I’d say that “optimizing expected money is the only thing Bob cares about” is an example, not an assumption or conclusion. If you want to argue that agents should care about ergodicity regardless of their utility function, then you need to argue that to the agent whose utility function is linear in money (and has no other terms, which I assumed but didn’t state in the previous comment).
Such an agent is indifferent between a certainty of 1025 dollars, and a near-certainty of 0 dollars with a 10−67 chance of 1092 dollars. That’s simply what it means to have that utility function. If you think this agent, in the current hypothetical scenario, should bet Kelly to get ergodicity, then I think you just aren’t taking seriously what it means to have a utility function that’s linear in money.
In the limit as the number of bets goes to infinity
I spoke about limits and infinity in my conversation with Ben, my guess is it’s not worth me rehashing what I said there. Though I will add that I could make someone whose utility is log in money—i.e. someone who’d normally bet Kelly—behave similarly.
Not with quite the same setup. But I can offer them a sequence of bets such that with near-certainty (p→1 as t→∞), they’d eventually end up with $0.01 and then stop betting because they’ll under no circumstances risk going down to $0.
These bets can’t be of the form “payout is some fixed multiple of your stake and you get to choose your stake”, but I think it would work if I do “payout is exponential in your stake”. Or I could just say “minimum stake is your entire bankroll minus $0.01”—if I offer high enough payouts each time, they’ll take these bets, over and over, until they’re down to their last cent. Each time they’d prefer a smaller bet for less money, but if I’m not offering that they’d rather take the bet I am offering than not bet at all.
Also,
It’s weird to me that something of measure 0 probability can swamp the entirety of the rest of the probability.
The Dirac delta has this property too, and IIUC it’s a fairly standard tool.
Here were talking something that’s weird in a different way, and perhaps weird in a way that’s harder to deal with. But again I think that’s more because of infinity than because of utility functions that are linear in money.
I’d say that “optimizing expected money is the only thing Bob cares about” is an example, not an assumption or conclusion. If you want to argue that agents should care about ergodicity regardless of their utility function, then you need to argue that to the agent whose utility function is linear in money (and has no other terms, which I assumed but didn’t state in the previous comment).
Such an agent is indifferent between a certainty of 1025 dollars, and a near-certainty of 0 dollars with a 10−67 chance of 1092 dollars. That’s simply what it means to have that utility function. If you think this agent, in the current hypothetical scenario, should bet Kelly to get ergodicity, then I think you just aren’t taking seriously what it means to have a utility function that’s linear in money.
I spoke about limits and infinity in my conversation with Ben, my guess is it’s not worth me rehashing what I said there. Though I will add that I could make someone whose utility is log in money—i.e. someone who’d normally bet Kelly—behave similarly.
Not with quite the same setup. But I can offer them a sequence of bets such that with near-certainty (p→1 as t→∞), they’d eventually end up with $0.01 and then stop betting because they’ll under no circumstances risk going down to $0.
These bets can’t be of the form “payout is some fixed multiple of your stake and you get to choose your stake”, but I think it would work if I do “payout is exponential in your stake”. Or I could just say “minimum stake is your entire bankroll minus $0.01”—if I offer high enough payouts each time, they’ll take these bets, over and over, until they’re down to their last cent. Each time they’d prefer a smaller bet for less money, but if I’m not offering that they’d rather take the bet I am offering than not bet at all.
Also,
The Dirac delta has this property too, and IIUC it’s a fairly standard tool.
Here were talking something that’s weird in a different way, and perhaps weird in a way that’s harder to deal with. But again I think that’s more because of infinity than because of utility functions that are linear in money.