If you keep playing until you’ve saved everyone or go broke, each resident gets a 0.51^10 = 1 in 840 chance of survival.
That’s if you bet everything every time. If you bet 1 gold coin every time, I think you get about a 4% chance of saving everyone. (Per http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GR.pdf I think it would be 1−49/511−(49/51)1024=0.039.) I don’t think you can do better than that without fractional coins—Kelly would let you get there faster-on-average, if you start betting multiple coins when your bankroll is high enough, but I think not more reliably.
The Kelly Criterion sneaks in the assumption that you want to maximize the geometric rate of return.
So, I’m not sure what the “Kelly Criterion” even is—a criterion is a means of judging something, so is the criterion “a bet of size f is better than any other bet size”? Or is it “the best bet size is the one that causes …”? Or something else? Wikipedia says the criterion is a formula, which sounds like not-a-criterion.… Without knowing what it is, I’m not sure whether I agree it sneaks in that assumption.
But to rephrase, it sounds like you’re saying “the argument for Kelly rests on wanting to maximize geometric rate of return, and if you don’t want to do that, Kelly might not be right for you”?
I kind of disagree. To me, the biggest argument to bet Kelly is that in the sufficiently far future, with very high probability, it gives you more money than any non-Kelly bettor. Which in many situations is equivalent to maximizing geometric rate of return, but not always; and it’s much more closely connected to what people care about.
It looks like with a 60% double-or-nothing, 100 rounds isn’t enough to reach “very high probability” (eyeballing, the Kelly bettor has more somewhere between 60 and 80% of the time?); but what does the graph look like at 1,000 rounds, or 1,000,000? I’d guess that at 1,000 rounds, almost everyone would prefer the Kelly distribution over the .4 Kelly one; and I’d be quite surprised if that wasn’t the case at 1,000,000.
(In the parasite example, there is no strategy that gives you this desideratum without fractional coins. Proof: any strategy that places at least one bet has at least a 49% chance of ending up with less money than the place-no-bets strategy; but that strategy has a 51% chance of ending up with less money than the place-exactly-one-bet strategy. But also, this desideratum is different from “maximize probability of saving everyone”.)
Most people also have a preference for reducing the risk of losing much of their starting capital. Betting a fraction of the Kelly Bet gives a slightly lower rate of return but a massively lower risk of ruin.
This seems to be a nonstandard definition of “ruin”. Under the framework you seem to be using here, neither the Kelly or the fractional-Kelly bettor will ever hit 0; they’ll always be able to keep playing, with a chance of getting back their original capital and then some.
We could decide that “ruin” means having less than some amount of money. Assuming the starting capital is not-ruined, it may be that the Kelly bettor will always have a higher risk of ever becoming ruined than a fractional-Kelly bettor. (Certainly, it’ll be higher than for the 0-Kelly bettor who never bets.) It may even be—I’m not sure about this—that at some very far future time, the Kelly bettor has a higher probability of currently being ruined. But if so it’ll be a higher very small probability. And a ruined Kelly bettor will be faster on average to become un-ruined than a ruined fractional-Kelly bettor.
Practically, 0.2 to 0.5 Kelly seems to satisfy most peoples’ real-world preferences.
That’s if you bet everything every time. If you bet 1 gold coin every time, I think you get about a 4% chance of saving everyone. (Per http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GR.pdf I think it would be 1−49/511−(49/51)1024=0.039.) I don’t think you can do better than that without fractional coins—Kelly would let you get there faster-on-average, if you start betting multiple coins when your bankroll is high enough, but I think not more reliably.
So, I’m not sure what the “Kelly Criterion” even is—a criterion is a means of judging something, so is the criterion “a bet of size f is better than any other bet size”? Or is it “the best bet size is the one that causes …”? Or something else? Wikipedia says the criterion is a formula, which sounds like not-a-criterion.… Without knowing what it is, I’m not sure whether I agree it sneaks in that assumption.
But to rephrase, it sounds like you’re saying “the argument for Kelly rests on wanting to maximize geometric rate of return, and if you don’t want to do that, Kelly might not be right for you”?
I kind of disagree. To me, the biggest argument to bet Kelly is that in the sufficiently far future, with very high probability, it gives you more money than any non-Kelly bettor. Which in many situations is equivalent to maximizing geometric rate of return, but not always; and it’s much more closely connected to what people care about.
It looks like with a 60% double-or-nothing, 100 rounds isn’t enough to reach “very high probability” (eyeballing, the Kelly bettor has more somewhere between 60 and 80% of the time?); but what does the graph look like at 1,000 rounds, or 1,000,000? I’d guess that at 1,000 rounds, almost everyone would prefer the Kelly distribution over the .4 Kelly one; and I’d be quite surprised if that wasn’t the case at 1,000,000.
(In the parasite example, there is no strategy that gives you this desideratum without fractional coins. Proof: any strategy that places at least one bet has at least a 49% chance of ending up with less money than the place-no-bets strategy; but that strategy has a 51% chance of ending up with less money than the place-exactly-one-bet strategy. But also, this desideratum is different from “maximize probability of saving everyone”.)
This seems to be a nonstandard definition of “ruin”. Under the framework you seem to be using here, neither the Kelly or the fractional-Kelly bettor will ever hit 0; they’ll always be able to keep playing, with a chance of getting back their original capital and then some.
We could decide that “ruin” means having less than some amount of money. Assuming the starting capital is not-ruined, it may be that the Kelly bettor will always have a higher risk of ever becoming ruined than a fractional-Kelly bettor. (Certainly, it’ll be higher than for the 0-Kelly bettor who never bets.) It may even be—I’m not sure about this—that at some very far future time, the Kelly bettor has a higher probability of currently being ruined. But if so it’ll be a higher very small probability. And a ruined Kelly bettor will be faster on average to become un-ruined than a ruined fractional-Kelly bettor.
In what situations does it seem this way to you?