Proper Betting: Mathematical Methods

• Proper betting is a mathematically disciplined approach that optimizes wager sizing and sequential inference via methods like the Kelly criterion.
• It integrates game-theoretic probability, statistical testing, and capital growth frameworks to manage drawdowns and compounding risks.
• Applied in areas such as sports betting and prediction markets, proper betting converts probabilistic advantages into coherent positions under liquidity and risk constraints.

Proper betting denotes a family of mathematically disciplined approaches to wagering, trading, and sequential inference. In repeated favorable games, it refers to stake selection that is appropriate for compounding wealth over time rather than for maximizing one-step expected payoff; in game-theoretic probability and statistical testing, it refers to nonnegative capital processes whose expected value under a null model is controlled; in market and forecasting applications, it refers to converting probabilistic edge, odds mispricing, or score advantage into coherent positions under explicit feasibility, liquidity, and risk constraints (Haghani et al., 2017, Shafer, 2019, Gu et al., 7 Jul 2026).

1. Conceptual foundations

The most persistent distinction in the literature is between fair-looking one-step optimization and proper repeated-play optimization. For a favorable even-money gamble with win probability $p>1/2$, maximizing one-flip expected wealth pushes the stake fraction $f$ to $1$, because

$\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$

is increasing in $f$. Proper betting in repeated play instead maximizes expected logarithmic growth,

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

thereby penalizing drawdowns and accounting for compounding (Haghani et al., 2017).

A second distinction concerns coherence. In the evidential literature, a proper bet is a nonnegative payoff $S$ priced at its null expected value, often normalized so that $\mathbb{E}_{H_0}[S]=1$. Realized capital $S(x)$ is then a betting score or e-value. Sequentially, products of such scores form nonnegative test supermartingales, so optional stopping and adaptive continuation remain valid through Ville’s inequality (Shafer, 2019).

A third distinction concerns forecast-to-action conversion. In modern prediction markets, “proper betting” has been formalized as a trading rule derived from a strictly proper scoring rule $S$, so that profitability tracks forecasting accuracy whenever liquidity is sufficient. In that setting, properness means more than internal coherence: it means that a forecaster’s score advantage over the market is converted into a position with a robust expected-profit guarantee (Gu et al., 7 Jul 2026).

2. Log-utility, Kelly sizing, and repeated favorable bets

The canonical formulation is the Kelly criterion. With even-money odds, win probability $f$0, loss probability $f$1, and bankroll fraction $f$2, the expected log-growth per bet is

$f$3

The first-order condition gives

$f$4

and concavity,

$f$5

makes this the unique maximizer. For general net odds $f$6, the growth function becomes

$f$7

In the 60% heads, even-money coin experiment, proper betting therefore meant staking $f$8 of current bankroll on heads each flip (Haghani et al., 2017).

The experiment exposed the gap between formal optimality and observed behavior. Subjects began with $f$9, 28% went bust to below $1$0; average payout was $1$1, and 18 players went all-in at least once. Tails betting was also common despite the stated 60% heads probability: 67% bet tails at least once, and 48% did so more than five times. The dominant observed errors were overbetting, erratic sizing, gambler’s fallacy, hot-hand effects, illusion of control, anchoring, sunk-cost bias, and Martingale-like loss chasing (Haghani et al., 2017).

The contrast with constant-fraction discipline was stark. With full Kelly $1$2,

$1$3

while expected wealth grows by a factor $1$4 per flip. With half Kelly $1$5,

$1$6

or about 75% of full Kelly’s log-growth with materially lower variance and drawdown risk. Simulations in the capped 60% coin game showed that constant fractions of 10–20% reached the $1$7 cap with probability about 95%, whereas both underbetting at 5% and overbetting at 40% reduced cap-hit probability to about 70% (Haghani et al., 2017).

3. Temporal correlation, finite horizons, and computational constraints

Classical Kelly assumes i.i.d. outcomes, but proper betting can be generalized to temporally correlated sequences. In the correlated even-money model, outcomes $1$8 satisfy

$1$9

For arbitrary joint dependence over a finite horizon,

$\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$0

so the optimal constant fraction is

$\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$1

In the simplest Markov case,

$\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$2

the paper gives a closed form

$\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$3

with $\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$4, $\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$5, and $\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$6. Time-varying feedback $\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$7 strictly improves ELG whenever $\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$8 varies over time (O'Brien et al., 2020).

Finite horizons change the objective even when the game is favorable. In a discrete casino model with integral chips, absorbing boundaries at $\mathbb{E}[W_1]=W_0\big(1+f(2p-1)\big)$9 and $f$0, and deadline $f$1, proper betting may mean maximizing the probability of reaching $f$2 by time $f$3, not maximizing asymptotic log growth. With state $f$4 and remaining time $f$5, the exact dynamic program is

$f$6

where $f$7. This discrete framework sharpens classical Dubins–Savage and Breiman results: bold play maximizes eventual success probability when $f$8, timid play is optimal when $f$9 and no deadline is imposed, and finite-horizon optimization can recommend neither pure bold nor pure timid play (Ekhad et al., 2011).

