Kelly Criterion Blog (Stanford)

Summary

This Stanford blog post provides one of the clearest accessible explanations of the Kelly criterion, walking through the original Kelly1956 paper's key insights with illuminating examples. The author explains the connection between Kelly's work and Bernoulli's earlier work on gambling, shows how Kelly extends the problem to series of bets, and derives the optimal strategy.

The post emphasizes that Kelly's key insight is maximizing the expected logarithm of wealth (E[ln(wealth)]) rather than expected wealth (E[wealth]), which accounts for risk aversion and the compounding effect of sequential bets. It also explains why log-wealth maximization is equivalent to expected utility maximization with log utility.

Key Concepts

  • Bernoulli's problem: Before Kelly, Bernoulli solved a single bet problem by maximizing expected log wealth. Kelly extended this to a sequence of bets where the gambler receives information before each bet.
  • Log-wealth utility: Maximizing E[ln(wealth)] is the same as expected utility maximization with log utility — Kelly's criterion is a specific case of utility theory
  • Information advantage: Kelly's gambler has an information channel giving noisy signals about outcomes. The gambler's edge comes from using this information better than the bookmaker.
  • Exponential growth: With Kelly betting, wealth grows as exp(N × G) where G is the growth rate and N is the number of bets
  • Risk of ruin: Full Kelly has ~33% chance of halving bankroll before doubling; half-Kelly reduces this to ~11%
  • Why fractional Kelly: Estimation error in probabilities, finite betting horizon, and volatility aversion all justify betting less than full Kelly

Key Examples from the Post

Even money bet with 60% win rate:
- Kelly fraction f* = 2 × 0.6 - 1 = 0.20 (20% of bankroll)
- Full Kelly: high growth but high volatility
- Half Kelly: ~65% of full Kelly growth with ~1/4 the volatility

Calculation showing the "don't bet" condition:
- Fair odds (b = 1): f = 2p - 1
- For p = 0.5: f = 0 → Kelly says don't bet at all
- This makes intuitive sense: no edge at fair odds

Wealth trajectory:
Starting with $1000, half-Kelly at60%/2.00 odds:
- After 10 bets: ~$1,085 (8.5% return)
- After 100 bets: ~$2,200 (120% return — but this is theoretical, real results vary)

Notes

  • This blog is the clearest non-academic explanation of Kelly — complements the existing kelly-criterion.md note with more narrative context
  • The Bernoulli → Kelly progression is particularly well explained — useful for understanding why Kelly is the "right" answer vs. ad-hoc betting strategies
  • The post's example showing why Kelly says "don't bet" at fair odds (p=0.5, b=1 → f*=0) is a key insight for understanding when to bet
  • The risk-of-ruin calculation (33% vs 11%) is a strong argument for half-Kelly in practice
  • For the World Cup model: the volatility of Kelly with only 64 bets is underappreciated — this blog helps explain why half-Kelly is the industry standard