Kelly betting

Slide the bet fraction f. Twenty gamblers each start with 1 and bet the fraction f of whatever they have on a coin that comes up heads with probability p = 0.6, paying even money. The left graph shows their wealth over 1,000 rounds on a log scale, with the median gambler in bold. The right graph is the long-run growth rate for every f at once. Find the f that makes the median climb fastest, then push f past 0.5 and watch a favourable game ruin everyone. The same coin flips are reused as you slide, so only your bet size changes.

bet fraction f 0.10 P(heads) p 0.60

Wealth of 20 gamblers, 1,000 rounds

each gamblermediantheory: e^g(f)·t

Growth rate per round, g(f)

g(f) = p ln(1+f) + (1−p) ln(1−f)median gambler's actual rate

What's going on

Betting a fixed fraction f means your wealth is multiplied by 1 + f on a win and 1 − f on a loss. After many rounds the order of wins and losses does not matter, only how many of each: wealth = (1 + f)^wins(1 − f)^losses. Take logs and divide by the number of rounds, and the law of large numbers hands you the growth rate g(f) = p ln(1 + f) + (1 − p) ln(1 − f). Wealth grows like e^g·t, so the sign of g decides everything: positive and you get rich, negative and you go broke, no matter how favourable the coin is.

Differentiate: g′(f) = p/(1 + f) − (1 − p)/(1 − f), which is zero at f* = 2p − 1. For p = 0.6 that is f* = 0.2, the Kelly fraction: bet your edge. The growth rate turns negative a little before 2f*. For p = 0.6, g drops below zero at about f = 0.39, not 0.4. The rule of thumb "above twice Kelly you shrink" is exact only for tiny edges. Betting more than Kelly does not just add risk, it lowers your typical growth, because the logarithm punishes big losses more than it rewards big wins.

Why the median and not the mean? The expected wealth is largest when you bet everything every time. The one gambler who wins all 1,000 rounds has 2¹⁰⁰⁰ times their stake. That happens with probability 0.6¹⁰⁰⁰, so the expected wealth is 2¹⁰⁰⁰ × 0.6¹⁰⁰⁰ = 1.2¹⁰⁰⁰. Every other gambler is at zero. Expected value is dominated by a miracle nobody will live to see. The median gambler, and in fact almost every gambler, experiences the growth rate g(f) instead. The orange theory line on the left is e^g(f)·t, and the bold median hugs it.

Sheet: How much to bet. Related demos: Galton board and growth and compounding.