Topic: Social Choice
With various voting methods, there is a method for individuals to vote, and then there is a method for combining the votes to determine the outcome of the election. For instance, with the Borda Count, all the candidates are ranked by the voters with the top ranked getting a number of points equal to the number of candidates and the bottom ranked getting 1 point. Then the votes are counted by counting the total points for each candidate, and the candidate with the most points wins. Range Voting is similar except the top ranked candidate can be assigned an arbitrary number of points usually determined by the ground rules of the election. Also 2 or more candidates can be assigned the same number of points. Then the points are counted for each candidate with the one getting the most points declared the winner.
Computer simulations by the Center for Range Voting have shown that Range Voting is superior to other methods in that it minimizes Bayesian Regret. Bayesian Regret measures the difference between the social utility produced by the candidate whom, if he had won, would have maximized social utility, and the social utility produced by the winner as computed by the voting system used, in this case, Range Voting. The social utility is the sum over the individual utilities of all the voters. This presupposes that there is a meaningful measure of individual utility which is a foregone conclusion as far as the Center for Range Voting is concerned.
I would argue that, although there are many definitions of utility and the definition of utility used by the Range Voters is basically preference utility, it is, nevertheless, a meaningful form of utility. Each voter's utility is essentially revealed by his vote. In Range Voting with a range from 1 to 100, for example, if a voter rated some candidate a 100 and then that candidate won the election, that voter's individual utility would have been maximized. The number 100 may not have any meaning in itself, but just because it is the maximum point value that can be assigned in this example, the voter would be considered to have achieved maximum individual utility. On the other hand, if a voter rated the winner of an election as a 1, minimum utility would have been achieved by that individual.
My suggestion is this. Instead of summing point values over all individuals for all the candidates and then declaring the winner as the one with the highest point total, compute the social utility which would be the sum of the individual utilities for each candidate and declare as the winner the candidate who maximized social utility. Obviously, this system would minimize Bayesian Regret over all other systems! An individual's utility for any candidate would correspond to the point value assigned to that candidate. This system could be used for Borda Voting, Approval Voting, Plurality Voting or Range Voting. In fact, Range Voting is the generalization of Borda, Approval and Plurality. Any voter could submit his vote as a Borda, Approval, Plurality or Range Vote within the confines of Range Voting. For instance, with a range from 1 to 100, if a voter wished to be a plurality voter all he would have to do is vote 100 for some candidate and 1 for all others. For Borda, he would equally space his point values from 1 to 100 and then assign them in order of his preferences. For Approval Voting he would assign 100 points to all those candidates he approved of and 1 point to all others. Finally, for Range Voting, he would distribute point values among the candidates corresponding to his preference intensities.
Another consideration is strategic voting. Some feel that Approval or Plurality Voting within Range Voting is strategic, that really the voter has an "honest" distribution of point values over the candidates but then maximizes some and minimizes others. But how do you know that, or, more to the point, how can you assume that for the purposes of computer simulations? Maybe the maximin voter truly feels that this vote represents his true utility distribution. All the voting "system" knows is what the voter reveals by his submitted vote. You really can't tell if the vote is a strategic vote or not, so why worry about it, and why berate some method because the social utility is assumed to be lower than it would have been if all voters had voted "honestly."
The maximum social utility that can be achieved is a function of the distribution of utilities among the individual voters, the domain, if you will. Some distributions (or elements of the domain) will produce a greater social utility than others. How much social utility that can possibly be achieved depends on the distribution of utilities among the voters.
Another objection is that the computation of the maximum social utility for any election is much more complex than simply counting up the points. This is true, but it can be done and it was done in the computer simulations done by the Center for Range Voting. Otherwise, it wouldn't have been possible to calculate Bayesian Regret. In fact, these calculations can be precomputed and stored much in the way Google precomputes search results in order to speed up the search process. In addition shortcuts in the computation process may be discovered.