I have been considering methods of evaluating distributions for quality in terms of providing many people with relatively high utilities and few people with relatively low utilities, i.e. placing a “utilitarian” measure on distributions. It is my belief that “minimax” methods are a mistake in terms of voting systems, since political and social landscapes for example are fairly dynamic—that is, I feel like it is OK for few voters to be very dissatisfied, as long as it isn’t the same voters who are very dissatisfied consistently or all the time, and as long as those voters were at liberty in engaging with the risks of dissatisfaction for the potential of reward. “The needs of the many outweigh the needs of the few,” but the “many” shouldn’t be some group that you can materially identify, if you catch my drift. All voters should have the liberty to engage with risks for potential rewards. If the rewards are real, then so are the risks.
But that’s enough waxing philosophical. My recent thoughts have been concerned with bounded score distributions, which is what we would be facing anyway with distributions of discrete scores over candidates. Anyway, suppose candidates are scored on a discrete scale from 0 to N. (By the way, if possible, we should try to get LaTeX formatting to work on the new forum at some point, not sure how that is done). Take a function
such that T(0)=0, T(N)=1, and A<B implies T(A)<T(B). This will constitute a mapping or “translation” of the discrete scores into the real interval [0,1].
Next, for a given score distribution, consider the cumulative distribution function (CDF) given by F:[0,1]—>[0,1], which maps X to the fraction of voters whose score S satisfies T(S)=< X. Note that this distribution function depends on our choice of T. Like all CDFs it is weakly increasing, and in this case it’s always the case that F(1)=1.
To measure a distribution, simply calculate the area beneath the curve of F from 0 to 1. The intuition is that the smaller this area is, the better the distribution. However, this measure depends on the choice of T. It is fairly easy to find distributions whose scores will swap in order according to different translation maps.
The experimental science comes in determining what an appropriate T should be to match with “normal” or “social” intuitions about which distributions are superior to which others. No matter what, it is going to be arbitrary, but ideally it would correspond nicely with what most people would expect most of the time. As a voting method, a single translation map T should be used to compare distributions—essentially it is the measuring instrument. T should be varied or calibrated to optimize “voter satisfaction.” (Which ironically is another matter of distributive justice). To calibrate T, one can simply have the algorithm running side by side with human distribution selections, keep a record of the human selections, and generate random maps for T until its winning distributions match those in the record.
A second, more complicated way of measuring distributions would be to try to take the “average distance” of points in the graph of F to the point (1,0). If this average is minimized, then F is 0 until a sharp jump up to 1 at the very end, which indicates that every voter has given a maximal score—this is the ideal distribution. The same occurs when the area under the curve is minimized, so this method of “average distance” is similar but is more complicated to calculate, so probably isn’t as neat. For either measure though, topologically speaking, lower measures will naturally tend to correspond with distributions that have simultaneously high means and low variances. More directly, they will approach the ideal distribution.
These give straightforward methods of selecting a distribution from a multitude—just select the distribution with the lowest measure. If there are any ties (which is extremely unlikely), then I don’t know, break the tie at random. Still, these methods are arbitrary. Nonetheless, they define a very broad class of voting systems that are simple to evaluate and more than just superficially utilitarian.
Again, any thoughts, suggestions, or concerns are welcome. Thanks for checking it out!