Re: [EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)
At 11:29 AM 3/28/2010, Chris Benham wrote: Jameson Quinn wrote (26 March 2010): snip Right now, I think MCV - that is, two-rank, equality-allowed Bucklin, with top-two runoffs if no candidate receives a majority of approvals in those two ranks - is my favorite proposal for practical implementation. snip I agree with him, except that I'd make it three-rank, unless there were fewer than four candidates. (i.e., four explicit on the ballot plus write-in). This allows full ranking of four candidates, which may be completely adequate, given the equal ranking allowed, even for very large candidate sets. (Three-rank Bucklin allows four actual ranks, when no rank is considered the bottom rank.) Jameson, What does MCV stand for? Does top-two runoffs mean a second trip to the polls? Yes, I'm sure. That's a runoff election. How are the candidates scored to determine the top two? Is it based on the candidates' scores after the second Bucklin round? Probably. I'm not sure that the limitation to two candidates in the runoff is good. Bucklin could surely handle three, so the runoff could be two-rank Bucklin, with top three, but using three ranks is harmless and allows full ranking with a write-in in the runoff. Yes. Most-approved top three would be one choice; another would be top-two plus a Condorcet winner, if one exists that is not in the top two. However, I'd probably prefer this algorithm: Condorcet winner, if apparent from ballots, will either win or be included in a runoff. If no candidate gains a majority, considering all Bucklin ranks, then a runoff would include a Condorcet winner, plus one or two of the most-approved candidates, as necessary to show two. So there are two selected candidates for the runoff. Write-ins would be allowed in the runoff, so the Bucklin runoff would then be two-rank Bucklin. The remaining question: what if there are two candidates gaining a majority, one candidate is leading in approvals, but the other is the Condorcet winner? In theory, a runoff should be held, but it might not be considered to be worth the cost. I'd want to study this case more closely, and I don't propose Condorcet analysis as part of the first implementations, but study of ballots later from real elections will ultimately reveal how significant this might be. In that event, yes, the runoff would be between the most-approved candidates (considering any Bucklin vote, at any rank, as an approval.) Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)
At 10:59 AM 4/2/2010, C.Benham wrote: Unfortunately these top-two runoff versions break MCA's compliance with Favorite Betrayal and Mono-raise. Top-rating your favourite F could cause F to displace your compromise C in the runoff with your greater-evil E, and then F loses to E when C would have beaten E. The problem arises with candidate elimination, but the analysis is incompletely. Sure, if all you do is top-rate F, bullet-voting, your vote could have this effect, with a two-candidate restriction. However, two candidates is an artificial restriction; some runoff systems use three, and repeated ballot, one of the most-used voting systems in the world (vote-for-one ballot, majority required), has no such problem, relying upon the desire of partisans to complete the election. If a majority do not want to complete the election, there you have it. Majority rule. Repeated ballot, majority rule, using approval ballots, suggests a very obvious strategy, and, as Robert's Rules of Order points out, each pool is informed by the results of the poll before. The first poll will tend strongly to bullet votes for voters with a significant favorite. Then the approval cutoff is raised. Bucklin collapses the first two or three rounds of this into a single ballot. So, in the scenario described, the voter has voted a poor strategy, and sees a bad result. TANSTAAFL. If the voter is concerned about that risk, the voter should certainly have added an approval for C. Favorite Betrayal assume that the favorite is raised above the compromise, which causes the voter to lose utility, but in Bucklin methods, the preference, before the end of counting, i.e., before any runoff choice is made, is not maintained, it is reduced to equal ranking. Bucklin is an Approval method, simply one that allows preference to be first asserted before being collapsed to equal ranking. Also, like plain Approval followed by a runoff between the two most approved candidates, it is *very* vulnerable to turkey-raising Push-over strategy. Voters who are fairly confident that their favourite can get into the final runoff have an incentive to also approve (or top-rate, depending on the version) all the candidates they are confident their favourite can pairwise beat in the runoff. It's tricky. All candidate elimination schemes are vulnerable to possible turkey-raising, but as the number of preserved candidates increases, it becomes more difficult. Candidate elimination is the culprit, period, and it's worst when the restriction is to two. This is a reason why I strongly suggest allowing at least one additional write-in approval in a runoff. This makes a two-round system a closer simulation of repeated ballot. It is possible, even, that the second round has no eliminations at all, other than withdrawals. Same ballot, but printed on the ballot are the results from the first round. Any significant write-ins are reported specifically. A new write-in is possible. The second ballot is then pure Bucklin, no runoff, plurality wins after all rounds are collapsed if no majority found in the process. What I strongly prefer to see is experimentation with Bucklin. We need more data to understand how the