Kelly Criterion: Optimal Bet Sizing

The Kelly Criterion is a mathematical formula developed by John L. Kelly Jr. at Bell Labs in 1956 for maximising the long-run growth rate of a bankroll when faced with a series of bets with positive expected value. It is used by professional gamblers, sports bettors, card counters, and financial investors.

The Formula

f* = (bp − q) / b

Where: - f* = fraction of bankroll to bet - b = net odds received on the bet (e.g., a 2:1 bet returns 2 for every 1 wagered, so b = 2) - p = probability of winning - q = probability of losing (1 − p)

Example: Card Counting Edge

A blackjack card counter in a favourable count calculates: - Win probability: p = 0.51 (51%) - Loss probability: q = 0.49 - Net odds: b = 1 (even-money bet)

f* = (1 × 0.51 − 0.49) / 1 = 0.02 = 2% of bankroll

The Kelly bet is 2% of current bankroll. This fraction changes as the count changes.

Why Kelly is Mathematically Optimal

Kelly maximises the expected value of the *logarithm* of wealth, which is equivalent to maximising long-run bankroll growth rate. Betting more than Kelly will eventually lead to ruin. Betting less than Kelly is sub-optimal but not ruinous.

Half-Kelly and Fractional Kelly

Practitioners often bet a fraction of the Kelly amount — commonly Half-Kelly (f*/2) or Quarter-Kelly — because: - Kelly calculations depend on precise edge estimates; errors lead to over-betting - Variance at Full Kelly is very high — extreme swings even during winning periods - Half-Kelly has approximately 75% of Full Kelly's growth rate with significantly lower variance

Casino Application

The Kelly Criterion is relevant in casino contexts only where genuine positive expectation exists: card counting in blackjack, expert sports betting, some promotional situations, and video poker with optimal strategy. In standard negative-edge games, Kelly would prescribe a bet of zero — the theoretically correct response to negative EV.