凯利公式

Definition and Formula

The Kelly Criterion, developed by John L. Kelly Jr. in 1956, is a bet-sizing formula that balances risk and reward to maximize the logarithm of wealth over time. The full Kelly formula for a bet with two outcomes is:
[ f^* = \frac{bp - q}{b} ] where ( f^* ) is the fraction of current bankroll to bet, ( b ) is the net odds received on the bet (e.g., ( b = 1 ) for a 1:1 payout), ( p ) is the probability of winning, and ( q = 1-p ) is the probability of losing. For a simple even-money bet, it reduces to ( f^* = 2p - 1 ).

Application in Poker

In poker, the Kelly Criterion is applied to bankroll management, particularly for cash games and tournaments. The “edge” in poker is the player’s expected value (EV) per hand or per tournament relative to the buy-in. For example, if a player has a 5% edge in a $100 tournament (expected profit of $5), the full Kelly bet size would be a fraction of the bankroll: ( f^* = \frac{EV}{\text{odds}} ), where odds represent the net multiplier of the buy-in. However, poker outcomes are not binary with fixed odds, so players often approximate using win rate and standard deviation.

The Kelly Criterion suggests betting a fraction of bankroll proportional to the edge. A common adaptation for poker is the “Kelly bankroll,” which recommends having at least 30-50 buy-ins for cash games or 100-300 buy-ins for tournaments to avoid ruin, based on half-Kelly or quarter-Kelly to reduce volatility.

Risks and Adjustments

Full Kelly betting maximizes growth but leads to huge short-term swings and a high risk of drawdown (e.g., 33% chance of losing half the bankroll). Most poker players use fractional Kelly (e.g., half-Kelly or quarter-Kelly) to reduce variance while still growing the bankroll. The formula assumes known probabilities and no transaction costs, which rarely hold in practice. Win rates are uncertain and vary over time, so conservative estimates are wise.

Additionally, Kelly does not account for life expenses or the need to preserve a minimum bankroll. Therefore, many professionals use a “Kelly bankroll” guideline: the required bankroll ( R ) for a given edge and risk of ruin can be estimated as ( R \approx \frac{\sigma^2}{2 \mu} ), where ( \mu ) is expected return and ( \sigma^2 ) is variance. This relates to Kelly by ensuring bet sizes stay within safe limits.

Conclusion

The Kelly Criterion offers a theoretically optimal approach to bet sizing for long-term growth. In poker, it provides a framework for bankroll management, but due to real-world uncertainties, fractional Kelly and conservative estimates are recommended. Understanding the balance between growth and risk is essential for sustained success.