Iterated Gambles and Expected Utility Theory

Utility is approximately the logarithm of money. Pretend otherwise, and you will get results that go against the intuition, duh.

Utility is linear to money only if we take such a small part of the logarithmic curve that it is more or less linear at the given interval. But this is something you cannot extrapolate to situations where the part of the logarithmic curve is significantly curved. Two examples of linearity:

1) You are a millionaire, so you more or less don't give a fuck about getting or not getting $1000. In such case you can treat small money as linear and choose B. If you are not a millionaire, imagine that it is about certainty of 24¢ versus 25% chance of $1.

2) You are an effective altruist and you want to donate all the money to a charity that saves human lives. If $1000 is very small compared with the charity budget, we can treat the number of human lives saved as a linear function of extra money given. (See: Circular Altruism.)

You're ignoring risk aversion. Just maximising expected utility does not take it into account and humans do care about risk. Your consciousness just discovered that your gut cares :-)

So one of the major issues I've identified with why our gut feelings don't always match with good expected utility models is that we don't live in a hypothetical universe. I typically use log utility of end state wealth to judge bets where I am fairly confident of my probability distributions as per Vaniver in another comment.

But there are reasons that even this doesn't really match with our gut.

Our "gut" has evolved to like truly sure things, and we have sayings like "a bird in the hand is worth two in the bush" partly because we are not very good at mapping probability distributions, and because we can't always trust everything we are told by outside parties.

When presented with a real life monty haul bet like this, except in very strange and arbitrary circumstances, we usually have reason to be more confident of our probability map on the sure bet than on the unsure one.

If someone has the $240 in cash in their hand, and is saying that if you take option B, they will hand it you right now and you can see it, you can usually be pretty sure that if you take option B you will get the money -- there is no way they can deny you the money without it being obvious that they have plain and simply lied to you and are completely untrustworthy.

OTOH, if you take the uncertain option -- how sure can you really be that the game is fair? How will the chance be determined? The person setting up the game understands this better than you, and may know tricks they are not telling you. If the real chance is much lower than promised, how will you be able to tell? If they have no intention of paying you for a "win", how could you tell?

The more uncertainty is promised, the more uncertainty we will and should have in our trust and other unknown considerations. That's a general rule of real life bets that's summed up more perfectly than I ever could have in this famous quote from Guys and Dolls:

"One of these days in your travels, a guy is going to show you a brand new deck of cards on which the seal is not yet broken. Then this guy is going to offer to bet you that he can make the jack of spades jump out of this brand new deck of cards and squirt cider in your ear. But, son, do not accept this bet, because as sure as you stand there, you’re going to wind up with an ear full of cider."

So for these reasons, this gamble, where the difference in expected value is fairly small compared to the value of the sure win -- even though a log expected utility curve says to take the risk at almost any reasonable level of rich country wealth, unless you have a short term liquidity crunch -- I'd probably take the 240. The only situations under which I would even consider taking the best are ones where I was very confident in my estimate of the probability distribution (we're at a casino poker table and I have calculated the odds myself for example), and either already have nearly complete trust or don't require significant trust in the other bettor/game master to make the numbers work.

In the hypothetical where we can assume complete trust and knowledge of the probability distribution, then yes I take the gamble. The reason my gut doesn't like this, is because we almost never have that level of trust and knowledge in real life except in artificial circumstances.

If you were actually offered this gamble in real life, which option would you take?

That reminds me of me calculating the expected value of Ethereum around half a year ago and writing on LW about it. I was pretty certain that there's a >50% chance of 10x returns. However I was to lazy to actually go and buy Ethereum.

Was it because of risk aversion? I don't think so, it was pure lazinss.

Dollars are fungible. If you expect to sometimes trade off risk vs. expected value, then you can aggregate all such events and act as if choosing your strategy for many or all of them at once.

So even if you don't think you'll be offered this particular gamble, as long as you expect to encounter other vaguely similar decisions, you should choose more high-risk, high-reward options. Unless, as others have pointed out, you can't afford to take the financial risk in the short term.

Iteration isn't the only thing you're changing - you're also increasing the top payout and smoothing the results to reduce the probability of getting nothing. This is going to trigger your biases differently.

As you note, money does not actually translate to utility very well. For a whole lot of people, the idea of $240 feels almost as good as the idea of $1000.

It would help to play with the problem and translate it to something more linear in utility. But such things are really hard to find, almost everything gets weighted weirdly by brains. In fact, if you phrase it as "certain death for 250 people vs 24% of death of 1000" you often get different answers than "certain saving of 24 people vs 25% of saving 1000".