method works in reality, we do have historicasl data that indicates it worked well. (The common assertion is that, in some circumstances, only a small percentage of voters added additional approvals, but these were, I believe, primary elections, and first preferences will stand out even more in the minds of the voters. In the Bucklin elections where I've seen actual round data, there was hefty voting in the lower ranks. That in a one-round Bucklin system most voters, stable conditions -- not the first election, but after voters become accustomed to it -- may bullet vote, is not a failure of the system, rather it shows preference strength for the favorite.) Also some people might object that parties that run a pair of clones have an advantage over parties that run a single candidate. Of course. But this doesn't work if the second ballot isn't restricted. And it runs into the reality of political election process: two candidates means divided attention and campaigning, two names to raise in familiarity and approval, not just one. It can seriously backfire. What seems clear to me is that a runoff system using Bucklin-ER, three-rank, for both ballots and a liberal inclusion in the second poll, incentivizes voting a true, sincere Range ballot, particularly in the first poll. Approval is tied to the utility to the voter of avoiding a runoff. If the voter prefers a runoff to any result that does not elect the favorite, the voter will bullet vote, a sincere and accurate vote. The Range Ballot for three-rank Bucklin is one with ratings of 4, 3, 2, 0, with 0 being the default. The rating of 2 is the rating of equal or better than the expected result. In the second round, the voter has better
Re: [EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)
Jameson Quinn wrote (28 March 2010): / What does MCV stand for? // / Ooops. I garbled your term, didn't I? It's supposed to be Majority Choice Approval, not Majority Choice Voting. Majority Choice Approval was invented and introduced a few years ago by Forest Simmons, and I think he coined the term. For a short time I endorsed it as something simple that meets Favorite Betrayal, the voted 3-slot ballot version of Majority for Solid Coalitions and Mono-raise. / Does top-two runoffs mean a second trip to the polls? // / Yes. I regard this as an advantage. If the situation is divisive enough to prevent a majority choice in two rounds of approval, then a further period of campaigning is a healthy thing. It's the only way to guarantee a majority. (I don't think that mandating full ranking counts as a true majority). / // How are the candidates scored to determine the top two? Is it based on the //candidates' scores after the second Bucklin round? // /That's the simplest answer, and I'd support it. It's also the best answer with honest voters. Actually, the best answer for discouraging strategy is to use the two first-round winners. That tends to discourage strategic bullet voting, since expanding your second-round approval can not keep your favorite candidate from a runoff. Unfortunately these top-two runoff versions break MCA's compliance with Favorite Betrayal and Mono-raise. Top-rating your favourite F could cause F to displace your compromise C in the runoff with your greater-evil E, and then F loses to E when C would have beaten E. Also, like plain Approval followed by a runoff between the two most approved candidates, it is *very* vulnerable to turkey-raising Push-over strategy. Voters who are fairly confident that their favourite can get into the final runoff have an incentive to also approve (or top-rate, depending on the version) all the candidates they are confident their favourite can pairwise beat in the runoff. The Push-over incentive is stronger than it is in normal TTR, because the strategists don't have to abandon their favourite in the first round (and so taking a much greater risk, if there are too many trying the strategy, of their favourite not getting into the final without their votes when without their strategising their favourite would have got into the second round and won it). Also some people might object that parties that run a pair of clones have an advantage over parties that run a single candidate. From your (Jameson's) earlier (26 March 2010) message, I gather you consider likely to elect the CW a big positive. For something simple then, why not 3-slot Condorcet//Approval? *Voters give each candidate a Top, Middle or Bottom rating. Default rating is Bottom. If one candidate X (based on these maybe constrained ballots) pairwise beats all others, elect X. Otherwise, interpreting Top and Middle rating as approval, elect the most approved candidate.* Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)
Jameson Quinn wrote (26 March 2010): snip Right now, I think MCV - that is, two-rank, equality-allowed Bucklin, with top-two runoffs if no candidate receives a majority of approvals in those two ranks - is my favorite proposal for practical implementation. snip Jameson, What does MCV stand for? Does top-two runoffs mean a second trip to the polls? How are the candidates scored to determine the top two? Is it based on the candidates' scores after the second Bucklin round? Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)
