I suggest explicitly stepping outside of an expected-utility framework here.
EV seems fine. You just need to treat it as the multi-stage decision problem it is, and solve the MDP/POMDP. One of the points of my Kelly coin-flip exercises is that the longer the horizon, and the closer you are to the median path, the more Kelly-like optimal decisions look, but the optimal choices looks very unKelly-like as you approach boundaries like the winnings cap (you 'coast in', betting much less than the naive Kelly calculation would suggest, to 'lock in' hitting the cap) or you are far behind when you start to run out of turns (since you won't lose much if you go bankrupt and the opportunity cost decreases the closer you get to the end of the game, the more greedy +EV maximization is optimal so you can extract as much as possible, so you engage in wild 'overbetting' from the KC perspective, which is unaware the game is about to end).
Ah, yeah, this looks pretty close to what I was looking for.
OK, so if I'm understanding correctly, the basic idea is EV maximization with a cap on total possible winnings? (Which makes sense -- there's only ever so much money to win.)
So is the claim that this approaches Kelly in the limit of simultaneously increasing cap and horizon?
The Kelly criterion, as a bet-sizing optimum, makes a few assumptions, which are not true in most humans.
It's a little unclear whether the log utility is an assumption or a result of the bankrupcy-is-death assumption. The original paper, http://www.herrold.com/brokerage/kelly.pdf , says:
The gambler introduced here follows an essentially different criterion from the classical gambler. At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with 926 the bell system technical journal, july 1956 the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies. Suppose the situation were different; for example, suppose the gambler’s wife allowed him to bet one dollar each week but not to reinvest his winnings. He should then maximize his expectation (expected value of capital) on each bet. He would bet all his available capital (one dollar) on the event yielding the highest expectation. With probability one he would get ahead of anyone dividing his money differently.
This all implies that the special-case is the last wager you will ever make. And from there the more complicated cases of the penultimate wager and the probabilistic-finite cases. I don't know how big the chain needs to get to converge to Kelly being the optimum, but since it's compatible with logarithmic utility of money in the first place, for some agents it'll be the same regardless.
My strong suspicion is that Kelly always applies if your terminal utility function for money is logarithmic. But I don't see how that could be - the marginal amount of money/resources you'll control at death is tiny compared to all resources in the universe, so your utility for any margin under consideration should be close to linear.
My strong suspicion is that Kelly always applies if your terminal utility function for money is logarithmic. But I don't see how that could be - the marginal amount of money/resources you'll control at death is tiny compared to all resources in the universe, so your utility for any margin under consideration should be close to linear.
If the amount is tiny, and your utility is log resources, then that puts us close to the origin, where the derivative of the logarithm is very high, and reducing very quickly.
But logarithm can still look nearly linear if the differences we can make are sufficiently small in relation to the total.
As far as I can tell, the fact that you only ever control a very small proportion of the total wealth in the universe isn't something we need to consider here.
No matter what your wealth is, someone with log utility will treat a prospect of doubling their money to be exactly as good as it would be bad to have their wealth cut in half, right?
makes a few assumptions, which are not true in most humans
Right, it would be interesting to take a Kelly-like approach while relaxing those assumptions.
No, the number of iterations is irrelevant. You can derive Kelly by trying to maximize your expected log wealth for a single bet. If you care about wealth instead of log wealth, then just bet the house every opportunity you get.
A bigger issue with Kelly is that it doesn't account for future income and debt streams. There should be an easy fix for that, but I need to think a bit.
It's important that we can derive Kelly that way, but if that were the only derivation, it would not be so interesting. It begs the question: why log wealth?
The derivation that does something interesting to pin down Kelly in particular is the one where we take the limit in iterations.
Several prediction markets have recently offered a bet at around 62¢ which superforecasters assign around 85% probability. This resulted in a rare temptation for me to Kelly bet. Calculating the Kelly formula, I found that I was supposed to put 60% of my bankroll on this.
Is this assuming you take the 85% number directly as your credence?
On my post weird things about money, Bunthut made the following comment:
I think the interesting question is what to do when you expect many more, but only finitely many rounds. It seems like Kelly should somehow gradually transition, until it recommends normal utility maximization in the case of only a single round happening ever. Log utility doesn't do this. I'm not sure I have anything that does though, so maybe it's unfair to ask it from you, but still it seems like a core part of the idea, that the Kelly strategy comes from the compounding, is lost.
Kelly betting seems somehow less justified when we're not doing it a bunch of times. If I were making bets left and right, I would feel more inclined to use Kelly; I could visualize the growth-maximizing behavior, and know that if I trusted my own probability assessments, I'd see that growth curve with high probability.
Several prediction markets have recently offered a bet at around 62¢ which superforecasters assign around 85% probability. This resulted in a rare temptation for me to Kelly bet. Calculating the Kelly formula, I found that I was supposed to put 60% of my bankroll on this.
Now, 60% of my savings seems like a lot. But am I really more risk-averse than Kelly? It struck me that if I were to do this sort of thing all the time, I would feel more strongly justified using Kelly.
Bunthut is suggesting a generalization of Kelly which accounts for the number of times we expect to iterate investment. In Bunthut's suggestion, we would get less risk-averse as number of iterations dropped, approaching expectation maximization. This would reflect the idea that the Kelly criterion arises because of long-term performance over many iterations, and normal expectation maximization is the right thing to do in single-shot scenarios.
But I sort of suspect this "origin of Kelly" story is wrong. So I'm also interested in number-iteration formulas which reach different conclusions.
The obvious route is to modulate Kelly by the probability that the result will be close to the median case. With arbitrarily many iterations, we are virtually certain that the fraction of bets which pay out approaches their probabilities of paying out, which is the classic argument in favor of Kelly. But with less iterations, we are less sure. So, how might one use that to modulate betting behavior?
I suggest explicitly stepping outside of an expected-utility framework here. The classic justification for Kelly is very divorced from expected utility, so I doubt you're going to find a really appealing generalization via an expected-utility route.