# Removing a degree of freedom from relative utilitarianism

Relative utilitarianism takes utilities as reported by agents, rescales them so that, for each agent, the alternatives have utility values between 0 and 1, and then chooses an alternative to maximize the (weighted) sum of the rescaled utilities. This forces an equality of sorts upon the agents as their utilities are all bounded on the same range. However, they can still express the degree to which they prefer A to B (relative to their preference of B over C) by making the gap between A and B larger (or smaller) than that between B and C.

Has anybody come across a mechanism like this except that, instead of rescaling reported utilities to a given range, we simply take the ordinal preferences expressed by the reported utilities and assign predetermined utility values (respecting the order).

That is, with relative utilitarianism:

$$(10,4,2,0) \rightarrow (1,0.4,0.2,0) \text{ and } (100,9,1,0) \rightarrow (1,0.09,0.01,0)$$

What I want to do is the following:

$$(10,4,2,0) \rightarrow (10,7,5,2) \text{ and } (100,9,1,0) \rightarrow (10,7,5,2)$$

The numbers in $(10,7,5,2)$ are irrelevant (they could be between 0 and 1) -- the point is just that they're the same across the two cases. Basically, I'm just extracting a preference order from the reported cardinal utilities and then slapping predetermined utility values on to them, respecting the order but not magnitudes in differences. Then, subsequently, I would choose an alternative to maximize the sum of utilities.

Two related questions:

1. Have you seen this done anywhere, either explicitly or implicitly?
2. Supposing you haven't, what would you call this?
• I have not seen this. I don't see for what purpose can this be used. Without knowing this the best name would be 'ad hoc utility values to support the social choice that I wanted to get from the model'. – Giskard Nov 23 '15 at 23:42
• @denesp Haha -- I appreciate the cynicism, though I would prefer a slightly more positive, less normative title. It's value would be that it would allow me to use numerical optimization techniques (which need cardinal values) using only preference orderings. I am still implicitly doing interpersonal comparisons of utility, which is sketchy, but in a more controlled way: I'm letting agents tell me their preference orderings -- the rest I decide (in an ad hoc way, but hopefully compelling, manner). – Shane Nov 24 '15 at 1:01
• I see. What is the added benefit of giving ad hoc values instead of (1,2,3,4) reordered to reflect the preferences? – Giskard Nov 24 '15 at 7:21
• @denesp I, as the social planner, might want to impose differences in magnitude. So if I think the difference in value between somebody getting their first and second choice is smaller than the difference between second and third, I might choose (6,5,3,0). – Shane Nov 24 '15 at 15:42
• I expected something like that. I think that unless you provide some general scoring method your approach will be criticized as arbitrary or paternalistic. – Giskard Nov 24 '15 at 16:24

## 1 Answer

One of the major criticisms you hear about classical economic frameworks is that ordinal utility is the only valid form of utility, because cardinal utility is just putting arbitrary numbers onto people's preferences. The thing is, cardinal utility is really not as fraught as heterodox schools (or even Austrian economists) might expect. Cardinal utility is only an expression of preference up to positive affine transformations, so as long as a utility function maintains the order of all the preferences, the utility functions represent the same preferences; cardinal utility just allows for us to express convexity/concavity along with it.

So when I see a system of utility such as this, it just looks like a normal monotonic transformation into the range $[0,1]$. The only problem with it is that it will normalize all utility functions considered into that range, and you might lose some information about the strength of preferences relative to other agents to each other. I'm not exactly sure what it would be used for.

• Nooo @Oliv your answer was good too. – Kitsune Cavalry Nov 26 '15 at 3:49
• I'm not going to mark this as the accepted answer, because it doesn't really answer the question, but I agree with all of your sentiments and consider them relevant to the question. – Shane Dec 30 '15 at 2:29