The same work also gives exact linear systems for fixed strategies. If $g(f)=p\ln(1+f)+q\ln(1-f),$0 is the integral stake from bankroll $g(f)=p\ln(1+f)+q\ln(1-f),$1, then ultimate success probabilities satisfy

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

with $g(f)=p\ln(1+f)+q\ln(1-f),$3, $g(f)=p\ln(1+f)+q\ln(1-f),$4, while expected exit times satisfy

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

with $g(f)=p\ln(1+f)+q\ln(1-f),$6. Proper betting in this setting is therefore explicitly computational: it is the optimal action under the actual chip size, target, bankroll, and deadline constraints, not the continuous-horizon idealization (Ekhad et al., 2011).

4. Risk control, support restrictions, and departures from classical Kelly

A central complication is that classical Kelly can be either too aggressive or too conservative, depending on how risk is modeled. One source of conservatism is the support of returns. For scalar returns $g(f)=p\ln(1+f)+q\ln(1-f),$7 with support $g(f)=p\ln(1+f)+q\ln(1-f),$8, any optimizer of

$g(f)=p\ln(1+f)+q\ln(1-f),$9

must satisfy

$S$0

where $S$1. In vector form, the Restricted Betting Theorem states that any optimizing Kelly vector must satisfy

$S$2

where $S$3 is the support function of $S$4. If support is unbounded in both scalar directions, then $S$5; the same phenomenon appears in the normal-return example, where unbounded downside forces no betting at all despite positive mean (Hsieh et al., 2017).

This support sensitivity motivates empirical and approximate variants. Under a second-order Taylor approximation,

$S$6

the approximate scalar optimizer is

$S$7

Because this ignores the log-domain constraint, survivability must be reimposed. The exact scalar survival condition is

$S$8

and the paper proposes saturating $S$9 to this interval. It also derives explicit formulas for expected cumulative gain,

$\mathbb{E}_{H_0}[S]=1$0

and for the variance of cumulative gain and logarithmic growth under the approximate strategy (Hsieh, 2020).

A different line of work studies betting strategies that maximize growth subject to controlled fluctuations. In the fair-odds horse-race model, Kelly is $\mathbb{E}_{H_0}[S]=1$1, the null risk-free strategy is $\mathbb{E}_{H_0}[S]=1$2, and the paper optimizes

$\mathbb{E}_{H_0}[S]=1$3

The stationarity condition is

$\mathbb{E}_{H_0}[S]=1$4

A phase transition occurs at

$\mathbb{E}_{H_0}[S]=1$5

For $\mathbb{E}_{H_0}[S]=1$6, the null strategy is optimal; for $\mathbb{E}_{H_0}[S]=1$7, a risky mixed strategy is optimal. The paper also proves the uncertainty-relation-like bound

$\mathbb{E}_{H_0}[S]=1$8

which constrains achievable growth at a given fluctuation level (Dinis et al., 2020).

For applied sports betting, a large experimental review arrives at a complementary practical conclusion: pure Kelly and pure maximum-Sharpe strategies can be fragile under probability-estimation error, while fractional Kelly, drawdown-constrained Kelly, and distributionally robust Kelly substantially improve stability. The reported cross-sport recommendation is an adaptive variant of fractional Kelly tuned by a survival constraint on the 5th percentile of final wealth (Uhrín et al., 2021).

5. Odds, overround, coupons, and parlays

In sportsbook markets, proper betting begins with price normalization. Decimal odds $\mathbb{E}_{H_0}[S]=1$9 imply $S(x)$0; for a market with outcomes $S(x)$1, overround is

$S(x)$2

Expected return per unit stake under a bettor’s own probability $S(x)$3 and decimal odds $S(x)$4 is

$S(x)$5

which is also the expected profit per unit stake. Lower overround makes a book more player-friendly, and line shopping across books can reduce the aggregate margin or, in rare cases, create a surebet when best-line implied probabilities sum to less than $S(x)$6 (Saha et al., 2017).

A more formal treatment views bookmaker offers as a set of desirable gambles. With fractional odds $S(x)$7, decimal odds $S(x)$8, and upper mass bound

$S(x)$9

the odds alone avoid sure loss if and only if

$S$0

Free coupons can destroy that coherence. If $S$1 is the first-free gamble induced by an initial qualifying bet and a winnings-only coupon, then the augmented book avoids sure loss if and only if

$S$2

If $S$3, the bookmaker incurs sure loss and the customer can compute the optimal guaranteed-gain portfolio via complementary slackness in a primal–dual linear program (Nakharutai et al., 2019).

