Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Ruining an expected-log-money maximizer, published by philh on August 21, 2023 on LessWrong.

Suppose you have a game where you can bet any amount of money. You have a 60% chance of doubling your stake and a 40% chance of losing it.

Consider agents Linda and Logan, and assume they both have £11. Linda has a utility function that's linear in money (and has no other terms), ULinda(m)=m. She'll bet all her money on this game. If she wins, she'll bet it again. And again, until eventually she loses and has no more money.

Logan has a utility function that's logarithmic in money, ULogan(m)=ln(m). He'll bet 20% of his bankroll every time, and his wealth will grow exponentially.

Some people take this as a reason to be Logan, not Linda. Why have a utility function that causes you to make bets that leave you eventually destitute, instead of a utility function that causes you to make bets that leave you rich?

In defense of Linda

I make three replies to this. Firstly, the utility function is not up for grabs! You should be very suspicious any time someone suggests changing how much you value something.

"Because if Linda had Logan's utility function, she'd be richer. She'd be doing better according to her current utility function." My second reply is that this is confused. Before the game begins, pick a time t. Ask Linda which distribution over wealth-at-time-t she'd prefer: the one she gets from playing her strategy, or Logan's strategy? She'll answer, hers: it has an expected wealth of £1.2t. Logan's only has an expected wealth of £1.04t.

And, at some future time, after she's gone bankrupt, ask Linda if she thinks any of her past decisions were mistakes, given what she knew at the time. She'll say no: she took the bet that maximized her expected wealth at every step, and one of them went against her, but that's life. Just think of how much money she'd have right now if it hadn't! (And nor had the next one, or the one after..) It was worth the risk.

You might ask "but what happens after the game finishes? With probability 1, Linda has no money, and Logan has infinite". But there is no after! Logan's never going to stop. You could consider various limits as t∞, but limits aren't always well-behaved2. And if you impose some stopping behavior on the game - a fixed or probabilistic round limit - then you'll find that Linda's strategy just uncontroversially gives her better payoffs (according to Linda) after the game than Logan's, when her probability of being bankrupt is only extremely close to 1.

Or, "but at some point Logan is going to be richer than Linda ever was! With probability 1, Logan will surpass Linda according to Linda's values." Yes, but you're comparing Logan's wealth at some point in time to Linda's wealth at some earlier point in time. And when Logan's wealth does surpass the amount she had when she lost it all, she can console herself with the knowledge that if she hadn't lost it all, she'd be raking it in right now. She's okay with that.

I suppose one thing you could do here is pretend you can fit infinite rounds of the game into a finite time. Then Linda has a choice to make: she can either maximize expected wealth at tn for all finite n, or she can maximize expected wealth at tω, the timestep immediately after all finite timesteps. We can wave our hands a lot and say that making her own bets would do the former and making Logan's bets would do the latter, though I don't endorse the way we're treating infinties here.

Even then, I think what we're saying is that Linda is underspecified. Suppose she's offered a loan, "I'll give you £1 now and you give me £2 in a week". Will she accept? I can imagine a Linda who'd accept and a Linda who'd reject, both of whom would still be expected-money maximizers, just taking the expectation at different times and/or expanding "mone...