When you know the terms of the bet (what probability of winning, and what payoff is offered), the Kelly criterion spits out a fraction of your bankroll to wager. That doesn’t support the result “a poor person should want to take one side, while a rich person should want to take the other”.
So what’s going on here?
Not a correct answer: “you don’t get to choose how much to wager. The payoffs on each side are fixed, you either pay in or you don’t.” True but doesn’t solve the problem. It might be that for one party, the stakes offered are higher than the optimal amount and for the other they’re lower. It might be that one party decides they don’t want to take the bet because of that. But the parties won’t decide to take opposite sides of it.
Let’s be concrete. Stick with one bad outcome. Insurance costs $600, and there’s a 1⁄3 chance of paying out $2000. Turning this into a bet is kind of subtle. At first I assumed that meant you’re staking $600 for a 1⁄3 chance of winning… $1400? $2000? But neither of those is right.
Case by case. If you take the insurance, then 2⁄3 of the time nothing happens and you just lose $600. 1⁄3 of the time the insurance kicks in, and you’re still out $600.
If you don’t take the insurance, then 2⁄3 of the time nothing happens and you have $0. 1⁄3 of the time something goes wrong and you’re out $2000.
So the baseline is taking insurance. The bet being offered is that you can not take it. You can put up stakes of $2000 for a 2⁄3 chance of winning $600 (plus your $2000 back).
Now from the insurance company’s perspective. The baseline and the bet are swapped: if they don’t offer you insurance, then nothing happens to their bankroll. The bet is when you do take insurance.
If they offer it and you accept, then 2⁄3 of the time they’re up $600 and 1⁄3 of the time they’re out $1400. So they wager $1400 for a 2⁄3 chance of winning $600 (plus their $1400 back).
So them offering insurance and you accepting it isn’t simply modeled as “two parties taking opposite sides of a bet”.
Oh, I think that also means that section is slightly wrong. You want to take insurance if
log(Wyou−P)>plog(Wyou−c)+(1−p)log(Wyou)
and the insurance company wants to offer it if
log(Wthem)<plog(Wthem+P−c)+(1−p)log(Wthem+P).
So define
V(W)=log(W−P)−(plog(W−c)+(1−p)log(W))
as you did above. Appendix B suggests that you’d take insurance if V(Wyou)>0 and they’d offer it if V(Wthem)<0. But in fact they’d offer it if V(Wthem+P)<0.
Here’s a puzzle about this that took me a while.
When you know the terms of the bet (what probability of winning, and what payoff is offered), the Kelly criterion spits out a fraction of your bankroll to wager. That doesn’t support the result “a poor person should want to take one side, while a rich person should want to take the other”.
So what’s going on here?
Not a correct answer: “you don’t get to choose how much to wager. The payoffs on each side are fixed, you either pay in or you don’t.” True but doesn’t solve the problem. It might be that for one party, the stakes offered are higher than the optimal amount and for the other they’re lower. It might be that one party decides they don’t want to take the bet because of that. But the parties won’t decide to take opposite sides of it.
Let’s be concrete. Stick with one bad outcome. Insurance costs $600, and there’s a 1⁄3 chance of paying out $2000. Turning this into a bet is kind of subtle. At first I assumed that meant you’re staking $600 for a 1⁄3 chance of winning… $1400? $2000? But neither of those is right.
Case by case. If you take the insurance, then 2⁄3 of the time nothing happens and you just lose $600. 1⁄3 of the time the insurance kicks in, and you’re still out $600.
If you don’t take the insurance, then 2⁄3 of the time nothing happens and you have $0. 1⁄3 of the time something goes wrong and you’re out $2000.
So the baseline is taking insurance. The bet being offered is that you can not take it. You can put up stakes of $2000 for a 2⁄3 chance of winning $600 (plus your $2000 back).
Now from the insurance company’s perspective. The baseline and the bet are swapped: if they don’t offer you insurance, then nothing happens to their bankroll. The bet is when you do take insurance.
If they offer it and you accept, then 2⁄3 of the time they’re up $600 and 1⁄3 of the time they’re out $1400. So they wager $1400 for a 2⁄3 chance of winning $600 (plus their $1400 back).
So them offering insurance and you accepting it isn’t simply modeled as “two parties taking opposite sides of a bet”.
Oh, I think that also means that section is slightly wrong. You want to take insurance if
log(Wyou−P)>plog(Wyou−c)+(1−p)log(Wyou)and the insurance company wants to offer it if
log(Wthem)<plog(Wthem+P−c)+(1−p)log(Wthem+P).So define
V(W)=log(W−P)−(plog(W−c)+(1−p)log(W))as you did above. Appendix B suggests that you’d take insurance if V(Wyou)>0 and they’d offer it if V(Wthem)<0. But in fact they’d offer it if V(Wthem+P)<0.