Quote:
Originally Posted by Ganchrow
Kelly utility is the utility function that both implies and is implied by the conclusions of John Kelly's paper. While it's true that Kelly himself never appealed to the notion of utility per se, he did however make the a priori assumption that an agent acts to maximize the expected rate of growth of bankroll. This both implies and is implied by log utility. You've fully agreed with this.
But the point (and I believe I addressed this in my earlier post) is that losing one's entire bankroll needs to be infinitely bad for the Kelly conclusions to have force of logic. Why else would a player with a readily replenishable bankroll never choose to invest 100% of said bankroll (minimally defined) unless said bet won with certainty? Why else would a player choose inaction over a 99.9999% probability of multiplying his bankroll 50-fold with a 0.0001% probability of losing everything? Indeed why else would a player actually be willing to forfeit half his bankroll to avoid making such a monstrously +EV bet? Answer? Because a lost bankroll is infinitely bad.
A player can define his goals anyway he sees fit. But we should still be able analyze these goals using traditional utility theory and determine what preferences a player would need to have for them to be rational. To this end, I'm still waiting to hear of a nonpathological example of how a player's preferences would need to look for partial wealth Kelly to be rational.
If a player's goal were solely to maximize some subset of his bankroll for no reason other than "that's what he wanted to do" then sure -- partial wealth Kelly would work. But without making that rather contrived assumption -- I just don't see how one reaches partial wealth any other way.
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Well, I will concede that logarithmic utility breaks down at the extremes. Much of economic theory does. But I'm still frustrated, because I don't think I've conveyed the
idea behind what I'm trying to say yet.
Let's say a man is married to an absolute tyrant. (Don't we all know at least one man like this?) And let's say the tyrant is aware that the man has an inclination to gamble, and that the man is actually a quite skilled gambler. And the tryant says to the man "You may have $100 to gamble, but that's it. Ever. Even you collect cans and redeem them for the deposit, you're not throwing that money away on gambling. If you lose your $100, you never gamble again. Period". And let's say the man values money, but he values domestic tranquility far more, and is a coward, so he is unwilling to disobey the tyrant's orders, no matter the financial gain. And let's say he has ethical problems with murdering her or divorcing her or even gambling behind her back (he's very worried about his eternal soul). So again, no financial incentive could ever motivate him to do these things.
So he puts his $100 into an online sportsbook (matchbook would be a pretty good plan, given their lack of minimums). He analyzes his advantage. And now he has to decide how much to bet. He has 10,000 dollars in his checking account. What is rational behavior for this man? How should he size his bets?
Everyone's real world preferences extend far beyond money. No univariate utility function can even begin to address the following question:
How much shall I use to gamble, and how much shall I use on other things?