## Marcus Pivato's characterization of range voting

THEOREM claimed by Professor Marcus Pivato (2011): Given a finite set of candidates. Range Voting is the (1) "most expressive" voting rule which satisfies (2) "reinforcement," (3) "neutrality," (4) "overwhelming majority," and (5) "does not admit minority overrides."

What do all those numbered quoted terms mean? To define them (in reverse order):

(5) "Admits minority overrides" means a single voter can always change the election outcome.

(4) "Overwhelming majority" means that if you take some subset S of the voters V and make M copies of S, then for all sufficiently large M, i.e. in the limit M→∞, the winner of the full set of voters with all those copies added will always be the same as the winner using just the voters in S alone – except that the voters in V-S can play the role of breaking ties among the S-only winners.

(3) "Neutrality" means all candidates are treated equally (permutation equivariance, i.e. renaming the candidates has no effect on the winners other than the obvious renaming).

(2) "Reinforcement" means if set A of voters would yield some election result, and disjoint set B of voters would yield the same election result, then the union-set A∪B should too.

(1) "Expressiveness" comparison is done via subset. That is, voting method J is "at least as expressive" as voting method K if the possible votes with J form a superset of the votes with K and the election results using the two methods are the same provided we stay within the common subset.

## Source

Marcus Pivato: Variable population voting rules, June 2011 (42 pages). Later published in Journal of Mathematical Economics 49,3 (May 2013) 210-221.

Marcus Pivato: Formal utilitarianism and range voting, July 2013 (17 pages). Later published in Mathematical Social Sciences 67,1 (Jan. 2014) 50-56.