Bet Sizing Optimizer

The search for the optimal wager

When you have a mathematical edge (such as in sports betting or value play), your goal is to grow your bankroll as fast as possible. However, doubling your bet size does not double your long-term growth. Because gains and losses compound geometrically, larger bets increase your volatility, which eventually drags your geometric growth rate into the negative.

This optimizer uses the principles of the Kelly Criterion but adds a critical safety feature: a **Ruin Tolerance Cap**. It maps your expected growth rate ($g(f)$) across different fractions of your bankroll ($f$) and identifies the precise size that maximizes wealth accumulation while keeping your probability of going broke below your comfortable threshold.

The math: Geometric growth vs. ruin

To find the optimal bet size, the verifier evaluates two competing mathematical functions:

1. The Geometric Growth Rate (g(f))

This function calculates the expected logarithmic growth rate of your bankroll per bet as a function of your bet fraction ($f$):

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

Where:

• $f$: The fraction of your bankroll wagered on each round.
• $p$: Your probability of winning.
• $q$: Your probability of losing ($1 – p$).
• $b$: The decimal payout odds (e.g., $b = 1$ for even money).

2. The Geometric Ruin Approximation

To ensure you do not go broke chasing peak growth, the tool estimates your long-term probability of hitting ruin:

P(Ruin) ≈ exp(-2 * |μ| * B / σ²)

Where $mu$ is your average edge, $B$ is your bankroll size in units, and $sigma^2$ is the variance.

Step-by-step audit: Setting your stakes

Suppose you have a $5,000 bankroll and have identified a sports betting edge where you have a 55% chance of winning an even-money bet ($p = 0.55$, $b = 1$, edge = 10%). You want to find the optimal size with a maximum ruin limit of 5%:

1. Enter your total bankroll ($5,000) and the game metrics ($p = 0.55$, $b = 1$).
2. Set your maximum acceptable Ruin Probability to 5%.
3. Click “Verify.” The optimizer will sweep fractions from $f = 0.01$ to $f = 0.30$.
4. Analyze the growth curve. You will see that full Kelly recommends betting 10% ($500) per game for maximum growth, but this carries a high ruin risk.
5. The optimizer will locate the “constrained optimal” bet (e.g., 5% or $250) which achieves 75% of the maximum growth rate but reduces your risk of ruin to under 5%.

Frequently asked questions