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 13 of the time I’m going home with nothing- I’ve effectively lost that $20000 I wanted, and 13 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.