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Perplexed comments on Utility is unintuitive - Less Wrong
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I used to think I understood this stuff, but now jsteinhardt has me confused. Could you, or someone else familiar with economic orthodoxy, please tell me whether the following is a correct summary of the official position?
A lottery ticket offers one chance in a thousand to win a prize of $1,000,000. The ticket has an expected value of $1000. If you turn down a chance to purchase such a ticket for $900 you are said to be money risk averse.
A rational person can be money risk averse.
The "explanation" for this risk aversion in a rational person is that the person judges that money has decreasing marginal utility with wealth. That is, the person (rationally) judges that $1,000,000 is not 1000 times as good (useful) as $1000. An extra dollar means less to a rich man, than to a poor man.
This shifting relationship between money and utility can be expressed by a "utility function". For example, it may be the case for this particular rational individual that one util corresponds to $1. But $1000 corresponds to 800 utils and $1,000,000 corresponds to 640,000 utils.
And the rationality of not buying the lottery ticket can be seen by considering the transaction in utility units. The ticket costs 800 utils, but the expected utility of the ticket is only 640 utils. A rational, expected utility maximizing agent will not play this lottery.
ETA: One thing I forgot to insert at this point. How do we create a utility function for an agent? I.e. how do we know that $1,000,000 is only worth 640,000 utils to him. We do so by offering a lottery ticket paying $1,000,000 and then adjusting the odds until he is willing to pay $1 (equal to 1 util by definition) for the ticket. In this case, he buys the ticket when the odds improve to 640,000 to 1.
Now imagine a lottery paying 1,000,000 utils, again with 0.001 probability of winning. The ticket costs 900 utils. An agent who turns down the chance to buy this ticket could be called utility risk averse.
An agent who is utility risk averse is irrational. By definition. Money risk aversion can be rational, but that is explained by diminishing utility of money. There is no such thing as diminishing utility of utility.
That is my understanding of the orthodox position. Now, the question that jsteinhardt asks is whether it is not time to challenge that orthodoxy. In effect, he is asking us to change our definition of "rational". (It is obvious, of course, that humans are not always "rational" by this definition - it is even true that they have biases which make them systematically deviate from rationality, for reasons which seem reasonable to them. But this, by itself, is not reason to change our definition of "rationality".)
Recall that the way we rationalized away money risk aversion was to claim that money units become less useful as our wealth increases. Is there some rationalization which shows that utility units become less pleasing as happiness increases? Strikes me as a question worth looking into.
If we define a utility function the way you recommend (which I don't know if it's standard to do so, but it seems reasonable), then you're just not ever going to have utility risk averse individuals. By definition.
If a lottery pays 1M utils with 0.001 probability of winning, and the ticket costs 900 utils, an agent just wouldn't turn it down. If the agent did turn it down, this means that the lottery wasn't actually worth 1M utils, but less, because that's how we determine how much the lottery is worth in the first place.
It is, however, possible that the utility function is bounded and can never reach 1M utils. This, I think, may lead to some confusion here: in that case, the agent would turn down a lottery with a ticket price of 1000 and a probability of winning of 0.1%, no matter the payoff. This seems to imply that he turns down the 1M lottery, but it isn't irrational in this case.
I'm really enjoying the contrast between your comment and mine.
It's not every day that the same comment can elicit "By definition, this just can't be true of anyone" and "Yeah, I think this is true of me."
Yeah, the utility lottery is a bizarre lottery. For one thing, even if it's only conducted in monetary payoffs, both the price of the ticket and the amount of money you win depends on your overall well-being. In particular, if you're on the edge of starvation, the ticket would become close to (but not quite) free.
I can't imagine how it could be conducted in monetary payoffs, at least without a restrictive upper bound. Not only does the added utility of money decrease with scale, but you can only get so much utility out of money in a finite economy.
I'd be a bit surprised if, outside a certain range, utilons can be described as a function of money at all.
That's the issue of the usefulness of the Axiom of Independence - I believe.
You can drop that - though you are still usually left with expected utility maximisation.
Then you become a money pump.
It is the most commonly dropped axiom. Dropping it has the advantage of allowing you use the framework to model a wider range of intelligent agents - increasing the scope of the model.
What is the issue? Where, in my account, does AoI come into play? And why do you suggest that AoI only sometimes makes a difference?
My comments about independence were triggered by:
The independence axiom says "no" - I think - though it is "just" an axiom.
For the last question, if you drop axioms you are still usually left with expected utility maximisation - though it depends on exactly how much you drop at once. Maybe it will just be utility maximisation that is left - for example.
Well, for what it's worth: if I'm living my life in the 1000-utils range and am content there, and I have previously lived in the 100-utils range and it really really really sucked, I think I'd turn down that ticket.
That is to say, under some scenarios I am utility risk averse.
I'm not exactly sure where to go from there, though.
Having 1000 utils is by definition exactly 10 times better than having 100 utils. If this relationship does not hold, you are talking about something other than utils.
I don't see how that is a response to what I said, so I am probably missing your point entirely. If you're in the mood to back up some inferential steps, I might appreciate it.
Utils aren't just another form of currency. They're a currency that we've adjusted the value of so that we're exactly twice as happy with two utils than with one. Hence if the 1000 utils range is contentment, the 100 utils range can't possibly really really suck, instead it's exactly 10 times worse than contentment.
Thinking about them as a currency is in general misleading, since they're not fungible - what is worth 1 util to you isn't necessarily worth 1 util to anyone else.
I understand that part, as far as it goes.
So, by definition, if I'm content with 1000 utils, then given 100 utils I'm a tenth of content. (Cononet?) And being cononet is, again by definition, not bad enough that a 99% chance of it is something I could rationally choose to avoid, even if it meant giving up a 1% chance at being gloriously conthousandt.
