@lime you could also have a persistence diagram that shows the support level of each candidate at every possible cutoff. This produces “score proportion” profiles that indicate the fraction of voters who score each candidate at least a given score. It’s possible to define a dynamic threshold or even an integral across all thresholds.
Posts made by cfrank
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RE: Mutual Majorities in Score
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RE: State constitutions that require “a plurality of the votes” or the “highest,” “largest,” or “greatest” number of votes.
@gregw that’s a good question, I think that would be a contingency clause. I’m no lawyer and I don’t know much of anything about those or the limitations about how they can be structured.
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RE: State constitutions that require “a plurality of the votes” or the “highest,” “largest,” or “greatest” number of votes.
@Lime I agree with @Jack-Waugh. If we’re going to succeed in making any technical arguments then we will have to work with clear definitions and can’t afford to be loose with interpretations. It’s also dangerous territory to even bring up certain terms in the context of a legal argument, because terms that were previously undefined and may have left some room for interpretation are then liable to collapse into a narrower scope that sets a precedent. That means we have to be very careful, because if it gets screwed up once, it will be all that much harder to unscrew it in the future.
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RE: State constitutions that require “a plurality of the votes” or the “highest,” “largest,” or “greatest” number of votes.
@lime well, people who oppose reform (aka those in power) will find ways to concoct detailed arguments against the adoption of any reform proposal, and questionable constitutionality is a low bar.
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RE: State constitutions that require “a plurality of the votes” or the “highest,” “largest,” or “greatest” number of votes.
@jack-waugh that will work if everybody does it. However, it’s likely that people will not go through with that unless they have the same kind of discipline it takes not to constantly check stock values.
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RE: State constitutions that require “a plurality of the votes” or the “highest,” “largest,” or “greatest” number of votes.
@gregw I honestly think approval voting is one of the best footholds we have for moving forward with actual voting reform. Especially considering the language you reference in the constitutions, it fits the bill. It may not be perfect but it’s miles ahead of choose-one voting and would have dramatically positive consequences if adopted. While IRV is a flawed incremental change, approval voting would be a real game changer.
Score will certainly be more questionable than approval in terms of the constitutional language. I think these kinds of practical constraints are likely to focus reformer support for approval.
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RE: Cycle Cancellation//Condorcet
@lime since we would be iterating through the edges that belong to cycles in order of ascending weight, any edge under consideration would automatically be the minimum weight edge of every cycle it is a part of. I think maybe it wasn't clear that equivalently, we can ask for each edge, "Is there some cycle to which this edge belongs?" If yes, it is in the search space of edges, and then we identify the edges in the search space of minimum weight.
Generally, whether an edge (u,v) belongs to a cycle can be checked by removing the edge and doing a (depth-first) search for a path from u to v, since (u,v) belongs to a cycle if and only if such a path exists.
But in any case it's just a concept, similar to Young's method and Dodgson's method, since it tries to perturb the ballot set in a conservative way until a Condorcet winner emerges.
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RE: Toward A Second Vote On Voting Systems
@jack-waugh I think it would be best to have a multifaceted vote on voting systems. Here is the ballot format I suggest, since it can be flexibly transformed into ballots that are compatible with other systems.
Once the candidate voting systems under consideration are chosen, each voter should submit a ballot assigning each candidate an independent integer score ranging from 0 to 100. For scaling purposes, two pseudo-candidates who cannot win will be introduced to the election as well, one automatically receiving a 0 on every ballot, and the other automatically receiving a 100 on every ballot. Each voter will also submit an integer from 0 to 100 as their approval threshold, and scores above that threshold will count as an approval.
This way, we can examine the winner under various different systems using self-consistent ballots. We can also see which voting systems if any end up electing themselves, just for curiosity.
If the election gets organized well, I’ll participate.
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RE: Cycle Cancellation//Condorcet
@lime yes they’re different. I’ll have to understand split cycle.
Still, the concept of shrinking the weakest edges in a common cycle into single nodes is interesting to me just mathematically. If the collapsed nodes are labeled as candidate subsets, you get a new marginal victory graph between over candidate subsets, and applying the same algorithm to these “meta nodes” by aggregating marginal victories between them can eliminate groups of candidates. I don’t know if that’s equivalent to any known method. It is a Condorcet method and satisfies the Condorcet loser criterion, which means it can’t be Minmax.
Smith//Minmax is also somewhat interesting. Is the following equivalent to split cycle voting?:
- If there is a Condorcet winner, elect them.
- If there is a Condorcet loser, remove them, repeating until there are no Condorcet losers.
- Of all edges that belong to cycles, identify those of minimum weight. Create a new tournament without those edges.
- Repeat step (3) on the newest tournament until no cycles remain. Then remove all candidates who are defeated (preserve only the undefeated candidates).
- Return to the original graph including only the undefeated candidates. Repeat from (1).
