Suppose you had a nice metric on a candidate space for measuring the disparity between members of the electorate, and you wanted to use the metric to find the most representative member of the electorate. How would you use that metric? There are many possibilities, but let's consider three of them that are fairly natural and easy to understand: Let V be the set of voters, and let K be the subset of V consisting of the candidates who managed to get their names on the ballot. 1. Elect the candidate k that minimizes the total distance to the other voters, i.e. elect argmin{TotDist(k,V)|k in K} where TotDist(k,V) is the total distance from k to the other members of V. 2. Elect the candidate k closest to the most representative voter, namely argmin{TotDist(v,V)|v in V}. This is analogous to the Condorcet dictum, "elect the candidate closest to the voter median." 3. Find the point p in the space of possible ballots that minimizes the total distance to the actual ballots. Then elect the most preferred candidate on that idealized ballot. Number 3 strongly depends on the Universal Domain requirement that the only admissible information for determining the winner of the election is the ordinal information contained in the ballots of the voters,. So if our method depends on a distance metric between voters, that metric must be completely determined by the voters' ordinal ballots. In the case of Kemeny-Young, method three is the method used. The metric in question is the Kendall-tau distance between rankings. The idealized ballot ranking that determines the winner may or may not be one of the actual ballot rankings. The computational complexity of K-Y is not inherent in the Kendall-tau distance calculations. Rather it stems from the sheer number of possible idealized ballots. So methods one and two have available (low degree) polynomial time counting procedures, despite their fundamental use of the Kendall-tau metric. [The same goes for my de-cloned version of K-Y based on a clone free version of the Kendall-tau metric.] The winning idealized ballot not only decides the election winner, it also suggests an entire "social order" or "finish order" of the candidates. That's nice, and some applications may require it, but when the number of candidates is large, the computational burden has to be forestalled one way or another. More to the point ...the idealized finish order is not needed in single winner elections. Either of the first two methods above works great. In the extremely rare case where they do not agree, you can elect the pairwise winner of the two methods. What got me thinking about this was Colin Champion's question about a metric characterization of multi-winner/PR methods. ¿How would you make use of a decent metric on candidate/voter/ballot space to compose (like a musician) a good multi-winner method? ... or for starters, to choose between two or more proposed posible winning "slates" ....?

On 16.02.2022 08:58, Forest Simmons wrote: > Suppose you had a nice metric on a candidate space for measuring the > disparity between members of the electorate, and you wanted to use the > metric to find the most representative member of the electorate. How > would you use that metric? > > There are many possibilities, but let's consider three of them that are > fairly natural and easy to understand: > > Let V be the set of voters, and let K be the subset of V consisting of > the candidates who managed to get their names on the ballot.  > > 1. Elect the candidate k that minimizes the total distance to the  other > voters, i.e. elect argmin{TotDist(k,V)|k in K} where TotDist(k,V) is the > total distance from k to the other members of V. > > 2. Elect the candidate k closest to the most representative voter, > namely argmin{TotDist(v,V)|v in V}. This is analogous to the Condorcet > dictum, "elect the candidate closest to the voter median." These approaches make use of the ballots that the candidates cast, right? That would seem to be more in the spirit of Asset than an ordinary voting method. If there's a secret ballot, then the candidates are incentivized to lie on their ballots; and if the candidates' ballots are not secret, then the candidates have to predict the election accurately while still making their ballots pull the outcome in their direction. There's nothing wrong as such with the second - it fits with the Asset principle of offloading the strategic burden on the candidates themselves - but one should be aware that it's not an ordinary voting method and couldn't be used for choosing what movie to watch, budget item to fund, or similar. ... That makes me think, though. Could one say that the distinguishing feature of proxy and Asset-type elections is that they violate anonymity? This because the proxies are privileged in the sense that their votes are amplified by however many other voters decide to back them. Perhaps not so much for traditional Asset, where you first choose a "parliament" that then negotiates until some winner crosses a threshold. That method has multiple rounds: first voting for assets, then the parliamentary formation, then the final winner is decided; and it doesn't violate anonymity in either round. But a one-shot Asset-like where you either designate a proxy (whose vote you'll copy) or vote directly... would. I think. -km

