Range-Approval Hybrid Voting Model

Various voting methods have been proposed. The key is to find one that best satisfies certain criteria. In 1951, Kenneth Arrow presented a proof that no rational and ethical voting method existed according to the criteria he proposed. This has come to be called Arrow’s Impossibility Theorem. Actually, Arrow’s Theorem is more general in that it applies not just to voting methods but to economic systems as well. This more general field is called Social Choice. Range Voting has been shown to escape Arrow’s Impossibility Theorem since it is based on ratings rather than rankings of candidates. Arrow postulated that the voters ranked all the candidates from first to last and then the choice of winner was based on those rankings. Range Voting is based on giving a numerical rating to each candidate between a maximum and a minimum value, and, by virtue of this construct, is able to escape Arrow’s Theorem.

Therefore, Range Voting possesses some rational and ethical properties that are worth mentioning. It positively associates individual and social values. That is, if some voter raises his rating for some candidate, that candidate will do at least as good in the final results as he would have done before the rating was raised.  If he was a winner before the raised rating, he would remain a winner afterwards, and if he was a non-winner before, he might be a winner after. Positive associativity is a very desirable condition and not all voting methods manifest it. It is easy to see that range voting is positively associative because social ratings are computed as the sum over individual ratings. Therefore, if some voter raises (lowers) his rating for a particular candidate, that candidate’s social rating will increase (decrease).

Another desirable condition is called Independence of Irrelevant Alternatives. This states that the electoral results must be consistent if one or more candidates drops out after the votes have been received. In other words if A wins the contest based on ratings or rankings over n candidates, A should also win over fewer candidates. This is not true, for example, for the Borda count whose rankings change based on the number of candidates. However, it is true for range voting because the ranking is not relative to any of the other candidates. Therefore, the ranking would remain the same if some candidates dropped out of the race.

So range voting satisfies Arrow’s criteria including others that won’t be gone into here. Although range escapes Arrow’s Impossibility Theorem, there is one other criterion to consider and that is the Gibbard-Satterthwaite Theorem. Gibbard came up with his result in 1973, and Satterthwaite came up with a similar result in 1975. Briefly, what this theorem states is that, if a voter has knowledge of how the other voters will vote, he can misrepresent his own vote to gain an advantge for his preferred candidates. In other words, he can vote insincerely in order to manipulate the final outcome. This is an undesirable property in a voting method. Ideally, all voters would be encouraged to vote sincerely, and the outcome would represent a true expression of the sincere values of the electorate. Especially in range voting where a sincere vote represents the utility of the particular candidate for the voter, it is desirable for the outcome to represent a sincere expression of the social utility. For utility we could substitute happiness or satisfaction.

It has long been known, however, that if utilities can take on values between 0 and 1, that a voter should assign his most preferred candidate a value of 1 and his least preferred a value of 0. This represents in some sense an insincere vote if his most preferred was sincerely 0.8 and his least preferred was sincerely 0.2, for example. Nevertheless, in order for a voter to manifest the full power or potential of his vote and not suffer a disadvantage with repect to other voters, it is recommended that each voter vote this way. Or each voter could vote sincerely, and the system could adjust or normalize every vote before proceeding to the tally. Note that with this system which could be considered a range hybrid, it wouldn’t matter whether or not a voter normalized his vote. If he did, fine; if he didn’t, the system would do it for him. There is nothing to be gained or lost  by the individual voter normalizing his vote or leaving it unnormalized.

We might ask if there is a more general method which could be used to transform sincere individual votes in such a way as to maximize the “power” of the vote. In other words what strategy should an individual use to get the maximum potential out of his vote. This could be considered insincere voting, but it could also be considered using the vote for maximum effrectiveness. And if there is such a way to manipulate one’s vote, could every individual do it, thus maximizing the potential of each individual vote. Would this also tend to nullify any advantage a particular individual or individuals might gain if only they used the method and all others voted sincerely. Furthermore, could this transformation be applied by the voting system itself in such a way as to maximize the potential of each individual vote, nullify the advantage(s) of any particular voter or subset of voters and make it possible that the best any voter could do would be to vote sincerely? This then would be the ideal situation: each voter could do no better than to vote sincerely and no voter could vote strategically in order to gain an advantage.

There are two cases to consider: (1) when there is no a prior information about how the voters will vote; and (2) when there is a priori information. First we consider the case in which there is no a priori information available. In this case Warren D Smith and John Lawrence have proven that the best way to vote strategically is in accordance with a technique called mean-based thresholding. The voter computes the mean of his sincere range ratings, then assigns each candidate with a sincere rating greater than the mean a rating of 1 and each candidate with a rating less than the mean a rating of 0. If a candidate has a rating equal to the mean, he can be assigned a rating of ½. Warren states his theorm as follows:

Theorem: Mean-based thresholding is optimal range-voting strategy in the limit of a large number of other voters, each random independent full-range.

Warren has also stated this theorem this way: …with a large number of other voters, all of whom vote randomly (hence all candidates equally likely to win), your strategically-best approval vote is to approve those candidates whose utility exceeds T, and disapprove the others, and your best choice of that threshold T is the arithmetic mean utility (for you) of all the candidates.

Well, obviously, if one voter can optimize the power of his vote this way, so can all the others. Furthermore, the voting system itself could provide the transformation so that every voter’s voting power would be maximized and no voter would have an advantage over any other. In that case, there would be no advantage for a voter to vote strategically or insincerely, and the best each voter could do would be to state his utilities sincerely! I call this voting system Range-Approval Hybrid since Approval Voting is a method in which all candidates are assigned either a 1 or a 0. Approval lacks a method, however, for determining how to assign the 1s and 0s, and range voting is susceptible to manipulation or insincere  voting. As a hybrid, they overcome both Arrow’s Impossibility Theorem and Gibbard-Satterthwaite’s Manipulability Theorem!

Positive associativity of the hybrid is easy to see, and Independence of Irrelevant Alternatives still holds because, although the threshold could vary if some of the candidates dropped out, the new threshold computed by the system in that case would be the same as that computed by each individual.

Now the question is is there a transformation that could be applied when there is a priori information available. Warren D. Smith has proved a more general theorem regarding thresholding when all candidates are not equally likely to win. So the threshold can be calculated by the individual voter given the various probabilities of winning for each candidate, and then the sincere range votes or utilities can be assigned values of either 1 or 0 approval style. Again this could be done by the system itself, thus nullifying any advantage an individual voter or subset of voters might have and guaranteeing that the best any voter could do would be to vote sincerely. Therefore, Gibbard-Satterthwaite has been overcome in general, and the voting system is very stable.

Warren has concluded empirically that “mean-based thresholding is the best strategy here when the number V of other voters is sufficiently large”. Warren’s empirical results are for one particular voter and not when all voters use mean-based thresholding. The question is if all voters used mean-based thresholding, would the empirical results be superior to other voting systems, and also would the results maximize social utility and not just individual utility as Warren has shown. Put another way, how close would the social utility be if each voter voted sincerely to the social utility  manifested by the Range-Approval Hybrid model over many random trials? There would probably be some loss of social utility which would be the trade-off for gaining a voting system that was not subject to manipulation. With a slight tweak in Warren’s software, this could be easily computed.