I first came across the Kelly criterion in a paper by Haghani & Dewey where they give contestants a coin with a 60% chance to show heads, £25 and 300 flips to maximise their money. Well when I first read the question my performance would have likely been about as good as the average player, which wasn’t very good. In comes the Kelly criterion, this is basically a way to bet optimally given that you know the odds of winning. (F) is the fraction of your endowment you should bet and (p) is the probability fo winning. The formula is incredibly complex so get ready for it.
Yep, that’s all there is to it. So for the game in the paper you should bet (2*0.6)-1=0.2 or 20% of whatever you are holding every go. Now there are a few caveats to this. The first one is that the formula assumes you are risk neutral. Which is basically saying that given two bets you will always go for the one with the highest expected payoff regardless of the spread of how much you could win/lose. So the formula doesn’t actually maximise the value of a gamble for most people who would want to bet a lower amount per go, but really that doesn’t seem like such a problem when just admiring the formula. The other caveat is that the formula above is for even bets, ones where you stand to win as much as you lose per go. If we want to account for this we use:
Where (R) is the ratio of amount you win when you win vs the amount you lose when you lose ( if you win twice as much as you lose this is 2 ). A few last remarks about the formula and how it is used. The Kelly criterion pops up a bit in Edward Thorp’s autobiography and one comment seemed, at least to me, to be interesting. He went to Las Vegas with the backing of two guys to go test out his card counting system and they gave him a $100,000 or so bank roll. You may think this lends itself perfectly to the formula and in almost every way it does, however, Edward Thorp realised that given the situation he was in it was not just the maths but the men that mattered. He asked them what they would likely do if when they got to the casino he lost $95,000 leaving them only $5,000. They, like many others I’m sure, said they would probably cut their losses and pull out then to which Dr Thorp mentioned that that meant he in fact didn’t have a $100,000 pool to draw from but rather a $95,000 or even smaller amount and this would be crucial for getting his calculations correct. The idea that when you are given a range to spend it is always useful to figure out the ‘true’ range is something I think is worth keeping in mind.