I think the Kelly criterion doesn't apply as widely as people think it does. Its derivation is based on on maximising the growth rate of your fortune. But this is equivalent to assuming that money has logarithmic utility for you. If you don't value money in a logarithmic way then you shouldn't use the Kelly criterion.
Personally I feel that my utility function is sublogarithmic. If I'm just spending on myself then beyond a certain point additional money makes me absolutely no happier. Note that the usual justification of progressive taxation also assumes sublogarithmic utility. So based on this we should be more conservative than Kelly.
On the other hand, if I plan to give money to charity then my utility function is almost linear. Big charites can absorb a lot of money without becoming less effective. So in this case you should be maximally aggressive, betting everything at every opportunity.
Sometimes people say that because the Kelly criterion maximises growth rate it will be the best "in the long run" even if your utility function isn't logarithmic. But I've never seen any evidence of this. Does anybody know of a toy model where you can prove the Kelly criterion is optimal even if your utility is linear?
You are misunderstanding. Optimizing log wealth isn't because you have logarithmic utility... It's because doing so maximizes growth, which is more important that immediate return in a game of repeated investment. Kelly maximizes your long term absolute wealth in such games.
Edit: I see you last paragraph acknowledging but doubting this claim. Demonstrating an optimal strategy in a stochastic game is hard to make toy-like, because you have to somehow describe both the probability distributions and the entire range of strategies someone could take. In lieu, the end of this article does a good job of listing simple explanations of what that (provably) optimal strategy property entails: https://www.stat.berkeley.edu/~aldous/157/Papers/Good_Bad_Ke...
>Sometimes people say that because the Kelly criterion maximises growth rate it will be the best "in the long run" even if your utility function isn't logarithmic. But I've never seen any evidence of this. Does anybody know of a toy model where you can prove the Kelly criterion is optimal even if your utility is linear?
Basically it says that if you are making bets where money_(i+1) = f_i(money_i, x_i), such that your money always remains above zero, then you can apply the product form of the law of large numbers http://www.jams.or.jp/scm/contents/e-2006-6/2006-60.pdf
That means that over a long enough period any betting strategy that maximizes the geometric mean of the rates will beat any other bet with probability approaching 1. p(money(optimal strategy) > money(other strategy)) --> 1.
If your utility is monotonic (x > y implies that u(x) > u(y)) then I think this also implies that p(utility(money(optimal strategy)) > utility(money(other strategy))) --> 1.
Basically, you are eventually almost sure to have more money with this strategy than any other. If more money implies more utility, then you are eventually almost sure to have more utility with this strategy than any other.
> this also implies that p(utility(money(optimal strategy)) > utility(money(other strategy))) --> 1
That's true (at least as long as the other strategy also bets a constant proportion of wealth each turn). But it doesn't mean that Kelly is optimal. It could be that in the (increasingly unlikely) cases when the other strategy beats Kelly, the utility produced by the other strategy is much greater than that produced by Kelly. Then the other strategy could still be better overall.
Here's a simplified example of what's going on: consider the bet where you get a penny with probability 1-1/n, and otherwise you lose $(2^n). I think this bet gets worse and worse larger n gets, but the probability of having higher utility if you take the bet tends to 1.
>That's true (at least as long as the other strategy also bets a constant proportion of wealth each turn).
Does it require that assumption? I don't think it even requires identically distributed returns on each component bet. It just needs money_(i+1) = f(money_i, x_i) to have strictly positive support right? Then you can just push it into log space and apply the law of large numbers, telling you to maximize the expected log of f(money_i, x_i) wrt x_i. Nothing about fractions or constant fractions shows up in that derivation.
The usual statement of the Kelly criterion is about fractions, but the more general question is whether maximizing expected log is (eventually) optimal, which seems to only require being able to apply the law of large numbers in log space.
Imagine we started with $100 and were betting on a fair coin with fair odds. There's no edge so Kelly says to bet $0, and hence the Kelly strategy stays at $100 forever. You can give yourself a very high chance of beating the Kelly strategy if you use a Martingale strategy. Bet $0.01. If you win then you have $100.01 and you should not bet again. If you lose then bet $0.02 next time, and then $0.04 and then $0.08 doubling each time until you win. When you win you will go up to $100.01, which beats the Kelly strategy. So Martingale beats Kelly unless you go bankrupt by losing too many times in a row before getting a win, which only happens with a very small probability. So there are strategies that have a high probability of beating the Kelly criterion.
For simplicity I gave the above example in the degenerate case where the edge is zero. But I think the analogous strategy works in all cases. If you aim for just a penny over the Kelly strategy then you have a high probability of success.
Fair point. And while it's too late to edit my post, I think I found the flaw in the math.
Just because X converges in probability to x, f(X) doesn't necessarily converge in probability to f(x). If that were so, then logarithmic utility would be sound, but it isn't.
Coin flipping was an example. One must estimate the probabilities for their actual bets and run the numbers for a specific opportunity. You'll very quickly realize that getting probabilities in real-world situations is difficult. But that doesn't negate the lessons from the concept.
Not sure I follow. Unless I'm misunderstanding, wouldn't Kelly apply especially for your case? As in, Kelly is concerned with avoiding absorbing barriers and increasing bets when they're on "house money" (given a known edge, of course).
Kelly isn't about either of those things. It's a consequence of Kelly that it never hits the absorbing barrier, but lots of other strategies also have that property. What I'm saying is that among all of those strategies, Kelly isn't optimal.
Personally I feel that my utility function is sublogarithmic. If I'm just spending on myself then beyond a certain point additional money makes me absolutely no happier. Note that the usual justification of progressive taxation also assumes sublogarithmic utility. So based on this we should be more conservative than Kelly.
On the other hand, if I plan to give money to charity then my utility function is almost linear. Big charites can absorb a lot of money without becoming less effective. So in this case you should be maximally aggressive, betting everything at every opportunity.
Sometimes people say that because the Kelly criterion maximises growth rate it will be the best "in the long run" even if your utility function isn't logarithmic. But I've never seen any evidence of this. Does anybody know of a toy model where you can prove the Kelly criterion is optimal even if your utility is linear?