Kelly Log Growth

Worked Example

Select Win probability = 0.75 and Net win odds b = 1. A win multiplies the current size by 1 + f, while a loss multiplies it by 1 - f. Compare the selectable fractions 1/4 and 1/2 using natural logarithms:

g(f) = p * ln(1 + b*f) + (1 - p) * ln(1 - f)

1. At f = 1/4, the multipliers are 1.25 and 0.75. The win contribution is 0.75 ln(1.25) = 0.167357663486 nats and the loss contribution is 0.25 ln(0.75) = -0.071920518113 nats. Their sum is 0.095437145373 nats.
2. At f = 1/2, the multipliers are 1.5 and 0.5. The contributions are 0.304098831081 and -0.173286795140 nats, giving 0.130812035941 nats.
3. The derivative is p*b/(1 + b*f) - (1 - p)/(1 - f). Its zero is f = (p*(b + 1) - 1)/b = 1/2 here. The second derivative is negative in the interior, so this is the continuous maximum.
4. For these same inputs, the one-step arithmetic expected multiplier is 1 + 0.5*f, which is maximized at f = 1. This differs from logarithmic growth: at f = 1 the positive-probability loss has zero multiplier and makes expected log growth negative infinity.

Under the declared bounds 0 ≤ f ≤ 1, the continuous optimum is max(0, (p*(b + 1) - 1)/b). The exact selectable fractions include every possible optimum for the module's fixed probability and odds choices. At p = 1 the zero-probability loss term is omitted, g(f) = ln(1 + b*f), and the maximum is at f = 1.

Interactive Experiment

The Kelly Log Growth module allows you to explore the concave growth curve by varying the hypothetical win probability, the net win odds, and the staked fraction.

1. Set Win probability to 0.75, and Net win odds b to 1. Notice the continuous optimal fraction f* updates to 0.5.
2. Select a Staked fraction f of 1. With Win probability still 0.75, Expected Log Growth displays Negative infinity: a loss has positive probability and its multiplier is zero.

Limitations

This mathematical example fixes hypothetical probabilities and payouts and assumes repeated independent outcomes with those same values. In real-world scenarios, such as financial markets or sports betting, true probabilities are unknown and must be estimated. Overestimating your edge (the win probability or odds) can lead to staking too large a fraction, drastically increasing the risk of ruin.

The model ignores discrete bankroll effects, transaction costs, and limits on minimum or maximum bet sizes. The graph uses a disclosed finite vertical window; when p is less than 1, its negative-infinity all-in endpoint is not a finite plotted point.

References

Original Kelly 1956 paper, pp919-920, which introduced the criterion of maximizing the expected value of the logarithm of wealth.
https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf