On maximising expected value
One thing that tends to bug me about discourse round here is the notion that maximising one’s expectation is the be all and end all of decisions one should make. Given any problem, one should look at it, and pick the course that maximising one’s expectation. This usually ignores two major problems: what if my utility is non-linear, and what about risk aversion?
Let’s take an example: I bump into Omega, who offers me a choice: I can take a certain 1 unit of utility, or have a 1 in 10 million chance of getting 1 billion utility. The naive expectation maximiser will take that chance: after all, their expectation will be 100 units of utility, which is much better than a measly one! In all likelihood, our maximiser will walk away with nothing. It’s certainly true that if this is repeated enough then we would expect our maximiser to be doing better… but a simple calculation reveals that it will have to occur around 7 million times for our maximiser to have a greater than 0.5 chance of having actually won (once or more times).
This is a problem with tiny probabilities and large utilities: some justifications of cryonics have run along the lines of a Pascal’s wager, where a small monetary investment gives one a massive utility, so large, in fact, that no matter how small the probability of cryonics it makes sense to invest. But if the probability becomes small enough, then I’ve probably just wasted a lot of money. After all, we only get to live once (Note that I am aware that some people have much higher probability estimates for cryonics, which is fine: I’m addressing those who do not).
Without multiple repetition, risk aversion is, I would argue, an extremely sensible strategy for utility maximisation. Of course if one believes that one will be faced with a similar choice multiple times, then one can revert back to utility maximisation. As to when one should do this, I would probably encourage one to revert when the number of repetitions, N, is large enough so that the probability of an event occurring at least once has passed some threshold, p, decided by the user. Certainly p should probably be higher than 0.5.
Lets now take another example: I am on Deal or No Deal, and there are three boxes left: $100000, $25000 and $.01. The banker has just given me a deal of $20000 (no doubt to much audience booing). Should I take that? Expected gains maximisation says certainly not! After all my expectation is more than double that offer! Risk aversion could be applied here, but I’ve actually got a good chance (0.66) of beating the bankers offer, so maybe its worth it? Except… well if I had $20000 dollars there’s quite a lot I could do with that- perhaps its enough to get a mortgage on a house, or pay for that dream holiday I’ve always wanted. Sure, $100000 would be nice, but 1⁄3 of the time I’m going home with nothing- I’ve effectively lost that $20000 I wanted, and 1⁄3 of the time I’m only getting $5000 more, which isn’t going to make a huge difference to me.
Different amounts of money are valued very differently. The first million one earns will be quite a bit more important than the second million, and so on. Again, this is a perfectly reasonable criteria to have: the first amount of money lets us pay for things we need, the second for things we want, for a crude comparison. Yes, the banker is going to offer us below our expected gains, but his expectation is based on us valuing totals all the same. If that first $20,000 is what I really want, the utility of higher sums may be much smaller than one might consider. So again, naively maximising expectation could leave one disappointed.
Obviously defining one’s nonlinearity may be difficult. One of the advantages of trying to work out ones expected utility is it allows us to overwrite our brains, which don’t necessarily always think very clearly about our expected gains, and allow us to do much better overall. But if we don’t define our function carefully enough, then we are cheating ourselves. While I am not claiming that instinct is always correct about what will make us happy in the long run, using to simple a method to try and overwrite ourselves will not help.
If you value safety, you should simply include that in the utilify function you are considering.
That is if you equate money with utility. However, these things have different names for a good reason. Look into diminishing marginal utility for more details.
Your post shows that you don’t understand what you are criticising very well. Perhaps try harder to find a sympathetic interpretation.
I think you are missing a “don’t” there.
Which would give you a utility function where U(10% chance of 10 utiles) < 1 utile
Utility functions usually map from expected future states to utilities. You seem to be doing something else—since you have utilities in the arguments to the utility function. Put some dollars in there instead and you get:
...which is absolutely fine and correctly represents risk aversion.
No, though I was using 10 utiles as shorthand for “an event that, were it to occur, would give you 10 utiles”. So without that shorthand it would be something like:
Let A and B be two future states and assume without loss of generality that U(A) = 0 utiles and U(B) = 10 utiles. Then if U(10% chance of B, 90% chance of A) < 1 utile.
But that would have been ugly in the context.
This could be the same utility function that I am talking about, but it could also be one of a risk neutral agent with a diminishing marginal utility for money.
Those are intimately-linked concepts, as I understand it:
http://en.wikipedia.org/wiki/Marginal_utility#Revival
The Deal Or No Deal example is misleading if you are actually interested in discussing expected utility.
Dollars are not utility, a point which you usually seem familiar with except in producing your (unintentional, I hope) straw man:
Naively maximizing expectation of what? You are naively maximizing expectation of dollars—this is not the same thing as naively maximizing expectation of utils. Concerns have been raised elsewhere, which I have not had a chance to look at in sufficient detail, about the latter—but your objection here clearly does not apply.
And the naïve expectation maximiser would make a correct decision. Billion utils are so great that they are worth spending one util even against such astronomical odds. In most sensible approaches this is how utilities are defined: A has n-times greater utility than B iff you are considering certain B equally valuable as a gamble with 1/n chance of getting A.
