sensitivity, misspecification, and practical risk control

A central practical issue is sensitivity to estimation error. In the binary net-odds model,

$q=1-p$02

in the asymptotic-variance treatment, while another source reports for the same binary formula and $q=1-p$03 that better odds increase $q=1-p$04 with

$q=1-p$05

The first derivative with respect to $q=1-p$06 agrees across treatments: small errors in $q=1-p$07 can produce proportionally larger errors in the stake, especially at even odds where $q=1-p$08 (Hakobyan et al., 23 Mar 2025, Lototsky et al., 2020). Near the fair boundary $q=1-p$09, small upward errors in $q=1-p$10 can move the prescribed stake from zero to materially positive values, while small downward errors truncate the optimal action to no bet under no shorting/no leverage (Hakobyan et al., 23 Mar 2025).

A second-order local regret formula quantifies the cost of probability misspecification. If $q=1-p$11 is estimated as $q=1-p$12 and $q=1-p$13, then

$q=1-p$14

In that approximation, the local growth loss depends primarily on the curvature in $q=1-p$15, not directly on $q=1-p$16, although the feasible stake and boundary exposure still do depend on $q=1-p$17 (Lillo et al., 26 Aug 2025).

Several practical mitigations recur across the literature. Fractional Kelly is the simplest: $q=1-p$18 It sacrifices some expected log-growth in exchange for reduced variance and greater robustness to parameter error (Lototsky et al., 2020). In empirical sports-betting experiments, an adaptive variant of fractional Kelly was tuned by grid search to maximize median final wealth subject to the safety condition $q=1-p$19 of final wealth, and this KellyFrac variant achieved zero ruin across horse racing, basketball, and football datasets in the reported study (Uhrín et al., 2021).

Other risk-control schemes include maximum bet caps, drawdown-constrained Kelly, and distributionally robust Kelly. The drawdown-constrained formulation in the sports-betting review uses the convex approximation

$q=1-p$20

to limit the probability of falling below a wealth threshold. Distributionally robust Kelly replaces the estimated probability vector by a box ambiguity set and solves a worst-case log-utility problem over that set (Uhrín et al., 2021). These are not part of the classical binary-choice formula, but they arise as practical corrections when the formal assumptions behind full Kelly are considered unrealistic.

The option-based binomial extension pursues robustness differently. In a binomial stock–bond model with a European put, the paper proves that a fairly priced option does not improve log-optimal growth under correct specification, but a convex mixture of two Kelly-with-option portfolios can be robust to estimation risk in the “max-of-two” sense: asymptotically, the mixture achieves the better realized growth of the two constituent option-augmented Kelly portfolios under the true environment (Lillo et al., 26 Aug 2025). This suggests that robustness can be engineered structurally rather than only by scalar stake shrinkage.

6. Generalizations and adjacent formulations

The binary-choice Kelly criterion is the base case for several generalizations. If payoffs on wins are random rather than constant, with nonnegative payoff variable $q=1-p$21, the expected log-growth becomes

$q=1-p$22

and the optimal fraction solves the fundamental integral equation

$q=1-p$23

The resulting optimal fraction is smaller than the classical constant-payoff Kelly fraction computed using the average payoff $q=1-p$24, with equality only in the degenerate constant-payoff case (Pérez-Marco, 2014). A plausible implication is that payoff uncertainty itself acts as an endogenous conservatism mechanism through the concavity of $q=1-p$25.

Another extension replaces log utility by a general concave utility function and allows extraneous wealth. In a one-step binary gamble, the interior first-order condition becomes

$q=1-p$26

For log utility this yields

$q=1-p$27

while CRRA and CARA utilities produce different closed-form stakes. The same work shows that, in an IID binomial tree with concave utility, the optimal local action at a node depends only on the node’s state and primitives, enabling an $q=1-p$28 dynamic program rather than exponential path enumeration (Viswanathan, 2016).

The continuous-time analogue arises in diffusion models with return process $q=1-p$29. There the log-growth and asymptotic variance are

$q=1-p$30

the continuous-time Kelly fraction is $q=1-p$31 (or $q=1-p$32 with risk-free rate $q=1-p$33), and the ridge-optimal fraction becomes

$q=1-p$34

The discrete binary case and diffusion case share the same qualitative structure—concave growth objective, volatility increasing with stake, and fractionalization under variance penalization—although the binary model retains nonlinear wealth multipliers and hard boundaries under no shorting/no leverage (Hakobyan et al., 23 Mar 2025, Lototsky et al., 2020).

Finally, the single-bet binary formula is no longer sufficient when multiple binary bets are placed simultaneously. In the multivariate case with $q=1-p$35 simultaneous wagers, the Kelly objective involves the joint outcome space of size $q=1-p$36. Recent work replaces explicit enumeration by an integral-transform formulation for independent bets, reducing objective evaluation from $q=1-p$37 to $q=1-p$38 per quadrature node, and studies lower and upper bounds via decomposition subproblems (Tepelyan et al., 27 Apr 2026). This establishes the single binary criterion as the tractable primitive from which higher-dimensional Kelly optimization departs.

7. Prediction markets, forecast evaluation, and interpretation

In binary prediction markets with Kelly bettors, the criterion determines both individual trade size and market aggregation. If agent $q=1-p$39 has belief $q=1-p$40 and normalized wealth $q=1-p$41, the competitive equilibrium price is

$q=1-p$42

After a realized outcome, wealth updates proportionally to likelihood ratios: $q=1-p$43 Thus wealth reweighting is exactly Bayesian, and the next-period market price is the posterior wealth-weighted average of beliefs. With fractional Kelly, the equilibrium price becomes a wealth-and-confidence-weighted average,

$q=1-p$44

and each trader behaves as if holding the effective belief

$q=1-p$45

(Beygelzimer et al., 2012).

A newer forecast-evaluation framework treats each probabilistic model as a canonical Kelly bettor whose bankroll evolves in real time as probabilities are updated before a single binary outcome resolves. For a model that already holds win shares $q=1-p$46 and cash bankroll $q=1-p$47, the generalized Kelly fraction is

$q=1-p$48

Cash updates as

$q=1-p$49

win shares update as

$q=1-p$50

and the market-clearing consensus probability in the binary case is

$q=1-p$51

The framework interprets

$q=1-p$52

as real-time model credibility and emphasizes a Bayesian analogue in which bankroll is a proxy for posterior credibility (Beuoy, 10 Feb 2026).

Simulation results reported in that study show that Kelly-bankroll evaluation can distinguish correct from incorrect time-updating binary models more accurately than average log-loss or Brier score in several settings, including faulty recency bias and random-walk miscalibration (Beuoy, 10 Feb 2026). This suggests that the binary Kelly criterion functions not only as a staking prescription but also as an evaluation functional for sequential probabilistic forecasts, especially when the timing and confidence of updates matter.

Across these formulations, the central structure remains unchanged: a binary event, multiplicative wealth, a log-based objective, and an optimal fraction determined by the discrepancy between subjective probability and implied odds. The classical formula

$q=1-p$53

is therefore both a specific betting rule and the seed from which broader theories of risk-sensitive growth, Bayesian aggregation, and probabilistic forecast evaluation are constructed.