El mié., 16 de feb. de 2022 4:33 a. m., Kristofer Munsterhjelm < km_elmet@t-online.de> escribió: > On 16.02.2022 08:58, Forest Simmons wrote: > > Suppose you had a nice metric on a candidate space for measuring the > > disparity between members of the electorate, and you wanted to use the > > metric to find the most representative member of the electorate. How > > would you use that metric? > > > > There are many possibilities, but let's consider three of them that are > > fairly natural and easy to understand: > > > > Let V be the set of voters, and let K be the subset of V consisting of > > the candidates who managed to get their names on the ballot. > > > > 1. Elect the candidate k that minimizes the total distance to the other > > voters, i.e. elect argmin{TotDist(k,V)|k in K} where TotDist(k,V) is the > > total distance from k to the other members of V. > > > > 2. Elect the candidate k closest to the most representative voter, > > namely argmin{TotDist(v,V)|v in V}. This is analogous to the Condorcet > > dictum, "elect the candidate closest to the voter median." > > These approaches make use of the ballots that the candidates cast, > right? That's a possibility, but not what I had in mind. For example, in method two, once you have found the secret anonymous ballot B that minimizes the total distance to the other ballots, just assume that ballot B's favorite is the candidate closest to ballot B. This ballot B could well be the ballot of one of the candidates, but there is no way of knowing that. This is the same reasoning that Kemeny-Young uses to identify the candidate closest to its idealized ballot. Another possibility is (for each candidate k) find the ballot B(k) that minimizes the total distance to the ballots that rank k as favorite. Then consider B(k) to be the position of candidate k in ballot space. That might be the best approach for multi-winner K-Y. Arguably either one of these inferences of the location of the winning candidate is just as good as the (computationally expensive) idealized ballot inference of Kemeny-Young. That would seem to be more in the spirit of Asset than anh > ordinary voting method. If there's a secret ballot, then the candidates > are incentivized to lie on their ballots; and if the candidates' ballots > are not secret, then the candidates have to predict the election > accurately while still making their ballots pull the outcome in their > direction. > > There's nothing wrong as such with the second - it fits with the Asset > principle of offloading the strategic burden on the candidates > themselves - but one should be aware that it's not an ordinary voting > method and couldn't be used for choosing what movie to watch, budget > item to fund, or similar. > > ... > > That makes me think, though. Could one say that the distinguishing > feature of proxy and Asset-type elections is that they violate > anonymity? This because the proxies are privileged in the sense that > their votes are amplified by however many other voters decide to back them. > > Perhaps not so much for traditional Asset, where you first choose a > "parliament" that then negotiates until some winner crosses a threshold. > That method has multiple rounds: first voting for assets, then the > parliamentary formation, then the final winner is decided; and it > doesn't violate anonymity in either round. > > But a one-shot Asset-like where you either designate a proxy (whose vote > you'll copy) or vote directly... would. I think. > > -km >

Kemeny-Young is a geometric method based on the Kendall-tau distance between rankings. If we combine this idea with the Universal Domain axiom that the only relevant voter information for determining the outcome of an election is the ordinal information contained in the ballot rankings of the candidates, then for all intents and purposes we can identify the space of voters with the space of candidate rankings. Let's see where this leads when we adopt the Kendall-tau distance as a metric on the ballot space. For example, consider the ballot profile 40 A>B>C 35 B>C>A 25 C>A>B The Kendall-tau distance between any pair of the three faction rankings is exactly two units of distance; it takes a minimum of two swaps to turn a permutation of three symbols into another such permutation of the same cyclical parity. So geometrically the three distinct ballot orders of the three respective factions represent voters at the three respective vertices of an equilateral triangle of side two unts. Note that the candidates themselves are supposed to be voters. Which of the three ballot orders gives A's position on the equilateral triangle? And which represent B and C? It's not hard to see by a process of elimination (and assumption of sincere ballot preferences) that A has to be at the A>B>C ballot position, B at the B>C>A position, and C at the C>A>B vertex of the equilateral triangle. If that is the case, how can we take seriously the ballot order of the 40 members of the A>B>C faction that says that they unanimously, strictly prefer candidate B to candidate C? All forty of them are equally close (a distance of 2 units) to candidates B and C. How is it that not even one of them prefers C to B? Evidently the Kendall-tau metric is not consistent with all geometric based affinities. So what do we do when our randomly generated factions give us a profile that