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Losing with the Kelly System

Is it possible to lose everything using Kelly system betting? In most real life situations the answer is yes. If there is a limit to how small your bet can be then if you drop below the limit you are ruined. It is quite possible for this to happen even when using the Kelly system. As a simple example look at a game where you either win or lose the amount bet and the minimum bet is one dollar. If you play the game \(n\) times with \(k\) wins and \(f\) is the Kelly fraction (the fraction of your bankroll that you bet on each game) then your bankroll after \(n\) games will be:


where \(a_0\) is your starting bankroll. If \(p\) is the probability of winning then the Kelly fraction is \(2p-1\) and the equation can also be written as:

\[a_n = a_0 2^n p^k (1-p)^{n-k}\]

After \(n\) games the Kelly system says your bet for the next game should be \(a_nf\) and if this is less than 1 you can't play anymore. Using the above equation you can find that this happens when the number of wins is less than

\[k < \frac{\log{1/f}-n\log{2(1-p)}-\log{a_0}}{\log{p}-\log{1-p}} = N\]

and the probability of this happening is

\[P(k < N) = \sum_{k=0}^{N-1}\binom{n}{k}p^k(1-p)^{n-k}\]

As an example suppose the game is very favorable with \(p=3/4\) and a Kelly fraction of \(f=1/2\). So you are betting half your bankroll in each game. The smallest your initial bankroll can be in this case is \(a_0=2\). Substituting the numbers into the above equation you find that when \(k<n\log{2}/\log{3}\) = 0.63092975n the game is over. If your percentage of wins drops below 63 percent you're finished. On average you can expect to win about 75 percent of the games since \(p=0.75\) but the variance is \(0.1875n\) so the chance of getting only 63 percent is not negligible. Using the above equation you can calculate the probability of ruin exactly. The following table shows the probability for the first 10 games

n P(k < nlog(2)/log(3))
1 0.25
2 0.4375
3 0.15625
4 0.26172
5 0.36719
6 0.16943
7 0.24359
8 0.32146
9 0.16573
10 0.22412

The probability does decrease as you play more games but only very slowly. The point is that it's possible to lose almost all your money even when playing a favorable game using the Kelly system. It's not very likely but it is possible.

For more information on the Kelly system see Bet Smart: The Kelly System for Gambling and Investing.

© 2010-2012 Stefan Hollos and Richard Hollos

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