The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market

Author: Edward O. Thorp Type: Handbook chapter (Elsevier, 2006; conference paper 1997, revised 2005) Raw: raw/THE KELLY CRITERION IN BLACKJACK SPORTS BETTING,2006-thorp (ingested).pdf

Thorp's definitive practitioner-mathematician treatment of Kelly sizing: derivation, ruin math, casino applications, a successful sports-betting system, and continuous-time portfolio Kelly.

The Central Problem

Once you have positive expectation, the remaining question is how much to bet. Two naive answers both fail:

Bold (maximize arithmetic expectation): bet everything each round → ruin is almost certain even with 51/49 odds.
Timid (minimize ruin probability): bet the minimum → survive but starve growth.

Kelly chooses the intermediate: maximize expected log wealth — equivalent to maximizing the long-run geometric growth rate (g(f)).

Coin-Toss Derivation

Even-money biased coin: win prob (p), lose prob (q = 1 - p). Bet fixed fraction (f) of bankroll each trial.

g(f) = p \log(1 + f) + q \log(1 - f), \qquad f^* = p - q \quad \text{(even money)}

For (p = 0.51): (f^* = 0.02) — risk 2% per flip. Proportional betting means technical "ruin" (hitting exactly zero) has probability 0; wealth can still approach arbitrarily small values under bad sequences.

Why Log Utility

Daniel Bernoulli used (\log x) on the St. Petersburg paradox. John Kelly (1956) showed remarkable asymptotic properties. Claude Shannon brought the paper to Thorp in 1960 while Thorp was building blackjack card-counting theory — Thorp then applied Kelly in Beat the Dealer (1962).

Key Asymptotic Properties (Kelly vs Other Strategies)

Kelly maximizes asymptotic growth rate.
Kelly minimizes expected time to reach large goals.
Kelly bettor's wealth eventually surpasses any essentially different fixed-fraction strategy with probability → 1.
But for finite (n), Kelly can lose to other strategies — the long run takes a long time.

Drawdowns and Fractional Kelly

Thorp reports cautious Kelly users find frequency of substantial bankroll reduction uncomfortably large at full Kelly. Moderate underbetting:

reduces drawdown discomfort
provides margin if edge is overestimated
sacrifices relatively little growth (half-Kelly keeps ~75% of max growth rate with ~50% of variance)

Section 7.3: if estimated edge (m_e) exceeds true edge (m_t), full Kelly can produce g ≤ 0 (wild oscillations or ruin). Assume true edge is below estimate; choose (f < f^*_e).

Half-Kelly trade: probability of doubling before halving rises from 2/3 to 8/9; probability of halving starting capital drops from 1/2 to 1/8 — in exchange for giving up ~25% of growth rate.

Blackjack

With even-money independent positive-expectation hands: bet fraction ≈ expectation. Real play adjusts down for: negative-expectation waiting bets, payoffs >1:1, multiple simultaneous hands, estimation error.

Sports Betting (1994 System)

Thorp describes a 101-day baseball betting system (first ~4.5 months of 1994) using Kelly-style sizing on identified edges — empirical proof the framework works outside cards.

Stock Market / Continuous Approximation

For continuous compounding with expected excess return (m - r) and variance (s^2):

f^* = \frac{m - r}{s^2}, \qquad g_\infty(f^*) = \frac{(m - r)^2}{2s^2} + r

Kelly investor dynamically reallocates as forecasts of (m), (r), (s) change. With leverage and taxes, required fractions shift — high tax rates amplify apparent optimal leverage in theory but practical leverage limits bite.

S&P rough example ((m=.11, s=.15, r=.06)): full Kelly (f^* \approx 2.2) — most investors use unlevered or fractional equivalents.

Overbetting Warning

Overbetting is always harmful — worse than underbetting. Combining overestimated edge with (f > f^*) can produce negative growth and ruin despite positive true edge.

Connections

Sources

raw/THE KELLY CRITERION IN BLACKJACK SPORTS BETTING,2006-thorp (ingested).pdf