The Kelly criterion takes into account the fact that "gaining $25000 is worse than not losing $25000". A game where you have a 50/50 chance of gaining $25k or losing $25k has a negative expectation in the log domain, so per the Kelly Criterion one would not bet on this game.
You are right that "your rate of return in a given game might depend on your the amount you bet", this is actually very common. Consider a stock market: buying 1000 shares and selling them a year later will generate less than 1000x the return of buying 1 share and selling it a year later (assuming the stock goes up), because you pay more per share to buy 1000 shares and make less per share when you sell 1000 (because the share price moves as you buy/sell).
Kelly Criterion maximizes the wealth in the long-run. It doesn’t take asymmetric utility into account. It just happens to coincide with log-utility. If there is a fixed amount of bets the Kelly criterion will be suboptimal, but as the number of bets grows the optimal strategy will asymptotically reach the Kelly criterion.
It does not. It maximizes log-wealth. Maximizing expected wealth gives a strategy which results in $0 a lot of the time but much much more than Kelly occasionally, achieving a higher average wealth. Kelly gives that up, getting far less expectation of actual wealth, but far more expectation of log-wealth (which is -inf at $0 so avoids the $0 results). If you don't believe this, pick any one scenario and actually do the math, take the limits, etc.
> It does not. It maximizes log-wealth. Maximizing expected wealth gives a strategy which results in $0 a lot of the time but much much more than Kelly occasionally, achieving a higher average wealth.
Log utility function is u(w) = log(w), that’s what it is called in economics. Surely maximizing log of wealth means you are maximizing log utility function.
> Kelly gives that up, getting far less expectation of actual wealth, but far more expectation of log-wealth (which is -inf at $0 so avoids the $0 results).
That’s a crucial misunderstanding of the Kelly’s result. He doesn’t give anything up. He showed than in a nonterminating game the strategy of betting as if you have log utility will give you superior results to any other strategy in terms of long-term wealth growth. What if the game is terminating? Well, in that case you need to know the utility function of the gambler to determine a superior strategy. But in a non-terminating game it is a bit irrelevant because almost all strategies will lead to infinite utility.
(If memory serves right), for bets that either win or double, the Kelly criterion is equivalent to maximize the median wins. Which coincidentally also maximizes expected log wins.
That's an interesting connection! I've heard risk managers talk about how people put too much faith in the mean outcome and don't focus enough on the median outcome. I didn't know there's a correspondence to the Kelly criterion.
You are right that "your rate of return in a given game might depend on your the amount you bet", this is actually very common. Consider a stock market: buying 1000 shares and selling them a year later will generate less than 1000x the return of buying 1 share and selling it a year later (assuming the stock goes up), because you pay more per share to buy 1000 shares and make less per share when you sell 1000 (because the share price moves as you buy/sell).
Related, I really enjoyed this treatment of the Kelly Criterion by Thorp and highly recommend it http://www.eecs.harvard.edu/cs286r/courses/fall12/papers/Tho...