Discrete Stochastic Processes

Lesson 9 of 10

Betting Systems and the Kelly Criterion

Doubling after every loss wins a dollar almost every time and loses everything once in a while; when you do have an edge, Kelly tells you what fraction of your bankroll to bet.

The doubling staircase

Bet $1 on a fair coin. If you lose, bet $2. Lose again and bet $4, and keep doubling until the first win. If that win comes on bet kk, it pays 2k−12^{k-1} after you have lost

1+2+⋯+2k−2=2k−1−1,1 + 2 + \cdots + 2^{k-2} = 2^{k-1} - 1,

so every run of the system ends exactly $1 ahead, which looks like free money.

Now give yourself a real bankroll of $1,023. That pays for ten bets, 1+2+⋯+5121 + 2 + \cdots + 512, and no eleventh. You walk away $1 up unless the coin goes against you ten times in a row, which has probability 2−10=1/10242^{-10} = 1/1024, and then you lose all $1,023:

E[profit]=10231024⋅1−11024⋅1023=0.E[\text{profit}] = \frac{1023}{1024}\cdot 1 - \frac{1}{1024}\cdot 1023 = 0.

The system wins 99.9% of the time and its expected profit is exactly zero. It swaps a long run of small wins for one rare cliff that wipes them all out. Run it every night and the cliff stops being rare: over 1,000 nights the chance of falling off at least once is 1−(1023/1024)1000≈62%1 - (1023/1024)^{1000} \approx 62\%.