Bet sizing and Kelly

Idea: A good bet can still ruin you

Wealth compounds: results multiply, and one bad factor cannot be undone.

• All in for $10$ rounds: you survive with ${\left(\frac{2}{5}\right)}^{10}\approx 0.0001$
• Expected wealth hides that almost every run ends at zero

Idea: Look at the growth

Take $5$ rounds with $2$ wins and $3$ losses, staking a fixed share of wealth.

• Stake $10\%$: $1.2^2\times 0.9^3\approx 1.050$
• Stake $50\%$: $2^2\times 0.5^3=\frac{1}{2}$
• Stake $100\%$: $3^2\times 0^3=0$

Formula: The Kelly fraction

$f\cdot =p-\frac{q}{b}$

Win chance p, loss chance q = 1 − p, win b per 1 staked. Zero or less: do not bet.

Example: Kelly for the bet

A bet pays $2$ per $1$ staked with probability $\frac{2}{5}$, else you lose the stake. What fraction should you stake?

1. $p=\frac{2}{5}$, $q=\frac{3}{5}$, $b=2$
2. Kelly: $\frac{2}{5}-\frac{\frac{3}{5}}{2}=\frac{1}{10}$

Idea: Contracts fit too

A contract costing c that pays $1$ wins $b=\frac{1-c}{c}$ per dollar spent.

• Cost $0.25$: $\frac{1-0.25}{0.25}=3$
• Then use $f\cdot =p-\frac{q}{b}$ as usual

Quick check: Kelly on a contract

A contract costs $0.25$ and pays $1$. You think it pays with chance $\frac{1}{2}$. What is the Kelly fraction?

Idea: Why practitioners stake less

Kelly trusts your p completely, and your p is an estimate.

• At about double Kelly the growth is nearly gone
• Half Kelly: about a quarter less growth, swings halve
• Half the stake, $\frac{21600}{5400}=4$ times less variance

Common mistake: The edge is not the stake

The edge here is $\frac{1}{5}$ per dollar; staking that much is already double Kelly.

Instead Size by $f\cdot =p-\frac{q}{b}$. The same edge at longer odds calls for a smaller stake.

Idea: Say it out loud

Expected value decides whether to bet; sizing decides how much.

• Give the Kelly fraction with p, q and b named
• Say you would stake half of it or less