I like the Kelly criterion response, which brings in gambler’s ruin and treats a more realistic setup (limited portfolio of resources). You don’t bet on extremely unlikely payoffs, because the increased risk of going bankrupt & being unable to make any bets ever again reduces your expected utility growth.
For example, to use the KC, if someone offers me a 100,000x payoff at 1⁄90,000 for \$1 and I have \$10, then the KC says that the fraction of my money to wager is (p * (b + 1) - 1) / b:
b is the net odds received on the wager (“b to 1”); that is, you could win \$b (on top of getting back your \$1 wagered) for a \$1 bet
So I would bet extraordinarily small amounts on each such opportunity and maybe, since 1/100th of a penny is smaller than the offered bet of \$1, I would wind up not betting at all on this opportunity despite being +EV.
But as my portfolio gets bigger, or the probability of payoff gets bigger and less exotic (lowering the probability of a long enough run of bad luck to ruin a gambler), I think the KC asymptotically turns into just expected-value maximization as you become more able to approximate an ensemble which can shrug off losses; somewhat like a very large Wall Street company which has enough reserves to play the odds almost indefinitely.
Using KC, which incorporates one’s mortality, as well as a reasonable utility function on money like a log curve (since clearly money does diminish), seems like it resolves a lot of problems with expected value.
I like the Kelly criterion response, which brings in gambler’s ruin and treats a more realistic setup (limited portfolio of resources). You don’t bet on extremely unlikely payoffs, because the increased risk of going bankrupt & being unable to make any bets ever again reduces your expected utility growth.
For example, to use the KC, if someone offers me a 100,000x payoff at 1⁄90,000 for \$1 and I have \$10, then the KC says that the fraction of my money to wager is
(p * (b + 1) - 1) / b:then:
((1/90000) * ((10*100000) + 1) - 1) / (10*100000)= 1.011112222e-05,10 * 1.011112222e-05= \$0.0001011112222So I would bet extraordinarily small amounts on each such opportunity and maybe, since 1/100th of a penny is smaller than the offered bet of \$1, I would wind up not betting at all on this opportunity despite being +EV. But as my portfolio gets bigger, or the probability of payoff gets bigger and less exotic (lowering the probability of a long enough run of bad luck to ruin a gambler), I think the KC asymptotically turns into just expected-value maximization as you become more able to approximate an ensemble which can shrug off losses; somewhat like a very large Wall Street company which has enough reserves to play the odds almost indefinitely.
Using KC, which incorporates one’s mortality, as well as a reasonable utility function on money like a log curve (since clearly money does diminish), seems like it resolves a lot of problems with expected value.
Not sure who first made this objection but one paper is http://rsta.royalsocietypublishing.org/content/369/1956/4913