The original justification for the Kelly criterion isn’t that it maximizes a utility function that’s logarithmic in wealth, but that it provides a strategy that, in the infinite limit, does better than any other strategy with probability 1. This doesn’t mean that it maximizes expected utility (as your examples for linear utility show), but it’s not obvious to me that the attractiveness of this property comes mainly from assigning infinite negative value to zero wealth, or that using the Kelly criterion is a similar error to the one Weitzman made.
Yes, if we have large populations of “all-in bettors” and Kelly bettors, then as the number of bets increase the all-in bettors lead in total wealth increases exponentially, while the probability of an all-in bettor being ahead of a Kelly bettor falls exponentially. And as you go to infinity the wealth multiplier of the all-in bettors goes to infinity, while the probability of an all-in bettor leading a Kelly bettor goes to zero. And that was the originally cited reasoning.
Now, one might be confused by the “beats any other constant bankroll allocation (but see the bottom paragraph) with probability 1” and think that it implies “bettors with this strategy will make more money on average than those using other strategies,” as it would in a finite case if every bettor using one strategy did better than any bettor using any other strategy.
But absent that confusion, why favor probability of being ahead over wealth unless one has an appropriate utility function? One route is log utility (for which Kelly is optimal), and I argued against it as psychologically unrealistic, but I agree there are others. Bounded utility functions would also prefer the Kelly outcome to the all-in outcome in the infinite limit, and are more plausible than log utility.
Also, consider strategies that don’t allocate a constant proportion in every bet, e.g. first do an all-in bet, then switch to Kelly. If the first bet has a 60% chance of tripling wealth and a 40% chance of losing everything, then the average, total, and median wealth of these mixed-strategy bettors will beat the Kelly bettors for any number of bets in a big population. These don’t necessarily come to mind when people hear loose descriptions of Kelly.
The original justification for the Kelly criterion isn’t that it maximizes a utility function that’s logarithmic in wealth, but that it provides a strategy that, in the infinite limit, does better than any other strategy with probability 1. This doesn’t mean that it maximizes expected utility (as your examples for linear utility show), but it’s not obvious to me that the attractiveness of this property comes mainly from assigning infinite negative value to zero wealth, or that using the Kelly criterion is a similar error to the one Weitzman made.
Yes, if we have large populations of “all-in bettors” and Kelly bettors, then as the number of bets increase the all-in bettors lead in total wealth increases exponentially, while the probability of an all-in bettor being ahead of a Kelly bettor falls exponentially. And as you go to infinity the wealth multiplier of the all-in bettors goes to infinity, while the probability of an all-in bettor leading a Kelly bettor goes to zero. And that was the originally cited reasoning.
Now, one might be confused by the “beats any other constant bankroll allocation (but see the bottom paragraph) with probability 1” and think that it implies “bettors with this strategy will make more money on average than those using other strategies,” as it would in a finite case if every bettor using one strategy did better than any bettor using any other strategy.
But absent that confusion, why favor probability of being ahead over wealth unless one has an appropriate utility function? One route is log utility (for which Kelly is optimal), and I argued against it as psychologically unrealistic, but I agree there are others. Bounded utility functions would also prefer the Kelly outcome to the all-in outcome in the infinite limit, and are more plausible than log utility.
Also, consider strategies that don’t allocate a constant proportion in every bet, e.g. first do an all-in bet, then switch to Kelly. If the first bet has a 60% chance of tripling wealth and a 40% chance of losing everything, then the average, total, and median wealth of these mixed-strategy bettors will beat the Kelly bettors for any number of bets in a big population. These don’t necessarily come to mind when people hear loose descriptions of Kelly.
Sure, I don’t see anything here to disagree with.