Kelly Criterion

Also known as: kelly, kelly betting, full kelly, fractional kelly

The bet-sizing rule that maximizes long-run growth of a bankroll given an edge; pros use a fraction of it for safety.

The Kelly criterion gives the bet fraction of your bankroll that maximizes the long-run growth rate of the roll. For a simple bet:

\[ f^{*} = \frac{\text{edge}}{\text{odds}} = \frac{bp - q}{b} \]

where \(p\) is win probability, \(q = 1-p\), and \(b\) the net odds received. Full Kelly maximizes expected log-bankroll — bet more and growth falls; bet enough over Kelly and you trend toward ruin even with an edge.

The catch is that full Kelly is wildly aggressive in practice. It assumes your edge is known exactly and tolerates gut-wrenching drawdowns — a full-Kelly bettor can routinely be down 50% of the roll. Real poker edges are estimated with error, and overestimating your edge means you're effectively betting over Kelly. So serious players bet fractional Kelly — typically half or quarter Kelly:

• Half Kelly keeps ~75% of the growth rate for far less than half the variance — an excellent trade.
• It builds in a margin for the fact that your true edge is uncertain and your standard deviation is only estimated.

This is the theoretical backbone of buy-in rules and risk of ruin: "30 buy-ins" is effectively a fractional-Kelly stance for typical cash edges and variance, chosen so drawdowns stay survivable and psychologically bearable.