IMO, part of instrumental rationality is recognizing that your intuitions of mapping outcomes to utility are wrong in a lot of cases. Money really is close to linear in value for small amounts, and you are living a suboptimal life if you don't override your instincts.

The option A could be justified if you take into account emotional utility. Therefore you might trade 10$ for avoiding the regret you will feel in 75% of cases if you choose option B. This could hold even more true for larger sums.

By the way, I would choose option B because I don't think this self-indulging attitude is beneficial in the long run.

Decision 5. Which of the following options do you prefer (choose one)? A. A sure gain of $240 B. 25% chance to gain $1,000 and 75% chance to gain nothing

This is mathematically (though not necessarily psychologically) equivalent to being given $240 for free and then being offered a bet that wins $760 with 25% chance and loses $240 with 75% chance. Let's leave the former aside. As for the latter, in many circumstances I would turn it down (even if I had several tens of thousand dollars to begin with) because I would take the other party's willingness to offer me that bet as evidence that there's something that they know and I don't.

Utlity isn't dollars, so people may be not-equating dollars and utility well.

Imagine a machine that created $100 every time you (and only you, you can't hire somebody to do it for you, or give it to somebody else) push a button; more, this is a magical $100 imbued with anti-munchkin charms (such that any investments purchased never gain value, any raw materials transformed or skills purchased remain at most the same value (so research is out), and any capital machines purchased with it provoke the same effect on any raw materials they themselves process, and so on and so forth; and no, burning the money or anything else for fuel doesn't subvert the charm, or even allow you to stop the heat death of the universe). Assuming you can push it three times per second, fifteen minutes of effort buys you a really nice car. An hour buys you a nice house. A solid workday buys you a nice mansion. A week, and you could have a decent private island. A few months and you might be able to have a house on the moon - but because of the anti-munchkin charms, it won't jump-start space exploration or the space industry or anything like that.

You have, in effect, as much personal utility as you want.

There are three things about this scenario:

First, you consume utility in this process, you don't create it. The anti-munchkin charms mean you are a parasite on society; the space travel you purchase is taking resources away from some other enterprise.

Second, even without the anti-munchkin charms wrt capital goods, money is a barter token, not utility in and of itself. This machine cannot be used to make society better off directly, it merely allows you to redistribute resources in an inefficient manner. Insofar as you might be able to make society better off, that is dependent on you being about to distribute resources more efficiently than they already are - and if you can recognize global market inefficiencies with sufficient clarity to correct them, there are more productive ways for you to spend your time than pushing a button that dispenses what is relatively chump change.

Third, the first thousand times you push the button will make a substantively larger difference in your life than the second thousand times. Each iterative $100 has diminishing returns over the previous $100, even without taking the inflationary effect this will have on society into account.

Like GuySrinivasan said, it depends on the stakes. With the numbers you have, I'd take the risky bet based on the higher expected utility.

Why I should choose the risky option every time seems intuitive: even though I don't expect to face the exactly same problem very often, life in general consists of many gambles. And the sum of those decisions is going to be higher with the risky option chosen consistently every time.

tl;dr: If you have less than ~$13k saved and have only enough income to meet expenses, picking B might legitimately make you sad even if it's correct.

I'd take B every time. But it depends on your financial situation. If the stakes are small relative to my reference wealth I maximize expected dollars, no regrets, and if you can't without regrets then maybe try playing poker for a while until you can be happy with good decisions that result in bad outcomes because of randomness. You may not make that exact decision again, but you make decisions like it plenty often.

If the stakes are large relative to my reference wealth then the situation changes for two reasons. One, I probably won't have the opportunity to take bets with stakes large relative to my wealth very often. Two, change in utility is no longer approximately proportional to change in dollars. Perhaps $240 is a non-trivial amount to you? For a hypothetical person living in an average American city with $50k saved and an annual income of $50k, an additional $240 is in no way life changing, so dU(+$240) ~= 0.24 dU(+$1000) and they should pick B. But with 1000x the stakes, it's entirely possible that dU(+$240,000) >> 0.25 dU(+$1,000,000).

Another way of looking at this is investing with Kelly Criterion (spherical cow but still useful), which says if you start with $50k and no other annual income and have the opportunity to periodically pay $24x for a 25% chance at $100x, you should start by betting ~$657 a pop for maximum growth rate, which is within shouting distance of the proposed $240 - and KC is well known to be too risky for individuals without massive wealth, as people actually have to spend money during low periods. This is proportional to wealth, so the breakeven wealth before you're sad that you have to bet so much at once ($240) is, under the too-risky KC, about $18k, which means actually it's probably like $10k-$15k.

I have very little intuition how this translates if, for example, you have heaps of student load debt and are still trying to finish education in hopes of obtaining a promised-but-who-knows well paying job in a few years.