# 12

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Utility functions are only defined up to an additive constant and a positive multiplier. For example, if we have a simple universe with only 3 possible states (X, Y, and Z), a utility function u such that u(X)=0, u(Y)=1, and u(Z)=3, and another utility function w such that w(X)=-1, w(Y)=1, and w(Z)=5, then as utility functions, u and w are identical, since w=2u-1.

Preference utilitarianism suggests maximizing the sum of everyone's utility function. But the fact that utility functions are invariant on multiplication by positive scalars makes this operation poorly defined. For example, suppose your utility function is u (as defined above), and the only other morally relevant agent has a utility function v such that v(X)=0, v(Y)=2000, and v(Z)=1000. He argues that according to utilitarianism, Y is the best state of the universe, since if you add each of your utility functions, you get (u+v)(X)=0, (u+v)(Y)=2001, and (u+v)(Z)=1003. You complain that he cheated by multiplying his utility function by a large number, and that if you treat v as v(X)=0, v(Y)=2, and v(Z)=1, then Z is the best state of the universe according to utilitarianism. There is no objective way to resolve this dispute, but anyone who wants to build a preference utilitarianism machine has to find a way to resolve such disputes that gives reasonable results.

I'm pretty sure that the idea of the previous two paragraphs has been talked about before, but I can't find where. [Edit: here and here]

Anyway, one might argue that if you are not a preference utilitarian, and not planning to build a friendly AI, you have little reason to care about this problem. If you just want to maximize your personal utility function, surely you don't need a solution to that problem, right?

Wrong! Unless you know exactly what your preferences are, which humans don't. If you're unsure whether or not u or v (as described above) describes your true preferences, and you assign a 50% probability to each, then you face the same problem that preference utilitarianism did in the previous example.

Humans are a lot better at getting ordinal utilities straight than they are at figuring out cardinal utilities, but even assuming that you know the order of your preferences, the problems remain. Let's say that, in another 3-state world (with states A, B, and C) you know you prefer B over A, and C over B, but you are uncertain between the possibilities that you prefer C over A by twice the margin that you prefer B over A, and that you prefer C over A by 10 times the margin that you prefer B over A. You assign a 50% probability to each. Now suppose you face a choice between B and a lottery that has a 20% chance of giving you C and an 80% chance of giving you A. If you define the utility of A as 0 utils and the utility of B as 1 util, then the utility values (in utils) are u1(A)=0, u1(B)=1, u1(C)=2, u2(A)=0, u2(B)=1, u2(C)=10, so the expected utility of choosing B is 1 util, and the expected utility of the lottery is .5*(.2*2 + .8*0) + .5*(.2*10 + .8*0) = 1.2 utils, so the lottery is better. But if you instead define the utility of A as 0 utils and the utility of C as 1 util, then u1(A)=0, u1(B)=.5, u1(C)=1, u2(A)=0, u2(B)=.1, and u2(C)=1, so the expected utility of B is .5*.5 + .5*.1 = .3 utils, and the expected utility of the lottery is .2*1 + .8*0 = .2 utils, so B is better. The result changes depending on how we define a util, even though we are modeling the same knowledge over preferences in each situation.

Anything with moral uncertainty, such as a value loading agent, needs to know how to add utility functions, not just utilitarians. I do not have a satisfactory solution to this, although I have come up with 2 attempted solutions, neither of which is entirely satisfactory.

My first idea was to normalize the standard deviation of each utility function to 1. For example, in the XYZ world, after normalizing u and v so that their values have standard deviation 1, we get (approximately) u(X)=0, u(Y)=.802, u(Z)=2.405, v(X)=0, v(Y)=2.449, v(Z)=1.225, so (u+v)(X)=0, (u+v)(Y)=3.251, and (u+v)(Z)=3.630. Z is thus declared the best option overall. However, if there are an infinite number of possible states, then this is impossible unless we have some sort of a priori probability distribution over the possible states. Even more frightening is the fact that this does not respect independence of irrelevant alternatives. Let's suppose that we find out that X is impossible. Good; no one wanted it anyway, so this shouldn't change anything, right? But if you exclude X and set Y as the 0 value for each utility function, then we get u(Y)=0, u(Z)=2, v(Y)=0, v(Z)=-2, (u+v)(Y)=0, (u+v)(Z)=0. The relative values of Y and Z in our preference aggregator changed even though all we did was exclude an option that everyone already agreed we should avoid.

Then it occurred to me that we have much more knowledge about the relative values of options that we are already quite familiar with, so it seems reasonable to assume that most of our moral uncertainty is about the value of options that we are not so familiar with. For example, in the ABC world, if you make decisions involving A and B all the time, but C is an unfamiliar option that you have not thought much about, it might be tempting to accept the first calculation, which gave B a value of 1 util and the lottery a value of 1.2 utils. This seems like a promising heuristic, but is difficult to formalize, and does not completely solve the problem. For instance, if both B and C are unfamiliar, then this heuristic does not have any advice to give.

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