Fractional Kelly for Esports: Exact Growth and Probability Error
Fractional Kelly starts with a question the calculator cannot answer: how credible is the probability you entered? A formula can transform an estimate into a stake without making the estimate accurate. In an esports model, a roster change, a map veto or a new patch can make historical inputs less relevant. There is no universal error range for that process. A smaller fraction reduces exposure to an error; it does not turn an uncertain or negative expectation into a reliable income stream.
The derivation below concerns a binary bet with no push, commission, tax or acceptance change. It considers one position in isolation and assumes its win probability is known. Expected logarithmic growth measures multiplicative wealth, rather than expected cash profit. For example, a gain of fifty percent followed by a loss of fifty percent leaves three quarters of the starting capital. Choosing a logarithmic objective gives the loss branch the weight needed to reflect that arithmetic.
Strict concavity makes the stationary point the unique interior maximum when both outcomes have positive probability. If the calculated fraction is negative, a bettor who can only back this selection uses zero, not a negative stake. A zero output can therefore be the useful result. The formula does not require every match to produce a bet. It also does not choose a universally acceptable financial loss for the person entering the numbers.
The isolated binary model
Let b equal decimal odds minus one, p the stipulated true win probability, q = 1 - p, and f the fraction of capital staked. The exact expected log growth and its derivative are:
Binary wager, no push, no fees; one isolated position
decimal odds O = b + 1; true probability p; q = 1 - p
bankroll fraction 0 <= f < 1
g(f) = p ln(1 + b f) + q ln(1 - f)
g'(f) = p b / (1 + b f) - q / (1 - f)
g''(f) = -p b^2 / (1 + b f)^2 - q / (1 - f)^2 < 0
f_star = (b p - q) / b = (p O - 1) / (O - 1)
no short selling: use max(0, f_star)
Hypothetical exact probability p = 0.525, odds O = 2.00
EV per unit = p O - 1 = 0.050000
full fraction = 0.050000; half = 0.025000; quarter = 0.012500
g(full) = 0.001250521355
g(half) = 0.000937662817
g(quarter) = 0.000546901451
g(half)/g(full) = 0.749817517
g(quarter)/g(full) = 0.437338754Note the structure of the numerator. b·p − q is your expected profit per unit staked, and the denominator b scales it by the payout. At even money, Kelly reduces to "stake your edge": a 5% edge means a 5% stake. At shorter odds the same edge produces a larger stake, because each win pays less and you need more exposure to harvest the edge. At longer odds the same edge produces a smaller stake, because the outcome distribution is more skewed.
Probability error and approximations
Half Kelly means half the fraction calculated from the same input probability. Quarter Kelly means one quarter. The familiar seventy-five percent growth claim is a small-stake approximation, not an identity of the exact logarithmic function. In the Taylor expansion, the quadratic coefficient uses the second moment of the per-unit return. Substituting its variance without stating an additional approximation changes the calculation. The exact example below keeps that distinction visible instead of presenting a smooth quadratic as the full growth curve.
The error example deliberately assumes two different probabilities. The estimated probability produces a thirteen percent stake, while the stipulated true probability supports only five percent under this objective. Evaluating the thirteen percent stake with the true probability gives negative expected log growth. That establishes overbetting within the example. It does not establish how often an esports model makes a four-point error, nor a probability of bankruptcy or maximum drawdown over five hundred bets.
The exponential of expected log wealth is a geometric summary. A finite-horizon median requires the distribution of the number of wins and the associated discrete wealth outcomes. Path-dependent questions require more: touching half the starting bankroll is different from ending below half, and maximum drawdown is measured from a running peak. No simulation of those events is supplied here. A table of precise ruin percentages without a stated boundary, horizon and reproducible model would imply evidence we do not have.
Second-order expansion about f = 0, return R in {b, -1}
mu = E[R] = p b - q
m2 = E[R^2] = p b^2 + q
g(f) approximately mu f - m2 f^2 / 2
quadratic optimum f_quad = mu / m2
quadratic growth ratio at c f_quad = 2 c - c^2
c = 0.50 -> 0.750000; c = 0.25 -> 0.437500
These are approximation ratios, not universal exact ratios.
Same hypothetical true p, mistaken estimate p_hat = 0.565
estimated full fraction = 0.130000
g(0.13; true p = 0.525) = -0.001985224803
500 g = -0.992612402
exp(500 g) = 0.370607250
This last number is exp(expected log wealth ratio), not an exact median.
Independent example O = 1.87, assumed p = 0.57, B = 10000
EV = 0.0659; full fraction = 0.0659 / 0.87
quarter stake = B * 0.25 * 0.0659 / 0.87 = 189.367816092
Round only the final stake to the platform's permitted increment.Exposure and the review record
Select a multiplier as an explicit risk policy and document why it suits the uncertainty of the particular model. Do not treat a quarter fraction or a three percent cap as an industry certification of safety. If the true edge is negative, shrinking the stake only reduces the expected loss in cash terms. Where the probability has no defensible basis, paper tracking or passing the market is more informative than choosing a smaller positive number to make the ticket feel controlled.
Price comparisons and margin removal serve different purposes. Compute the expected return and Kelly fraction against the actual price that can be accepted. A de-margined reference market offers a comparison estimate, not the true probability. Different removal methods can produce different estimates from the same quotes. Likewise, positive closing-line value records a favourable price comparison under a defined benchmark; it cannot certify the probability used to size the original bet or guarantee future profit.
Before committing capital, inspect the combined loss across open positions. A match winner, a handicap and a player prop can all be affected by the same series. Adding their separate Kelly stakes is not a portfolio solution. Enumerate plausible joint outcomes and check available cash as well as total bankroll. Money already locked in a different position is not a second source of spendable balance. Do not increase a new stake because an earlier ticket lost or because the day has a profit target.
Keep the calculation reproducible: timestamp, model version, estimated probability, accepted odds, bankroll definition, fraction, currency and rounding rule. Record any policy cap separately so a later review can distinguish a model change from a risk-policy change. The local bankroll tool can support the sizing and manual record, but this guide does not claim it imports CSV files, runs Monte Carlo paths or estimates your true probability. Check the fields actually available before planning a workflow around them.
Review calibration by comparing forecasts with outcomes in comparable probability buckets, while recording sample sizes and dependence between matches. Keep a later period aside when developing a model, and account for patches or roster changes rather than pooling incompatible periods. Neither a fixed number of bets nor a positive CLV threshold automatically earns a larger multiplier. A scheduled review should be able to conclude that the evidence is insufficient and that exposure should remain zero. That is a completed decision, not a failed calculation.
J. L. Kelly Jr. (1956), A New Interpretation of Information Rate
