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alyssavance comments on Extreme risks: when not to use expected utility - Less Wrong
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"To consider an analogous situation, imagine having to choose between a project that gave one util to each person on the planet, and one that handed slightly over twelve billion utils to a randomly chosen human and took away one util from everyone else. If there were trillions of such projects, then it wouldn’t matter what option you chose. But if you only had one shot, it would be peculiar to argue that there are no rational grounds to prefer one over the other, simply because the trillion-iterated versions are identical."
That's not the way expected utility works. Utility is simply a way of assigning numbers to our preferences; states with bigger numbers are better than states with smaller numbers by definition. If outcome A has six billion plus a few utilons, and outcome B has six billion plus a few utilons, then, under whichever utility function we're using, we are indifferent between A and B by definition. If we are not indifferent between A and B, then we must be using a different utility function.
To take one example, suppose we were faced with the choice between A, giving one dollar's worth of goods to every person in the world, or B, taking one dollar's worth of goods from every person in the world, and handing thirteen billion dollar's worth of goods to one randomly chosen person. The amount of goods in the world is the same in both cases. However, if I prefer A to B, then U(A) must be larger than U(B), as this is just a different way of saying the exact same thing.
Now, if each person has a different utility function, and we must find a way to aggregate them, that is indeed an interesting problem. However, in that case, one must be careful to refer to the utility function of persons A, B, C, etc., rather than just saying "utility", as this is an exceedingly easy way to get confused.
To take one example, suppose we were faced with the choice between A, giving one dollar's worth of goods to every person in the world, or B, taking one dollar's worth of goods from every person in the world, and handing thirteen billion dollar's worth of goods to one randomly chosen person. The amount of goods in the world is the same in both cases. However, if I prefer A to B, then U(A) must be larger than U(B), as this is just a different way of saying the exact same thing.
Precisely. However, I noted that if you had to do the same decision a trillion trillion times, the utility of both options are essentially the same. So it means that your utility does not simply sum in the naive way if you allow distribution or variance issues into the equation.
You are right that utility does not sum linearly, but there are much less confusing ways of demonstrating this. Eg., the law of decreasing marginal utility: one million dollars is not a million times as useful as one dollar, if you are an average middle-class American, because you start to run out of high-utility-to-cost-ratio things to buy.
Standard utility does sum linearly. If I offer you two chances at one util, it's implicit that the second util may have a higher dollar value if you got the first.
This argument shows that utilities that care about faireness or about variance do not sum linearly.
If you hold lottery A once, and it has utility B, that does not imply that if you hold lottery A X times, it must have a total utility of X times B. In most cases, if you want to perform X lotteries such that every lottery has the same utility, you will have to perform X different lotteries, because each lottery changes the initial conditions for the subsequent lottery. Eg., if I randomly give some person a million dollar's worth of stuff, this probably has some utility Q. However, if I hold the lottery a second time, it no longer has utility Q; it now has utility Q - epsilon, because there's slightly more stuff in the world, so adding a fixed amount of stuff matters less. If I want another lottery with utility Q, I must give away slightly more stuff the second time, and even more stuff the third time, and so on and so forth.
This sounds like equivocation; yes, the amount of money or stuff to be equally desirable may change over time, but that's precisely why we try to talk of utils. If there are X lotteries delivering Y utils, why is the total value not X*Y?
If you define your utility function such that each lottery has identical utility, then sure. However, your utility function also includes preferences based on fairness. If you think that a one-billionth chance of doing lottery A a billion times is better than doing lottery A once on grounds of fairness, then your utility function must assign a different utility to lottery #658,168,192 than lottery #1. You cannot simultaneously say that the two are equivalent in terms of utility and that one is preferable to the other on grounds of X; that is like trying to make A = 3 and A = 4 at the same time.