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O.P. here. The Kelly criterion is a product of the logic outlined in the section about utility being the log of wealth. In short, assuming a jackpot of under

O.P. here. The Kelly criterion is a product of the logic outlined in the section about utility being the log of wealth. In short, assuming a jackpot of under $1 trillion, you'd need to have a bankroll on the order of eight digits to justify spending $2 on a ticket under the Kelly criterion. Also, the Kelly criterion doesn't exactly apply here because this is more like a one-off event than a repeated game. But a lot of the same principles apply.

nullc 13y ago
Kelly criterion has nothing to do with the marginal utility you gain from more money, it's about the risk of ruin in an iterated game. And indeed, you can say that purchasing one ticket is a one-off event... but if you buy that one, why not every one that happens when the expectation is in your favor? Thus the risk of ruin becomes interesting to consider.
- jmharvey 13y ago
  The Kelly criterion is designed to optimize the long term growth of your bankroll in an iterated game. And the strategy that optimizes your bankroll in an iterated game is the one that maximizes the expected value of the log of your bankroll after each iteration. Strictly speaking, the Kelly criterion isn't concerned with maximizing expected utility, but the math works out the same.
RockyMcNuts 13y ago
not sure I agree with you. Kelly says, if there is a positive expected value, there is an optimal amount to bet, which maximizes the growth rate of your stack per bet, if you were to bet repeatedly. If you overbet, your expected growth rate is negative... because over the long run, by the time you hit the jackpot you've lost too much of your stack to get back to even. Basically, I think Kelly says you can't just go by expected value, ie your edge, you have to look at your stack. Overbetting turns a +EV bet into a -EV bet and eventual loss of your whole stack. So 1) I don't think Kelly is a product of log utility, it just requires that you are trying to maximize your growth rate (As an aside, in many settings people are much more risk averse than log utility implies; and in many settings people don't have consistent cardinal utility, a time-inconsistent utility like prospect theory is more predictive) and 2) since you are presented with interesting bets every day, if not this particular one, Kelly generally applies, since if you don't heed it you fall victim to gambler's curse.
- bluecalm 13y ago
  Kelly bet amount just maximizes log(bankroll). Growth rate, utility, outcomes in series of bets etc. are just interpretations/uses of this fact.
  - RockyMcNuts 13y ago
    hmmh... nitpick but log utility => maximizing the growth rate, but maximizing the growth rate maximizes utility for many different utility functions, in particular all constant relative risk aversion utility functions, of which log utility is one. so preference for maximizing growth does not imply log utility, could be linear utility too. (under a CRRA utility function, a risky income stream that varies between $1 and $2 gets the same risk aversion discount as one with a similar distribution that varies between $100 and $200, see e.g. http://ocw.mit.edu/courses/economics/14-123-microeconomic-theory-iii-spring-2010/lecture-notes/MIT14_123S10_notes04.pdf http://ocw.mit.edu/courses/economics/14-123-microeconomic-th... )
- ronaldx 13y ago
  > Kelly says, if there is a positive expected value, there is an optimal amount to bet Yes, and the optimal amount to bet here is for all practical purposes equal to zero. In short, you don't have a large enough stack to keep on taking this bet until it pays you back. I tend to agree with your wider point, that you can treat life as a long-term sequence of taking risky bets and use Kelly to approximate that. However, even if you did have an equivalent opportunity to this all day every day (which is itself hard to assert), Kelly is saying you definitely shouldn't take it here. If you don't have that repeated opportunity, Kelly-type thinking just leads to an even stronger conclusion - you definitely definitely shouldn't take it. Good luck to everyone if you're playing the lottery :) Until the numbers are drawn, you still have a chance ;)
  - RockyMcNuts 13y ago
    yes, agree with you and OP! just Kelly makes the OP's advice not to play stronger, even if you get +EV, unless you're a millionaire you will overbet and go broke before you win !

nullc 13y ago

Kelly criterion has nothing to do with the marginal utility you gain from more money, it's about the risk of ruin in an iterated game. And indeed, you can say that purchasing one ticket is a one-off event... but if you buy that one, why not every one that happens when the expectation is in your favor? Thus the risk of ruin becomes interesting to consider.

jmharvey 13y ago
The Kelly criterion is designed to optimize the long term growth of your bankroll in an iterated game. And the strategy that optimizes your bankroll in an iterated game is the one that maximizes the expected value of the log of your bankroll after each iteration. Strictly speaking, the Kelly criterion isn't concerned with maximizing expected utility, but the math works out the same.

RockyMcNuts 13y ago

not sure I agree with you. Kelly says, if there is a positive expected value, there is an optimal amount to bet, which maximizes the growth rate of your stack per bet, if you were to bet repeatedly. If you overbet, your expected growth rate is negative... because over the long run, by the time you hit the jackpot you've lost too much of your stack to get back to even. Basically, I think Kelly says you can't just go by expected value, ie your edge, you have to look at your stack. Overbetting turns a +EV bet into a -EV bet and eventual loss of your whole stack. So 1) I don't think Kelly is a product of log utility, it just requires that you are trying to maximize your growth rate (As an aside, in many settings people are much more risk averse than log utility implies; and in many settings people don't have consistent cardinal utility, a time-inconsistent utility like prospect theory is more predictive) and 2) since you are presented with interesting bets every day, if not this particular one, Kelly generally applies, since if you don't heed it you fall victim to gambler's curse.

bluecalm 13y ago
Kelly bet amount just maximizes log(bankroll). Growth rate, utility, outcomes in series of bets etc. are just interpretations/uses of this fact.
- RockyMcNuts 13y ago
  hmmh... nitpick but log utility => maximizing the growth rate, but maximizing the growth rate maximizes utility for many different utility functions, in particular all constant relative risk aversion utility functions, of which log utility is one. so preference for maximizing growth does not imply log utility, could be linear utility too. (under a CRRA utility function, a risky income stream that varies between $1 and $2 gets the same risk aversion discount as one with a similar distribution that varies between $100 and $200, see e.g. http://ocw.mit.edu/courses/economics/14-123-microeconomic-theory-iii-spring-2010/lecture-notes/MIT14_123S10_notes04.pdf http://ocw.mit.edu/courses/economics/14-123-microeconomic-th... )
ronaldx 13y ago
> Kelly says, if there is a positive expected value, there is an optimal amount to bet Yes, and the optimal amount to bet here is for all practical purposes equal to zero. In short, you don't have a large enough stack to keep on taking this bet until it pays you back. I tend to agree with your wider point, that you can treat life as a long-term sequence of taking risky bets and use Kelly to approximate that. However, even if you did have an equivalent opportunity to this all day every day (which is itself hard to assert), Kelly is saying you definitely shouldn't take it here. If you don't have that repeated opportunity, Kelly-type thinking just leads to an even stronger conclusion - you definitely definitely shouldn't take it. Good luck to everyone if you're playing the lottery :) Until the numbers are drawn, you still have a chance ;)
- RockyMcNuts 13y ago
  yes, agree with you and OP! just Kelly makes the OP's advice not to play stronger, even if you get +EV, unless you're a millionaire you will overbet and go broke before you win !