## Dhillon & Mertens's characterization of (normalized) range voting

THEOREM claimed by Dhillon & Mertens (1999): Normalized continuum-score Range Voting ("normalized" meaning that every voter scores her favorite with the maximum allowable score and her most-hated candidate with the minimum) is the unique voting system (in situations with at least 3 voters and at least 5 candidates) obeying the following axioms:

INDIV:
If all voters are totally indifferent, so is society (note: "society" means the output of the voting system).
NONT:
Society is not always indifferent.
NOILL:
If every voter is indifferent except for one voter, then society is not always opposed to her.
ANON:
Permuting the voters has no effect.
CONT:
Continuity. This axiom is rather technical. But it has in mind that voter preferences are somehow continuously variable. [You will need to consult the original paper to get the full details; I think they are trying to say voter preferences must form a compact set in a topological space, or something like that.]
IRA:
If a "candidate" which happens to be a lottery among some subset S of candidates, is removed, then the election result (as a full ordering among all candidates and lotteries among candidates in S) is unaltered.
MON:
Monotonicity.
• One possible monotonicity-axiom would be that if society is indifferent about X versus Y, and a new voter, who prefers X over Y, enters the picture, then the new augmented society should prefer X over Y.
• A second such axiom could be that if society regards X as better than or equal to Y, and a new voter comes who is indifferent about X versus Y, then the new augmented society should still regard X as better than or equal to Y.
• A third such axiom could be that if society regards X as better than or equal to Y, and a new voter comes who prefers X over Y, then the new augmented society should still regard X as better than or equal to Y.
It is messy to state Dhillon and Mertens's monotonicity axiom, which is why have not stated it (see their paper); but suffice it to say that it actually is weaker than any of these three axioms (i.e. any of them imply it, but the reverse implications do not hold).

Note: I am not necessarily vouching for the validity of this theorem. For that see the paper which states and proves it (referenced below with link to pdf file); I am merely transmitting the news. I have in fact been unable to fully read+digest their paper, in part due to my laziness and also in part due to the dogged determination of its authors to use Notation From Hell (some of which apparently is not even defined in their paper) and to refuse to use plain English. (See attempt to decipher.) That being said, I am convinced that Normalized Range Voting does obey all the criteria in the theorem. (That's the easy part of the theorem.) The hard and impressive part of their theorem is that nothing else does. Their conditions are all extremely weak and hence extremely unobjectionable, which is why this theorem is impressive. Incidentally, they probably could prove their result without needing to assume #candidates≥5 if they were willing to start from a slightly stronger, e.g, monotonicity axiom. Another nice thing (which was intentional) is that their conditions resemble very much the conditions in Arrow's impossibility theorem. Thus the Dhillon-Mertens theorem can be interpreted as saying that range voting comes closer than any other voting system, to satisfying Arrow's desiderata. (Actually, [unnormalized] range voting totally satisfies all of Arrow's desiderata – but only under some wordings of his Theorem...)

### Source

Amrita Dhillon & J-F. Mertens: Relative Utilitarianism, Econometrica 67,3 (May 1999) 471-498.

I thank Professor Marcus Pivato for bringing this paper to my attention. He wrote a paper "Twofold optimality of the Relative Utilitarian Bargaining Solution" Social Choice and Welfare 32,1 (January 2009) 79-92, in which he presents two ways in which Normalized Range Voting is a uniquely optimal procedure. It also seems to be a good paper and it builds on (or re-does better) some previously published results, by, e.g. Uzi Segal [Let's agree all dictators are equally bad, Journal of the Political Economy 108,3 (June 2000) 569-589].

There is also this other paper which I have not seen:

Amrita Dhillon: Extended Pareto rules and relative utilitarianism, Social Choice and Welfare 15,4 (August 1998) 521-542.