# Score Distribution Evaluation

Hi, a bit ago I suggested some ideas that I hoped might help in devising a “communitarian” voting system, wherein utility is supposed to be more equitably and socially distributed than in systems where utility can be hoarded by a majority at the expense even of large minorities due to majoritarian characteristics (perhaps even unintended ones) of the system itself. However, I realized that there were some strategic methods that render the suggestions I made seemingly infeasible. After certain considerations it became more clear to me why we probably do in fact need an impersonal, essentially robotic, one-time decision-making procedure, as opposed to one where people are involved in multiple steps—not only for efficiency, but to avoid strategy. Game theoretically, people need to be “under threat” of an impartial mechanism or arbiter, so as to prevent “synergistic” strategification.

To illustrate this argument, we can consider the “social experiment” I described. Here is a version that evolved after a bit more thought, though it has at least one fatal problem:

1. Voters score each candidate “independently,” say on an integer scale from 0 to 5.
2. From these scores, each candidate is assigned a distribution that gives the proportion of voters who fall into each score bin for that candidate—or at least the distribution is a reliable approximation of those proportions, in case random or strategic noise is introduced to mask candidates. For example, perhaps we can establish a fixed budget for information loss as measured by the Kullback-Leibler divergence from information theory:
https://en.m.wikipedia.org/wiki/Kullback–Leibler_divergence
3. Without knowing which distribution corresponds to which candidate, voters then cast a ballot (say, a plurality ballot) a second time, but this time for the distribution, not for the candidate.
4. The candidate with the winning distribution is declared the winner of the election.

The fatal flaw is essentially that rather than indicating their preferences ``honestly," voters can instead group together and try to assign their mutually preferred candidates characteristic distributions that they can reliably identify in the second round. For example, a large enough group of voters could flip their scales around, and have a 0 refer to their favorite candidate and a 5 refer to their least favorite, or they could simply agree on an arbitrary re-ordering.

Trying to mitigate this, several thoughts came to mind. I thought it might be suitable to have an elected council, who abstain from casting a ballot, transparently choose a distribution. That has the obvious problem of bribery, however, despite the transparency deterring anything more than slight foul play through public scrutiny (assuming sufficient auditing powers are in place). Otherwise, an algorithm will have to evaluate the distributions, which basically destroys the whole purpose of the system.

However, perhaps it is enough to have the threat of an impartial algorithm determining the winner from the distributions. For example, perhaps establishing a 50% chance that some algorithm will determine the winner rather than the voters themselves might deter strategic voting enough to the point that it is negligible.

Anyway, this brings us to the problem of evaluating and comparing score distributions. For example, how can we formalize which of these two distributions is preferred over the other?

___0—1—2—3–4–5
X : [4] [0] [0] [0] [0] [6]
Y : [0] [3] [2] [5] [0] [0]

My thinking is that it may have something to do with a combination of the expected score and the variance. For example, perhaps some metric or score of the form MEAN/exp(B*STDDEV/MEAN) with an appropriate value of the “equity” parameter B>=0 could be used to evaluate distributions. This formula seems to make sense for distributions over non-negative values, and after some more thinking it can be extended naturally to yield a metric for any distribution with a support that is bounded from below. In particular, consider first shifting the distribution so that the minimum value in the support is equal to 0, then apply the metric to the shifted distribution, and add the minimum value in the support of the old distribution to the metric. Furthermore, it may be possible to extend this metric to unbounded domains using limits of scores for leftward-truncated distributions, if the limits exist. Perhaps insisting on the existence of these limits for certain natural distributions will help us to modify the metric in a suitable way.

For now though, we can stick with left-bounded distributions and entertain the suggested model. We can find approximations for the appropriate value of B by comparing the hypothetical scores of distributions we have a hard time deciding between. For example, if we set the scores for the two distributions exemplified above to be equal (intuitively meaning that it is hard to decide which is better than the other), then B is approximately 0.6.

Regarding the model, frankly it is just an arbitrary, natural-seeming (to me) suggestion/heuristic. It was designed to conveniently satisfy the following properties:

1. For two bins, it indicates that a higher proportion of voters in the higher-valued bin will always yield a higher score, for all allowed values of B.
2. A distribution’s score is at most its expected value, and always has the same sign. When a distribution is just a fixed value, the metric is equal to that value, independent of B. If B=0, then the score reduces to the mean.
3. For a fixed mean and positive B, a smaller variance yields a higher score. For a fixed variance, a larger mean yields a higher score. (The use of the metric is to formalize comparisons of distributions with different means and different variances).
4. If the distribution is shifted and stretched, then the metric is shifted and stretched accordingly.

However, there are many families of metrics that will also satisfy analogues of these properties, though perhaps with more than just one parameter.

Regardless, here is a plot of the metric for distributions over two bins, one with value 0 and the other with value 1, where the x-axis gives the proportion of voters in the 1 bin, and where B=0.6:

The function is strictly increasing, as one ought to expect (and as is by design) according to property 1.

And here is a surface that gives the scores for distributions over bins of values 0, 1, and 2, where the horizontal axes give the proportions of voters in the two higher-valued bins for B=0.6 (guess which axis is which bin):

And here are the parameters for some isosurfaces, which give distributions that are more or less equally preferable in terms of this model:

For example, in this case, it looks like

_____0----1-----2
X :—[0]—[1]—[0]
Y : [1/4] [0] [3/4]

are more-or-less equally preferable. As another instance, the following two distributions are also supposed to be equally preferable to each other according to the model:

_____0------1-------2
X : [2/3]–[1/3]—[0]
Y : [4/5]—[0] —[1/5]

What do you think? If the model is accurate, people voting this way, when forced to choose, should choose X over Y roughly 50% of the time, and Y over X the other 50% of the time. These kinds of things can be put to the test.

Do you think there is a different metric that satisfies the same properties and that might be more suitable? I just chose a convenient one that fits the mold I was envisioning. A different metric might use something other than exponential decay, perhaps Gaussian decay for example, and a more sophisticated one might try to take skew or higher order moments or other properties of the characteristic function (Fourier transform) into account as well. As usual, any thoughts are welcome!

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Very cool new perspective!
I’ve chewed on the idea of giving each voters 2 votes: one to express preferences on candidates and one to express a preference on the vote-counting-method. So voters themselves could vote to use a method that favors consensus/centre candidates, FPTP or something else. Power to the voter instead of power to votingsystem-experts like us…

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Thank you! I think that’s an interesting concept too, though it begs the question of how the voting method is decided lol. Did you cast a ballot for your preferred voting methods here?: Last chance to vote for your favorite voting method!

It’s just a game I thought would be fun and interesting, especially to see the results. @RobBrown has been kind enough to be collecting the ballots.