In a score voting system, there is the issue of what scores to allow voters to submit, as well as of how to bring together the various score ballots and produce an election winner. As such, my goal here is to produce a score-based system with results that are independent of what particular score set is chosen, as long as the scores are ordinally ranked. My hypothesis is that, even with *unlabeled* scores, simply blank boxes on a line from left to right with an indication of positivity, would accrue meaning to voters who used such a system as will be described.

In a score system, an (S,P)-consensual candidate is one who is scored at least an S by at least a P-fraction of the electorate. It is easy to show that a candidate is (S,P)-consensual if and only if it is (R,Q)-consensual for all R less than or equal to S and all Q less than or equal to P. The statement that a candidate is (S,P)-consensual is essentially an argument supporting their fitness as the election winner---the strength of the argument increases as S and P increase independently.

Let X be the random variable that is the score assigned to candidate C by a voter chosen uniformly at random. For each candidate, there is an (S,P) production possibilities frontier of points corresponding to the boundary of obtainable (S,P) values such that the candidate is (S,P)-consensual---it is simply the plot of Prob(X>= S) against scores S.

A candidate C is (S,P)-dominated if there is another candidate K whose production possibilities engulfs that of C. A voting system is (S,P)-efficient if (S,P)-dominated candidates are precluded from winning an election. I think it is almost self-evident that any reasonable score-based system definitely needs to be (S,P)-efficient.

For an example of an (S,P)-efficient election strategy, consider the voting system that chooses a candidate with a maximal area bounded in (S,P)-space by their production-possibilities frontier. This precludes (S,P)-dominated candidates from winning the election. More generally, any measure over (S,P) space will allow us to assign a score to each candidate consisting of the measure contained within their production-possibilities frontier, and this is also (S,P)-efficient.

Another method that is independent of both the measure and the scores is given by first eliminating all (S,P)-dominated candidates, and then using a determined method of generating a "random" production-possibilities frontier in (S,P)-space and calculating the probability that a candidate's frontier will be engulfed by a randomly produced curve. The candidate with the lowest probability of being engulfed seems like a very reasonable choice, although the algorithm to produce the random frontier will vary what exactly this means in practice.

Here is one method to produce a "random" production-possibilities frontier: if there are N scores, generate N-1 real numbers in [0,1] chosen uniformly at random. Let these numbers be X(1), X(2), ..., X(N-1). Then the random frontier is given by X(1) at score 1, X(1)*X(2) at score 2, X(1)*X(2)*X(3) at score 3, etc. Then the probability that a frontier {(S1, P1), (S2,P2),...,(Sn,Pn)} is engulfed by a random frontier is

(1-P1)(1-P2/P1)(1-P3/P2)...(1-Pn/P(n-1))

However, this measure is problematic---if P1=1 for example, then the measure is automatically 0 independent of the other values of P(n), which definitely does not seem logically admissible. More generally it is zero if we ever have P(n)=P(n-1). Although events like that are rare, if we are looking for a system that is in line with our intuitive social understanding of what should make one frontier superior to another, we should look for an alternative that does not produce these aberrations.

An alternative measure could be the expected value of the fraction of vertices on the frontier that are not engulfed by a random frontier. There is an easily computed (in a computer) form for this fraction: if F(N,P)is given by

F(N,P):=1/(N-1)! * INTEGRAL{from t=0 to P}[(-log(t))^{N-1}]dt=1-gamma(N,-log(P))/(N-1)!

where gamma(s,x) is the lower incomplete Gamma function, then for a score system with N scores, the expected value of the fraction of vertices that will not be engulfed by a random frontier is

1/N * SUM{from k=1 to N} F(K,P(K))

This seems reasonable to me, but I am open to any concerns or questions. Even without first excluding (S,P)-dominated candidates, the method of maximizing this fraction is easily shown to be (S,P)-efficient, and it does not seem to produce any aberrations. Of course, this is based on a particular model of construction for random frontiers using uniformly-distributed i.i.d. random variables, so it is not canonical. Unfortunately, while the arbitrary nature of the scores has been eliminated, we have not been able to eliminate the arbitrary nature of the production of frontiers.

A more interesting variant of this would be a model that takes account of past values for Prob(X>= S) for each score S and where X is the score given to any fixed candidate C by a voter chosen uniformly at random. Let P(j)=Prob(X>=S(j)) over the population of past, recorded candidates in "relevant" elections using the same framework---i.e. with the same set of allowed scores and over the "same" population of voters. Then if we use the measure that is the expected value of the fraction of vertices that are not engulfed by a "random" frontier based on the past constructions of (S,P)-frontiers, a record can be kept for the distribution of the P values for each score from past candidates, and that data can be used to estimate the expected value of the fraction of unengulfed vertices. This is possible because the expected value operator is linear even over variables that are not independent.

Let each score S(K) have a CDF of P values as F{K}(p). Then there is an indicator random variable 1{K} for each vertex (S(K),P(K)) that is 0 if the vertex is engulfed and 1 otherwise. The sum of all of these indicator random variables is the number of vertices that are not engulfed. The expected value of this is the sum of the expected values of these indicator variables. The expected value of the indicator 1{K} is simply the probability that the vertex (S(K),P(K)) is not engulfed, which is simply F{K}(P(K)). Hence the expected number of unengulfed vertices is

SUM{from K=1 to N}F{K}(P(K))

(And the expected fraction is this divided by N+1)