Well, OK, if y'all say so. But I don't understand where those quantitative judgments are coming from.
Put another way: if the range I'm in right now is 1000 utils, and I want to estimate what range I was in during the month after my stroke, when things really really sucked... a state that I would not willingly accept a 99% chance of returning to... well, how do I estimate that?
I mean, I understand that 100 is too high a number, though I don't know how you calculated that, but what is the right number? 10? 1? -100? -10000? What equations apply?
Imagine this bet: If you win, you'll get to a point that is twice as good as the one you're at right now: 2000 utils. If you lose, you'll be at a point that sucked as much as that post-stroke month. What would the probabilty of winning have to be for you to be indifferent to this bet?
The utility of the awful state is then x in the equation 2000P(w) + x(1 - P(w)) = 1000, where P(w) is the probability of winning.
If x were 100, the bet would be worth taking if it offered odds better than 9 in 19. If you wouldn't take that bet, x is lower. On this scale, I suspect your utility for x is very, very far below 0.
OK... that question at least makes sense to me. Thank you.
Hm. Would I take, say, a 50% bet along those lines? I flip a coin, heads I have another stroke, tails things suddenly get as much better than they are now as now is better than then. Nope, I don't take that bet.
A 25% chance? Hm.
No, I don't take that bet.
A 5% chance? Hmmmmmmmm... that is tempting. And I'll probably always wonder what-if. But no, I don't think I take that bet.
A 1% chance? Yeah, OK. I probably take that bet.
So P(w) is somewhere between .01 and .05; probably closer to .01. Call it .015.
I think what I've probably just demonstrated is that I'm subject to cognitive biases involving small percentage chances... I think my mind is just rounding 1% to "close enough to zero as makes no difference."
But, OK, I guess utils can measure biased preferences as well as rational ones. All that matters is that it's my preference, right, not why it's my preference.
So, all right. 2000P(w) + x(1 - P(w)) = 1000 => 2000(.015) + x(1 - .015) = 1000 => 30 + .985x = 1000 => x=(1000-30)/.985 = ~985.
OK, cool. So my current condition is 1000 util, and my stroke condition (which really really sucks) is 985 utils.
What does that tell us?
You got your numbers flipped. P(w) is your chance of winning. You want
2000(.985) + x(1 - .985) = 1000 => 1970 + .015x = 1000 => x = (1000-1970)/.015 = -64,666.66...
That tells you that you really don't want to have another stroke. Which is hopefully unsurprising.
Ah! That makes far more sense. (That number seemed really implausible, but after triple-checking my math I shrugged my shoulders and went with it.)
OK. So, it no longer seems nearly so plausible that I'd turn down the original bet... it really does help me to have something concrete to attach these numbers to. Thanks.
And, yeah, that is profoundly unsurprising.
No, wait. Thinking about this some more, I realize I'm being goofy.
You offered me a series of bets about "twice as good as the one you're at right now: 2000 utils" vs "a point that sucked as much as that post-stroke month". I interpreted that as "I have another stroke" vs. "things suddenly get as much better than they are now as now is better than then" and evaluated those bets based on that interpretation.
But that was a false interpretation, and my results are internally inconsistent. If how-things-were-then is -64.5K, then 2000 is not as much better than they are now as now is better than then... they are merely 1/65th better. In which case I don't accept that bet, after all... a 1% chance of another stroke vs a 99% chance of a 1/65th improvement in my life is not nearly as compelling.
More generally, I accepted the initial statement that the state we labeled 2000 is "twice as good as" the state we labeled 1000, because that seemed to make sense when we were talking about numbers. But now that I'm trying to actually map those numbers to something, it's less clear to me that it makes sense.
I mean, it follows that my stroke was "-64 times worse" than how things are now, and... well, what does that even mean?
Sorry... I'm not trying to be a pedant here, I'm just trying to make sure I actually understand what we're talking about, and it's pretty clear that I don't.
Maybe, but I find it easier to fall for the opposite bias, the one known as "There's still a chance, right?"
(nods) Sadly, my succeptability to rounding very small probabilities up when I want them to be true is not inversely correlated with my succeptability to rounding very small probabilities down when I want to ignore them. Ain't motivated cognition grand?
I do find that I can subvert both of these failure modes by switching scales, though. That is, if I start thinking in "permil" rather than percent, all of a sudden a 1% chance (that is, a 10 permil chance) stops seeming quite so negligible.
So, let's say you have 1000 utils when you are offered the bet that Perplexed proposed. You have two possible choices:
You're saying that you prefer the option with an expected value of 1000 utils over the option with an expected value of 1100 utils. If we were talking about dollars, you could explain this by saying that you are risk averse, i.e. that the more dollars you have, the less you want each individual dollar. Utils are essentially a measurement of how much you want a particular outcome, so an outcome that is worth 1,000,100 utils is something you want 1000.1 times more than you want an outcome worth 1000 utils. If you don't want to take this bet, that means you don't actually want the 1,000,100 outcome enough for it to be worth 1,000,100 utils.
For the purposes of expected utility calculations, don't think of utils as a measure of happiness; think of them as a measure of the strength of preferences.
(shrug) Sure. As long as I don't try to understand utils as actually existing in the world, as being interpretable as anything, and I just treat them as numbers, then sure, I can do basic math as well as the next person.
And if all we're saying is that it's irrational to think that (.999 * 100) + (.001 * 1,000,100) <> 1100, well, OK, I agree with that, and I withdraw my earlier assertion about being utility risk averse. I agree with everyone else; that's pretty much impossible.
My difficulty arises when I try to unpack those numbers into something real... anything real. But it doesn't sound like that's actually part of the exercise.