What Is the Kelly Criterion?
The Kelly Criterion, developed by John L. Kelly Jr. in 1956, is a formula for sizing bets to maximize the long-run geometric growth rate of your bankroll. It is not about maximizing expected value in a single bet — it is about maximizing the logarithm of wealth over many repeated bets.
The original Kelly formula applies to binary outcomes (win or lose) with fixed odds:
$$f^* = \frac{bp - q}{b}$$Where:
- \(f^*\) = fraction of bankroll to wager
- \(b\) = net odds received (e.g., 2 means you win 2× your bet)
- \(p\) = probability of winning
- \(q = 1 - p\) = probability of losing
Simple Example: Coin Flip with Edge
Suppose you have a coin that lands heads 55% of the time (p = 0.55) and you are paid 1:1 (b = 1). Kelly says:
$$f^* = \frac{1 \times 0.55 - 0.45}{1} = \frac{0.10}{1} = 10\%$$You should bet 10% of your bankroll on each flip. Betting more than 10% shrinks your long-run growth rate despite having positive EV.
Kelly Formula for Poker
In poker, outcomes are not binary — you can win or lose many different amounts. The continuous Kelly formula for games with a known win rate and variance is:
$$f^* = \frac{\mu}{\sigma^2}$$Where:
- \(\mu\) = expected win rate per hand (or per session) in big blinds or dollars
- \(\sigma^2\) = variance per hand (or per session) in the same units squared
This gives the fraction of your bankroll to put at risk in a single session. In practice, \(f^*\) tells you the correct game stakes relative to your bankroll.
For example, if your win rate is 5 bb/100 and your variance is 100 bb²/100 hands (standard deviation of 10 bb/100), then:
$$f^* = \frac{5}{100} = 0.05 = 5\%$$You should play stakes where your total buy-in is at most 5% of your bankroll. At \$1/\$2 NL with a \$200 buy-in, this implies a minimum bankroll of \$4,000.
Three Worked Examples
Example 1: Coin Flip with 3:2 Odds
Setup: You win \$3 for every \$2 wagered (b = 1.5), win probability = 50% (p = 0.5).
$$f^* = \frac{1.5 \times 0.5 - 0.5}{1.5} = \frac{0.75 - 0.5}{1.5} = \frac{0.25}{1.5} = 16.7\%$$Bet 16.7% of bankroll per wager. With a \$10,000 bankroll, risk \$1,670 per bet.
Example 2: Poker Game with Known Win Rate
Setup: NL100 (\$0.50/\$1.00). Win rate = 8 bb/100 hands = \$8/100 hands. Variance = 2,500 bb²/100 = \$2,500/100 hands.
$$f^* = \frac{8}{2500} = 0.0032 = 0.32\%$$Only 0.32% of bankroll per 100-hand session should be at risk. With a typical 100-hand session buy-in of \$100 (100bb), the required bankroll is:
$$\text{Bankroll} = \frac{\$100}{0.0032} = \$31,250$$This seems enormous! This is why full Kelly is impractical for poker — the high variance requires a huge bankroll to be Kelly-optimal. Fractional Kelly is the answer.
Example 3: Tournament Buy-In Decision
Setup: A tournament costs \$500 to enter. Your estimated ROI (return on investment) is 20% (you average \$600 return on a \$500 buy-in). The variance of tournament results is very high — approximated as \$EV × (1/ROI) × k, where k ≈ 5 for standard variance.
Using the simplified Kelly for tournaments:
$$f^* \approx \frac{\text{ROI}}{\text{variance multiplier}} = \frac{0.20}{5} = 4\%$$The \$500 buy-in should represent at most 4% of your bankroll → minimum bankroll of \$12,500 for this tournament. Most professionals use 1–2% per tournament (half-Kelly or quarter-Kelly), suggesting a \$25,000–\$50,000 bankroll for consistent \$500 tournament play.
Fractional Kelly
Full Kelly maximizes long-run growth but produces terrifying variance — drawdowns of 50% or more are common. Most professionals use fractional Kelly: betting a fixed fraction of the Kelly optimal amount.
$$f_{\text{fractional}} = k \times f^*, \quad k \in (0, 1]$$Half Kelly (\(k = 0.5\)) produces approximately 75% of the growth rate of full Kelly, but with far less variance. Quarter Kelly (\(k = 0.25\)) is ultra-conservative but virtually eliminates ruin risk.
| Kelly Fraction | Growth Rate (% of Full Kelly) | Max Drawdown Risk | Recommended For |
|---|---|---|---|
| Full (1.0×) | 100% | Very High (~50%+) | Mathematical ideal only |
| Half (0.5×) | ~75% | Moderate (~25%) | Experienced pros |
| Quarter (0.25×) | ~44% | Low (~12%) | Most recreational players |
| Tenth (0.1×) | ~19% | Very Low | Ultra-conservative |
Bankroll Requirements Table
The following table shows the required bankrolls at various win rates and risk-of-ruin levels for NL cash games, using half-Kelly sizing (buy-in = 100bb).
| Win Rate (bb/100) | Std Dev (bb/100) | 5% RoR Bankroll | 1% RoR Bankroll |
|---|---|---|---|
| 2 | 80 | ~250 buy-ins | ~400 buy-ins |
| 5 | 80 | ~80 buy-ins | ~130 buy-ins |
| 8 | 80 | ~50 buy-ins | ~80 buy-ins |
| 10 | 80 | ~40 buy-ins | ~65 buy-ins |
| 15 | 80 | ~25 buy-ins | ~40 buy-ins |
The risk-of-ruin formula for a given bankroll \(B\) and half-Kelly sizing is approximately:
$$\text{RoR} \approx e^{-2 \mu B / \sigma^2}$$To find the required bankroll for a 5% RoR: set \(\text{RoR} = 0.05\), solve for \(B\): \(B = \frac{\sigma^2 \ln(20)}{2\mu}\).
Practical Usage
Apply Kelly thinking to three key poker decisions:
- Game Selection: Use Kelly to determine which stakes you can afford. If your bankroll supports Kelly-optimal play at NL100, moving to NL200 without doubling your bankroll is overbetting — it increases variance faster than growth.
- Shot-Taking: When moving up in stakes, a "shot" of 5–10 buy-ins at the new level is the Kelly-aware approach — large enough to get a sample, small enough to preserve the bankroll if it fails.
- Tournament Scheduling: The Kelly approach suggests playing more lower-buyin events and fewer high-buyin events unless your ROI is very high. The high variance of tournaments penalizes over-buyin decisions severely.