It probably seems wrong to you because you are unable to imagine how great billion utils are, or because you round the tiny probability to zero. It is easy to commit such a fallacy—it’s hard to imagine two things that differ in value billion times, and on the other hand quite easy to subconsciously conflate utilities with money, even if you know that their relation is non-linear (you are explicitly conflating utils and dollars in the second example). Having billion dollars is hardly much better than having hundred thousand dollars, so it would be silly to bet a hundred thousand against a billion with 1:10,000 odds of winning. But this is not true for utils.
Even without conflating utilities with money, it is difficult to imagine such a huge difference. The reasons are: first, our imagination of utilities is bounded (and some say that so is the utility function), second, our intuitive utility detection has finite resolution, and third, our probability imagination has finite resolution too. Now when I read the described scenario, my intuition translates “billion utils” to “the best thing I can imagine” (which is, for most people, something like having a great family and a lot of money and friends and a nice job), “one util” to “the least valuable non-zero gain” (say eating a small piece of chocolate) and perhaps even “chance 1 in 10,000,000” to “effectively zero”. Now it becomes “would you refrain from eating the chocolate for an effectively zero increase in chance of getting a really great family and a lot of money”, where the reasonable answer is of course “no”. Even without rounding the probabilities to zero it is unlikely that the best imaginable thing has ten million (or even billion) times greater utility than the smallest detectable utility amount; that would need us to be able to measure our utilities to 8 (or even 12) significant digits, which is clearly not the case.
It may be helpful to realise that some people, namely the lottery players, make similar rounding error with opposite consequences. A lottery player’s feelings translate “1:100,000,000 chance of winning” to the lowest imaginable non-zero probability, something like “perhaps once in life” or “1:1,000” and the player goes to buy the ticket.
http://en.wikipedia.org/wiki/Expected_utility_hypothesis#Expected_value_and_choice_under_risk—“In the presence of risky outcomes, a decision maker could use the expected value criterion as a rule of choice: higher expected value investments are simply the preferred ones. For example, suppose there is a gamble in which the probability of getting a $100 payment is 1 in 80 and the alternative, and far more likely, outcome, is getting nothing. Then the expected value of this gamble is $1.25. Given the choice between this gamble and a guaranteed payment of $1, by this simple expected value theory people would choose the $100-or-nothing gamble. However, under expected utility theory, some people would be risk averse enough to prefer the sure thing, even though it has a lower expected value, while other less risk averse people would still choose the riskier, higher-mean gamble.”
Also,
Realistic examples can make things easier to think about. Given the choice between getting a dollar for sure, or a 1 in 10 million chance of getting a guaranteed cure for cancer, which do you choose?
I deliberately do not use money here, because of confusions over non-linearity. I dislike your example because there, are, for me, qualitative differences between a cure for cancer and some amount of money. I was trying to make my example as non-emotive as possible.
IME it’s a lot easier to make these estimations if I calibrate my utils. Otherwise I’m just tossing labels around without ever dereferencing them.
If I assume, somewhat arbitrarily, that “1 unit of utility” is a just-noticeable utility difference at my current average utility… and I try to imagine what “1 billion utility” might actually be like, I have real trouble coming up with anything about which I don’t have strong emotions.
This isn’t terribly surprising, since emotions are tied pretty closely to value judgments in my brain.
Is it different for you?
You’re confusing expected outcome and expected utility. Nobody thinks you should maximize the utility of the expected outcome; rather you should maximize the expected utility of the outcome.
Yes, and expected gains maximization, which nobody advocates, is stupid, unlike expected utility maximization, which will take into account the fact that your utility function is probably not linear on money.
Are you sure no-one advocates it? Because I’ve observed people doing it more than once.
Can you give examples?
Google seems to be blissfully unaware of expected gains maximisation.
The big problem with risk aversion is that it violates translation invariance. If someone will give me $2 if a coin flip lands heads, I might go “well, I’m risk averse, so I want to avoid getting nothing, which means I would only pay like $.5 for this bet.” But if I’ve got even a couple dollars in my bank account, what does that do to the bet? Now it’s $100 vs. $102, and so where’s the risk to avert? I’ll pay $ .98!
The way to resolve this asymmetry with things like money or candy bars is to say “well, we value the first candy bar more than the 400th. I’d rather have a certainty of 1 candy bar than a 1 in 400 chance of 400”—that is, our “wanting” becomes a nonlinear function of the amount of stuff. But utility is, by its definition, a unit with such nonlinearities removed. If you only can really eat 3 candy bars, then your utility as a function of candy bars will reflect this perfectly—it will increase from one through 3 and then remain constant. Similarly, if you don’t really want $100,000 more than you want $20,000, utility can reflect this too, by increasing steeply at first and then leveling off. Utility is what you get after you’ve taken that stuff into account.
When nonlinearities are accounted for and solved away, there’s nothing that breaks translational symmetry—there’s no landmarks to be risk-averse relative to. It’s a bit difficult to grok, I know.