Another way of thinking about voting system flaws

Not really sure how to explain this other than with examples. Judging from the way the ballot is counted, a voting system appears to “assume” something about the voter that may not be warranted. This fact can help explain some of the system’s flaws.

  • C1 (FPTP) assumes/acts like you only like one candidate, and all others are equally bad.
  • IRV assumes that you have a clear favorite, and then you prefer your second choice slightly above your third, and so on. It is like a limit of a geometric series of utility: 1, ε, ε2, … as ε → 0.
    • The flaw of IRV not electing consensus winners is contained in this one, as IRV assumes any choice but your first is nearly tied with everything else.
  • Borda assumes that your preferences of the candidates are equally spaced. That is, you lose as much utility from going from 1st to 2nd as you do from going from 6th to 7th.
  • Condorcet assumes (paradoxically) that you value all “I prefer X over Y” relations as equally strong. (Which is actually contradictory, as if you think X>Y>Z, your preference for X over Z is clearly stronger than X>Y or Y>Z.)
  • Approval Voting assumes you value the candidates in exactly 2 tiers and have no opinion on candidate pairs within a tier.
  • Score Voting assumes that you… um… oh yeah: that you are not going to want to exaggerate if candidates drop out.
  • STAR Voting is like Score but it then assumes you will want to (or will not, if you give both the same score) exaggerate once the leading two are decided.
  • 3-2-1 Voting is like Approval, but with 3 tiers. (The rest of the system is more about assumptions about the “best winner” than the voters themselves. This one has another flaw, like Borda and some of the various Condorcet cycle rules: it feels like the system, rather than the voters, chose the winner.)

I think the only voting system that makes no assumptions would be that weird one on the Kialo discussion where your vote is

if <boolean expression on candidates>:
    <score vote>
elif <second boolean expression>:
    <possibly different score vote>

(Actually… never mind, that one assumes your internal preferences are representable as numbers and you do not believe any cycles X>Y>Z≥X.)

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It’s maybe possible to renormalize voters’ ballots after the election if someone they scored gets disqualified.

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So I think I’ve grokked one fundamental difference between Score and Condorcet: Score passes the logical property “the strength of your preference X>Z is equal to strength of X>Y plus strength of Y>Z”, whereas Condorcet falls (since it’d be 1=1+1). So Score is basically a more logically coherent, but possibly less satisfying version of a Condorcet method.

Worth mentioning that Approval is equivalent to Condorcet where a voter must rank every candidate either 1st or 2nd. In that special case, Condorcet methods pass the logical property above because there can only be 1 vote at most separating any set of candidates when looking at each sequential pair in a beatpath.

@Keith_Edmonds I think what I wrote here is connected to Vote Unitarity and maybe to Cauchy’s formula? A voter whose utilities are A5 B3 C2 will have the A vs. C contest only have 60% vote strength for A, which is equal to the 40% support for A vs. B plus the 20% support for B vs. C. Condorcet however gives 100% support in all of these pairwise matchups and therefore fails to add up.

The issue with Approval is precisely this: it falsely assumes that all voters always separate candidates into two tiers (“Good” and “Bad”), with the Good candidates all equal and all equally better than any Bad candidate.

Also, I was a bit wrong when I said

What I should have said was that STAR assumes both that your opinions fall on a range scale and that all preferences are equal strength (like Condorcet), but for different purposes.
A critic could claim that this inconsistency creates a worse system than either Score or Condorcet directly, but a supporter of STAR (like myself) could claim that each one compensates for the other’s weaknesses.

The thing about STAR is that it makes those assumptions for the voter i.e. that all of them want full power between the top two and don’t want it when deciding who the top two are. That’s why I consider it more coherent and better accomplishing STAR’s objective to allow a voter to indicate how they’d vote in every possible score runoff between candidates (or any feasible approximation of that), and then run a Condorcet method on it. Then voters can decide how utilitarian to be. Notably, this scheme could mean a voter is allowed to vote Favorite 5 Lesser Evil 1 and then optimize both candidates in score runoffs against the greater evil, just like STAR.
Though some might say STAR is really just a sneaky way to get to utilitarianism once many candidates are running by being more majoritarian when there are fewer, “easing” voters in until its too late.

Yes sort of but there is a little difference here. The fact that score form a group because it has the needed arithmatic properties is why it does not run into the issues ranking does. Ranking is not arithmatic. When I am speaking of the Cauchy formula I am taking this concept one step further. I am not just saying that you can add scores but that the addition has a defined form in terms of Utility. ie U(x+Y) = U(x) + U(y) . RRV for example does not do this since it uses the Harmonic function to map score to utility and this function does not obey the Cauchy formula.

Why has nobody in the voting theory community written up something making these connections?
Also, would ordinal utility satisfy something similar to Cauchy’s function? If a voter gets both their 1st and 2nd choice, for example, that seems to be trivially equal to getting both their 1st and 2nd choice.

Not really sure. Scientific advancement is a weird thing. Everybody adds their little piece to the puzzle. Given my background it seemed obvious to me but many other things were not. In any case I am glad it was missed because it gave me the chance to contribute with my Vote Unitarity stuff.

I asked Warren why he did not notice this possibility. Basically he had assumed that the Optimal function needed to be smooth. If you look at the optimal version of Vote Unitarity

You can see that it is quite clearly not a smooth function with the min() function being part of it. If you assume the linear function for Utility addition you clearly do not get a system which is “Proportional”.

No. You cannot do arithmetic with ranks so it can’t have the Group properties required for that to make sense. Number theory and Group theory might not be a class you have taken yet.