Proper betting with parlays has a distinct structure. For independent multi-outcome events with multiplicative parlay pricing, each event is first solved in isolation in the implicit-cash Kelly formulation,

$S$4

The simultaneous optimizer over the full menu of singles and parlays is then the outer product

$S$5

and terminal wealth factorizes across events. A parlay is active if and only if every selected leg is active in its one-event problem. When parlays are forbidden, the singles-only optimizer differs from the isolated eventwise solution only at cubic order, and the growth-rate loss is $S$6, which the paper identifies as an explanation of Whitrow’s empirical near-proportionality phenomenon (Long, 27 Mar 2026).

Empirical sports-betting systems operationalize these principles by combining market prices with model-based edge estimates. One such approach estimates $S$7 nonparametrically from historical point spreads, computes moneyline expected value, applies an $S$8-band around 50% win probability to reduce tail misestimation, and keeps only bets with expected value above a tuned threshold. With out-of-sample bootstrap validation, the reported moneyline ROI is positive across NFL, NBA, NCAAF, NCAAB, and WNBA, with, for example, 16.57% ROI on 567 NFL bets for the Simple model and 13.02% ROI on 284 NCAAB bets for the Weighted model (Ramesh et al., 2019).

6. Betting as statistical evidence and forecast recalibration

In statistical communication, proper betting replaces tail-area language by capital language. If $S$9 is a null density and $f$00 an alternative, then the likelihood ratio

$f$01

is a proper bet because $f$02. Sequentially,

$f$03

is a nonnegative test martingale, and Ville’s inequality gives

$f$04

This framework yields e-values, optional-stopping validity, multiplicative meta-analysis, and naturally sequential evidence accumulation (Shafer, 2019).

The broader betting interpretation of probability distinguishes two meanings of numerical degrees of belief. Under the odds-offering interpretation, a probability is a two-sided fair price for a ticket paying $f$05 on event $f$06. Under the capital-multiplication interpretation, associated with Ville, the operative judgment is that no admissible strategy can multiply the capital it risks by a large factor. Both interpretations justify conditioning for additive probabilities, but only the capital-multiplication interpretation extends to Dempster–Shafer belief functions and justifies Dempster’s rule of combination (Shafer, 2010).

Betting can also be used to improve probabilistic forecasts. Given a predictive distribution $f$07 and realized outcome $f$08, the probability integral transform

$f$09

should be IID Uniform$f$10 under correct calibration. The forecast-enhancement procedure defines a betting martingale on the $f$11, using calibrators

$f$12

and then reweights the original predictive density by the effective betting function $f$13: $f$14 The cumulative log-loss difference is exactly $f$15, where $f$16 is the martingale capital, so successful betting against PIT non-uniformity translates directly into improved log score (Vovk, 2021).

A related sequential-inference construction mixes linear betting fractions to obtain time-uniform empirical Bernstein LIL envelopes. With bounded martingale differences $f$17, cumulative sum $f$18, and empirical variance proxy $f$19, the resulting wealth lower bound and Ville’s inequality yield an anytime boundary whose dominant asymptotic scale is

$f$20

This suggests that proper betting is not only an evidential metaphor but a constructive mechanism for deriving self-normalized concentration results (Orabona, 21 May 2026).

7. Prediction markets, proper scoring rules, and sim-to-real evaluation

In prediction markets, proper betting has been formalized for general central limit order books. For a strictly proper scoring rule $f$21 with convex potential $f$22, market price vector $f$23, and forecaster distribution $f$24, the proper position is

$f$25

Its expected profit against ground truth $f$26 decomposes as

$f$27

where $f$28 is the Bregman divergence induced by $f$29 and $f$30 is liquidity loss. Thus, whenever the forecast outperforms the market under $f$31 and liquidity loss is covered by the divergence term, proper betting guarantees positive expected profit. The same paper proves a uniqueness result: any strategy with such a robust profitability guarantee must be essentially the same as $f$32 up to rescaling and constant shifts (Gu et al., 7 Jul 2026).

The construction is explicit under standard scores. For Brier,

$f$33

and for log score,

$f$34

The empirical results reported there state that proper betting is the only strategy that reliably converts AI forecasters’ accuracy into profit across thousands of real markets, and a live Kalshi deployment achieved $f$35 ROI with Sharpe ratio $f$36 (Gu et al., 7 Jul 2026).

A separate line of work applies betting to estimator design under scarce physical experimentation. In sim-to-real performance evaluation, the target is

$f$37

with $f$38. The paper defines a betting payoff

$f$39

wealth update $f$40, and bet-weighted estimator

$f$41

Theorem 1 gives an exact efficiency criterion: $f$42 which states that variance reduction must exceed squared bias penalty for the bet-weighted estimator to outperform Monte Carlo. Under a small-stakes Kelly expansion, the ideal stake is approximately

$f$43

and Theorem 3 gives an e-value diagnostic: $f$44 under the no-edge null $f$45. In this usage, proper betting is both an allocation rule and an anytime-valid diagnostic of whether simulator information is actually improving real-world estimation (Mahboob et al., 27 Apr 2026).