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RE: Cycle Cancellation//Condorcet
@lime would you also iteratively look for the weakest edge among all simple cycles? Does this end up being different from ranked pairs? It has a kind of a co-“ranked pairs” flavor. I think the algorithm would be:
- Set k=1.
- If there is a Condorcet winner, elect them.
- Identify the kth weakest edge. If it is part of a cycle, remove it. Otherwise, set k—>k+1.
- If there are cycles remaining, repeat from (1).
- Declare the final ranking.
It’s weird though if two edges are both the weakest and are both part of the same cycle. You could shrink all tied weakest edges in a common cycle to points and re-expand them later, effectively skipping that edge weight until the issue resolves, which would retain connectivity. That would be weird too, though.
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RE: rcvchangedalaska.com
@jack-waugh It's unfortunate that the spoiler effect occurred here. Still, the system is better than choose one, even if it has the occasional spoiler issue, which is so ubiquitous that trying to find it is like a fish trying to find water. It's also good that this website used a reasonable concept like Condorcet compliance to evaluate the election outcome. As long as good, rational arguments are being made, things are going in the right direction.
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RE: Two Currencies
@jack-waugh I don't know how practical this would be, but I understand the concept of trying to disentangle the labor and capital economies. I am not a labor economist, sociologist or Marxist/capitalist theorist by any means, but it seems to me that the central issue here is the negotiation of real equitable wages, and I think that introducing a less predictable signal of market efficiency in the form of a variable exchange rate between labor and capital currencies seems would likely obscure whatever gleanings of fairness might be observed.
I think capital and labor are inextricably connected. As I see it, they fulfill the specialized directive and performative roles of project completion. Philosophically, there is already a solution to the negotiation between these roles: law and unionization. My conception is that these institutions are supposed to correct directives that are maligned against equitable wages for labor workers while preserving the directive incentives of capitalists so that socially beneficial projects can be completed.
Also, just sociologically speaking, this kind of currency splitting would externally impose an objective class distinction between labor workers and capitalist entrepreneurs, which I don't think is a good thing.
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RE: Score Difference Stratified Condorcet
@lime I see, so this would be used in absence of a Condorcet winner like for a ranked pairs resolution?
I was trying to think about burial but I don’t think my method addresses it quite as I conceived. My reasoning was that, by replacing absolute score differences with their more robust percentiles, burial (and bullet voting) strategies will suffer from severely diminishing returns compared with less risky and more honest ballots. For example, burying a second-favorite below a turkey to support a first favorite probably won’t significantly improve the score percentile provided by that voter to the first favorite’s runoff with the second favorite, but will significantly improve the chances of the turkey winning. This makes dishonest burial more severely punished and risky, meaning that fewer rational voters will choose to do it. Also, the effects of the fraction that do will be significantly reduced, since they will not only be fewer in number, but the magnitude of their indicated score differences will be majorly reeled back upon being replaced by their percentiles relative to the more honest bulk.
At the same time, the method is not restricted to Condorcet compliance, since, for example, it is possible for a [1-sqrt(1/2)]~0.2928… fraction minority of voters to overrule a sqrt(1/2)~0.707… fraction majority as long as the whole minority has the top quantile of absolute score differences and all of them have the same sign. That is the smallest possible minority that can overrule a majority in this method. It’s in one sense a generalized, more flexible extension of some of the reasonable measures we already have in the legislative houses, where for example a supermajority (2/3) is required for certain decisions.
Alternatively, each absolute score difference percentile could be measured relative to the distribution of all absolute score differences across all differences. The data set would consist of N*K(K-1)/2 values where N is the number of voters and K is the number of candidates.
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RE: Score Difference Stratified Condorcet
@lime but if you take the median and positivity means majority rule, then it’s just an ordinary Condorcet method, right?
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RE: Score Difference Stratified Condorcet
@lime if we take the raw score difference, doesn’t that just become score voting? Or maybe I misunderstand your meaning. I think c1>c2 if and only if c1 has a higher score than c2. In score voting, c1 and c2 will each accrue from each ballot the average of their scores on that ballot, and the sum of the score differentials will be the only deciding factor between them.
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Score Difference Stratified Condorcet
Let each voter v score candidate c as v(c). For each pair of candidates (c1,c2), consider the distribution of absolute score differences as v varies uniformly over the voters:
|v(c1)-v(c2)|~f(c1,c2).
To determine the marginal victory of c1 over c2, count -p, 0, or +p depending on the sign of the score difference (-1, 0, +1) and the percentile p of the ballot’s absolute score difference relative to f(c1,c2).
Elect the “Condorcet winner” from this marginal victory graph if one exists. Otherwise, elect the ordinary Condorcet winner if one exists. Otherwise, elect the score winner.
