Entropy-Statistic-Weighted Approval Voting

cfrank

Let each voter submit an approval ballot.
Weight his approvals by

P*log(1/P)+(1-P)*log(1/(1-P))

where P is the fraction of candidates he approved.

Consider it and let me know what you think.
My reasoning is that this gives voters incentive not to bullet vote unless they really only want to support a single or a small number of candidates.

One tactic would be to risk supporting weakly supported candidates to inflate the weight of there true top preferences. This is potentially foolish though, since it will also risk greatly inflating the influence of minority groups. In the same sense, it encourages voters to consider acceptable candidates from those minority groups to support, I.e. hinging the risk by supporting minority candidates that are somewhat more worth the risk.

This tactic can also be made less viable by imposing an approval cutoff (say, V/C where V is the number of voters and C is the number of candidates, or something of that kind), and then restricting the statistic only to include candidates that passed the threshold.

Jack Waugh

I assume you will throw out the votes that approve everyone or no-one.

This weighting favors those who approve half the candidates over those who approve just one or all but one. What grounds are there for not weighting them equally?

cfrank

@jack-waugh actually, the votes that approve of everyone or disapprove of everyone naturally have an entropy statistic of zero (the entropy statistic is constrained to be a continuous function on the interval [0,1]), so they automatically don’t contribute to the election.

I wrote some more of the motivation above. But one motivation is that they provide the voting system with more information (measured by the entropy statistic) than other candidates do.

Also the base of the logarithm doesn’t matter, since it amounts to a uniform positive scaling.

https://en.m.wikipedia.org/wiki/Binary_entropy_function

Lime

@cfrank I considered a similar voting system (dividing score votes by the variance, so exaggerated ballots received lower weight). Ultimately I concluded it degenerates back to standard approval/score voting, except more confusing, because parties can nominate "decoy" candidates that voters can give a score of 50 to (or in this case, approve half of the decoy candidates).

Most modifications to approval/score voting run up against similar problems: any modification to approval/score voting must give up either participation or independence of irrelevant alternatives, the two properties that make these systems so appealing.

(Which isn't to say they can't be useful at all; STAR accepts a very small violation of IIA in exchange for improving voter honesty.)

cfrank

@lime I think this system is fairly different, since it is quantifying the information in each ballot, which mathematically is a more universal property—more concretely, it’s independent of how the values in the ballot are transformed under any reversible operation, meaning that it’s less subject to influence by arbitrary pre-processing choices.

And actually, in the context of approval voting, dividing by variance achieves the opposite effect of this system: it actually encourages bullet voting or sparsity of approval. This system symmetrically discourages sparsity of approval and sparsity of disapproval.

Jack Waugh

A vote against a candidate should receive full weight, the same as any vote that supports that candidate.

Lime

@cfrank Yes, I think you're right that entropy is a better choice of information measure. Using the variance wouldn't work for approval ballots.

I'm thinking more about similarities in the end result--we're trying to assign more weight to more informative ballots, which I think is a good idea in principle, but in practice we're left open to very easy manipulation by strategic nomination/voting for hopeless candidates.

Lime

Perhaps we could do better by weighting each candidate's entropy contribution by their number of approvals? So that way, voters who cast ballots that assigns only 1 approval and 9 disapprovals in the 10 highest candidates, then approve/disapprove of 50% of the bottom-ranked 200 candidates, are treated as low-entropy/information (rather than high-entropy/information) votes.

cfrank

@lime yes strategic approval for hopeless candidates is an issue. However, we could also do something like, for example, a raw approval rate threshold, and then restrict our approval election only to candidates above that threshold. This would tamp down on the effectiveness of that kind of tactical nomination. There could even be a dynamic elbow-point cutoff for approval-ranked candidates to remove candidates with low raw approval rates prior to the entropy computation.

For example, suppose that there is a ballot such as:

Then the raw approval counts ranked in descending order are

B[27] A[25] C[19] D[19] F[8] E[1] G[0]

Or as fractions, these are approximately

B[0.6] A[0.556] C[0.4222] D[0.4222] F[0.1778] E[0.0222] G[0.000]

A reasonable tail-end elbow detection method would detect E as the elbow point, and we could remove E and G from the election. Or, for example, a 1/7=0.14285... approval rate threshold (7 being the number of candidates) would accomplish the same.

This would leave candidates

B, A, C, D, F

Now we have ballots, with binary entropies H2 as

From these, we compute the final scores as

A[21.7835...] B[26.215666...] C[18.448...] D[18.448...] F[0.7219...]

and B (who was actually the original approval winner) still wins, and by a wider margin.

I do think there are problems with this method from the standpoint of majoritarianism. It seems plausible that a minority could obtain disproportionate voting power over a bullet majority.

If the ballots were, for example,

51: A | B C D E F G
25: G F | A B C D E
12: E D C | A B F G
12: B | A C D E F G

then I think we have a problem.

Toby Pereira

@cfrank said in Entropy-Statistic-Weighted Approval Voting:

Let each voter submit an approval ballot.
Weight his approvals by
P*log(1/P)+(1-P)*log(1/(1-P))
where P is the fraction of candidates he approved.
Consider it and let me know what you think.

What is the mathematics/motivation behind this particular formula? I don't think we've been given much to go on.

Edit - But anyway, it seems that basically you get more weight per approval if you approve more candidates, though I'm not sure where this formula comes from other than it's possibly something to do with entropy.

But obviously it's a bad idea. There's no reason to punish people who approve fewer candidates, and it encourages cloning.

Jack Waugh

It rewards those who approve close to half the candidates and punishes those who disapprove just one or who disapprove all but one.

Toby Pereira

@jack-waugh Right, I see. Though I think it's no better or worse than what I thought it was. Just arbitrarily favouring voters who approve a particular number of candidates. And it encourages putting up clones or non-entities accordingly.

Jack Waugh

I don't approve of it.

cfrank

@toby-pereira yes you’re right, it was just a thought that occurred to me when I was thinking about how to discourage bullet approvals, but it has irreconcilable flaws that are now apparent.

Toby Pereira

@cfrank While I don't think it would be a good method in practice, there may still be some theoretical interest in it. Perhaps another way of implementing it would be to have the peak at the mean number of approved candidates, rather than half.

Instead of looking at the number of states for each voter/candidate being two (approved or not approved), if e.g. 1/3 of voters approve a candidate, we could see it as one state for approved and two for not approved. In such a case I think the highest entropy state would be for a voter to approve the mean number of candidates.

Lime

@toby-pereira said in Entropy-Statistic-Weighted Approval Voting:

While I don't think it would be a good method in practice

The 2 most popular voting systems in practice are IRV and plurality. Anything is a good method in practice