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## Main Question or Discussion Point

I'm just now learning about the Kelly betting system, and I can't wrap my mind around the result that sometimes you should bet a fraction of your money on a bet with negative expected value. Here is an application of the system when you can bet a fixed fraction on any combination of mutually exclusive outcomes, reinvesting your winnings (if any) each time:

The same four horses run a long series of races on the same track. You know from tracking past results for a long time that the win percentages for the horses are

horse 1: 28%

horse 2: 55%

horse 3: 15%

horse 4: 2%

A bookmaker offers the following odds on the horses:

horse 1 at 3:1

horse 2 at 1:1

horse 3 at 5:1

horse 4 at 7:1

If past winning percentages can be taken as the probability that a horse wins, then bets on horses 1 and 2 have positive expectation, while bets on horses 3 and 4 have negative expectation. So you would think one would bet on horse1, or maybe horses 1 and 2, avoiding 3 and 4. But according to the Kelly strategy you do best in the long run if you bet 22% of your current wealth on horse 1, 43% on horse 2, and 11% on horse 3. The remaining 24% is not bet. I simulated this strategy and it works, beating strategies where you don't bet on horse 3. I understand the mathematical derivation of the formula, but I don't understand how it could be true. It is just too counterintuitive. Any thoughts?

Also, notice that there is a "take", in the sense that the odds add up to more than 100%. So there are no cancelling bets in which you get back exactly what you bet, no matter what the outcome.

The same four horses run a long series of races on the same track. You know from tracking past results for a long time that the win percentages for the horses are

horse 1: 28%

horse 2: 55%

horse 3: 15%

horse 4: 2%

A bookmaker offers the following odds on the horses:

horse 1 at 3:1

horse 2 at 1:1

horse 3 at 5:1

horse 4 at 7:1

If past winning percentages can be taken as the probability that a horse wins, then bets on horses 1 and 2 have positive expectation, while bets on horses 3 and 4 have negative expectation. So you would think one would bet on horse1, or maybe horses 1 and 2, avoiding 3 and 4. But according to the Kelly strategy you do best in the long run if you bet 22% of your current wealth on horse 1, 43% on horse 2, and 11% on horse 3. The remaining 24% is not bet. I simulated this strategy and it works, beating strategies where you don't bet on horse 3. I understand the mathematical derivation of the formula, but I don't understand how it could be true. It is just too counterintuitive. Any thoughts?

Also, notice that there is a "take", in the sense that the odds add up to more than 100%. So there are no cancelling bets in which you get back exactly what you bet, no matter what the outcome.