This method attempts to strike a balance between strict ordinal preference and cardinal preference—it becomes possible for a sufficiently passionate and sufficiently large minority to overrule a sufficiently dispassionate and sufficiently small majority.
It represents a more general class of methods that operate in a similar way, with perhaps a different activation function for score difference percentiles.
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RE: Symmetric Quantile-Normalized Score
@toby-pereira I believe burial is significantly more risky with SQNV than with Borda, due to the general non-linear and unknown nature of the final normalized positional scores. Burying a second-favorite front runner below the top half of one’s ranks significantly increases the risk that a turkey will win, and “half burying” the second-favorite (placing them just above the half way mark) still may provide a fair deal of support for that candidate. It’s really risky to place turkeys above the half way mark unless absolutely necessary for the same reason.
This of course doesn’t necessarily stop turkey raising and burial from happening, but the non-linearity introduces significantly more severe deterrence against burial and turkey raising for a rational voter. At the same time, SQNV frequently elects the Condorcet winner (I would say “more frequently than score” for example, but that would be based anecdotally on the examples I’ve observed).
I think one step for analysis would be to see if SQNV elects the Condorcet winner in cases where Borda does not. My guess is that this happens quite often, which would be a heuristic indication that SQNV generally has better burial resistance than Borda.
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Weird idea about Borda/SQNV and Condorcet to potentially mitigate Burial
By a Borda system, I mean a positional score system, possibly with equal rankings, but where candidates with equal rankings will be given the mean of the scores assigned to each rank.
Consider this weird idea:
(0) Let X be the set of viable candidates.
(1) Among the candidates in X, identify the positional score (or SQNV) winner and the Condorcet winner. If they are the same, or if no Condorcet winner exists, elect the positional score winner.
(2) Otherwise, remove the Condorcet winner from X, and on each ballot, increase the rank of each candidate scored below the Condorcet winner by 1. Repeat from step (1) until a winner is identified.
I haven’t investigated how this would work, but the idea intrigues me. My thinking is that it greatly increases the risk that a turkey-raised candidates (under which a true preferred candidate is buried) will win if they exist, which discourages burial, but without resorting to a strict Condorcet method.
In this method, the Condorcet winner is challenged to achieve high ranking positions in addition to having marginal victories over all other candidates, otherwise it will lose out to another candidate.
I have a very simple example where burial of a second-favorite front runner causes a (least favorite) turkey to win, I’ll post it here soon. But actually, the mismatch of the Condorcet winner and the SQNV winner didn’t occur, so SQNV itself made the burial risky.
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RE: Symmetric Quantile-Normalized Score
@toby-pereira I will have to compare with Borda. You’re right that burial definitely seems to be an issue. I think clone behavior is mitigated to a large degree by the normalization, but that remains to be seen in practice.
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RE: Symmetric Quantile-Normalized Score
@toby-pereira yes, that’s exactly right. I didn’t conceptualize it that way initially, but it is another way of describing the same adjustment.
With (m+1) candidates, and assuming k <= (m-k), the computation by the formula for your example would be as follows:
If q[k]=0.20, and q[m-k]=0.90, then
q’[k]=(q[k]+(1-q[m-k]))/2
=(0.2+0.1)/2=0.15and
q’[m-k]=(q[m-k]+(1-q[k]))/2
=(0.9+0.8)/2=0.85exactly as you said. The general result is proved here:
Proof:
By definition, for each k among 0,1,2,…m,
q’[k]:= 1/2(q[k]+(1-q[m-k]))
which is also
(q’[k]-0)= 1/2((q[k]-0)+(1-q[m-k])).
Also, you can check the following symmetry condition (which is true by design):
q’[k]+q’[m-k]=1
—> (q’[k]-0)=(1-q’[m-k]).Substitution into the previous result gives
(1-q’[m-k])= 1/2((q[k]-0)+(1-q[m-k])).
QED
Also, rearranging the symmetry condition, we find
(q’[m-k]-1/2)=(1/2-q’[k]).
With substitution of the definition of q’[m-k] on the left hand side, this is also
=1/2(q[m-k]+(1-q[k]))-1/2
=1/2(q[m-k]+(1-q[k])-1)
=1/2((q[m-k]-1/2)+(1/2-q[k])).So you can average the difference from the middle rather than the ends, the result is the same.
In fact, for any p in the interval [0,1],
(q’[m-k]-p)=((1-p)-q’[k])
which by substitution is
=1/2(q[m-k]+(1-q[k]))-p
=1/2(q[m-k]+(1-q[k])-2p)
=1/2((q[m-k]-p)+((1-p)-q[k])).I have the intuition that it might be tactically advantageous to cast a pre-symmetrized score ballot with a shape that the voter finds most equitable as a representation of their preferences. I haven’t thought that through fully, it may not make a difference. Effectively a voter is voting for candidates and for the final positional scores that will be used.