is inconsistent with the Kendall-tau metric? Suppose that it was generated by random points in Euclidean space with a Euclidean L_2 metric or some other vector space metric. Then what do we say when it is not consistent with the Kendall-tau metric? Which do we trust more ... L_p metrics or Kendall-tau? The best answer to that depends on how seriously we take the Universal Domain axiom. Kendall-tau is topologically compatible with Universal Domain, while the vector space metrics are not. But, you object, Kemeny-Young, based on Kendall-tau is clone dependent. Two replies:(1) so are the L_p metrics in most ballot profile generation contexts. (2) Kendall-tau is easily de-cloned while completely retaining its UD compatible topology For clarity let me reiterate: no vector space metric is topologically compatible with Universal Domain. Personally, I'm not afraid of coloring outside of the lines ... most of my favorite methods are hybrids that violate UD ... but UD purists need to know the inherit UD inconsistency in geometrically generating ranked choice ballot profiles with UD incompatible topology ... namely flat vector space topology as opposed to the curved space topology of permutation/ranking space, which resides on the boundary of an (n-1) dimensional ball when there are n candidates. I hope this "abstract nonsense" (as Professor Klaus Bichtler used to glowingly call it) has helped to connect certain conceptual categories in relation to each other. Our understanding grows as our internal network of such connections becomes more complete ... especially as the degrees of separation from familiar intuitive concepts decrease. El mié., 16 de feb. de 2022 4:47 p. m., Forest Simmons < forest.simmons21@gmail.com> escribió: > > > El mié., 16 de feb. de 2022 4:33 a. m., Kristofer Munsterhjelm < > km_elmet@t-online.de> escribió: > >> On 16.02.2022 08:58, Forest Simmons wrote: >> > Suppose you had a nice metric on a candidate space for measuring the >> > disparity between members of the electorate, and you wanted to use the >> > metric to find the most representative member of the electorate. How >> > would you use that metric? >> > >> > There are many possibilities, but let's consider three of them that are >> > fairly natural and easy to understand: >> > >> > Let V be the set of voters, and let K be the subset of V consisting of >> > the candidates who managed to get their names on the ballot. >> > >> > 1. Elect the candidate k that minimizes the total distance to the other >> > voters, i.e. elect argmin{TotDist(k,V)|k in K} where TotDist(k,V) is the >> > total distance from k to the other members of V. >> > >> > 2. Elect the candidate k closest to the most representative voter, >> > namely argmin{TotDist(v,V)|v in V}. This is analogous to the Condorcet >> > dictum, "elect the candidate closest to the voter median." >> >> These approaches make use of the ballots that the candidates cast, >> right? > > > That's a possibility, but not what I had in mind. For example, in method > two, once you have found the secret anonymous ballot B that minimizes the > total distance to the other ballots, just assume that ballot B's favorite > is the candidate closest to ballot B. This ballot B could well be the > ballot of one of the candidates, but there is no way of knowing that. This > is the same reasoning that Kemeny-Young uses to identify the candidate > closest to its idealized ballot. > > Another possibility is (for each candidate k) find the ballot B(k) that > minimizes the total distance to the ballots that rank k as favorite. Then > consider B(k) to be the position of candidate k in ballot space. That might > be the best approach for multi-winner K-Y. > > Arguably either one of these inferences of the location of the winning > candidate is just as good as the (computationally expensive) idealized > ballot inference of Kemeny-Young. > > That would seem to be more in the spirit of Asset than anh >> ordinary voting method. If there's a secret ballot, then the candidates >> are incentivized to lie on their ballots; and if the candidates' ballots >> are not secret, then the candidates have to predict the election >> accurately while still making their ballots pull the outcome in their >> direction. >> >> There's nothing wrong as such with the second - it fits with the Asset >> principle of offloading the strategic burden on the candidates >> themselves - but one should be aware that it's not an ordinary voting >> method and couldn't be used for choosing what movie to watch, budget >> item to fund, or similar. >> >> ... >> >> That makes me think, though. Could one say that the distinguishing >> feature of proxy and Asset-type elections is that they violate >> anonymity? This because the proxies are privileged in the sense that >> their votes are amplified by however many other voters decide to back >> them. >> >> Perhaps not so much for traditional Asset, where you first choose a >> "parliament" that then negotiates until some winner crosses a threshold. >> That method has multiple rounds: first voting for assets, then the >> parliamentary formation, then the final winner is decided; and it >> doesn't violate anonymity in either round. >> >> But a one-shot Asset-like where you either designate a proxy (whose vote >> you'll copy) or vote directly... would. I think. >> >> -km >> >