2010/3/28 Chris Benham cbenha...@yahoo.com.au Jameson Quinn wrote (26 March 2010): snip Right now, I think MCV - that is, two-rank, equality-allowed Bucklin, with top-two runoffs if no candidate receives a majority of approvals in those two ranks - is my favorite proposal for practical implementation. snip Jameson, What does MCV stand for? Ooops. I garbled your term, didn't I? It's supposed to be Majority Choice Approval, not Majority Choice Voting. Does top-two runoffs mean a second trip to the polls? Yes. I regard this as an advantage. If the situation is divisive enough to prevent a majority choice in two rounds of approval, then a further period of campaigning is a healthy thing. It's the only way to guarantee a majority. (I don't think that mandating full ranking counts as a true majority). How are the candidates scored to determine the top two? Is it based on the candidates' scores after the second Bucklin round? That's the simplest answer, and I'd support it. It's also the best answer with honest voters. Actually, the best answer for discouraging strategy is to use the two first-round winners. That tends to discourage strategic bullet voting, since expanding your second-round approval can not keep your favorite candidate from a runoff. As a compromise between these two, I would run the first-round approval winner against the second-round winner. If these are the same, it probably shows that people are voting to narrowly; to discourage this from happening, in that case you use the two first-round winners. But these are details. I'd strongly support any of these systems, whichever one had more support from other activists. Chris Benham Jameson Quinn Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)
The Condorcet How? discussion got me to thinking (again) about how it's good to have similar proposals for single-winner and proportional systems. So I'd like to argue that MCV is an excellent single-winner system, then suggest a multiwinner generalization which is also attractive. Right now, I think MCV - that is, two-rank, equality-allowed Bucklin, with top-two runoffs if no candidate receives a majority of approvals in those two ranks - is my favorite proposal for practical implementation. While it's not the theoretical best method, it does the best on the following practical criteria, in order of importance to me: 1. Attainable (simple to explain, simple to vote, close to systems which have already been used in government, runs easily on existing equipment) 2. Strategy resistant (Unlike Range, Approval, or even 3-rank Bucklin, when there's a known CW, voting doesn't feel strategic[0]; and unlike all Condorcet systems and many others, all optimal strategies are semi-honest, which avoids pathologies.) [1] 3. Good honest results (by Bayesian Regret [2], or by Condorcet Criterion [3], or by Monotonicity, or other measures. 4. Summable (thus easily recountable, sampleable, etc. - this is important for confidence and legitimacy.) So, what would be a multiwinner variant of MCV which preserves these advantages as much as possible?[4] First, I'll describe a multiwinner system based on multiround Bucklin, then I'll explain how to patch it for two rounds (as the possible runoff patches MCV). The system I have come up with is STV-like - that is, candidates are elected one-by-one with droop quotas, which uses up a droop quota of votes. As long as there's a candidate with more than a Droop quota of approvals, elect that candidate. All ballots which approve that candidate are used up proportionally (multiplied by a factor of (a-D)/a, where a is the number of approvals that the candidate has, and D is the Droop quota). This fully defines the result. When no more candidates have a droop quota of approvals, proceed to the next round: re-weight each ballot to 1, add the next-lower category of candidates to the approved set for all ballots, and go through the list of already-elected candidates in order, re-discounting as if they'd just been elected. How can you ensure electing a full slate? Let A be the minimum number of approvals per ballot in a round, C be the number of candidates, and S be the number of seats. If there is a situation which elects only N candidates, then there's a situation where the same candidates are elected and no voter approves more than A-N candidates outside that set (that is, all voters approve the winners; these votes are free in terms of not electing more candidates). You want to be sure that S candidates are elected; so to prove it, assume the contrary, that only S-1 or fewer candidates are elected. So if A-S+1 approvals per vote, over C-S+1 candidates, with (S-1)/(S+1) of the votes used up - that is, 2/(S+1) of them remaining - is enough to ensure another Droop quota, then that's the contradiction that D's the QED. That is, 2(A-S+1)/(C-S+1)(S+1)1/(S+1). Solve for A: A(C+S-1)/2. Thus, if you require at least that many approvals (not counting write-ins) - one for each seat, and half of the rest - from each voter in the last round, you are guaranteed to elect a full slate. (I'll discuss ways to avoid this burden and thus finish in 2 rounds, below). In order to count this election without keeping an exponentially large number of piles of (possibly fractional) ballots, you can keep aggregate information about the remaining votes for in a summable C^2 half-matrix per round: M(x,y) says how many remaining votes approve both X and Y. Of course, this matrix is symmetrical around the x=y diagonal, which gives the one-candidate approval. This matrix does not have the same information as all ballots for the round; that would be exponential. However, it does keep enough information to give results. When you eliminate a droop quota (D votes) for candidate x, you find the used up votes factor u=(D/M(x,x)). All cells involving that candidate are multiplied by the remainder factor r=1-u, and, for y != x, M(y,y) is replaced by M(y,y) - u*M(x,y). This system is proportional: a group of N droop quotas who approve only the same MN candidates in round 1 will, obviously, elect N of them - their ballots will go to no one else, and the round will not end until N droop quotas have been deducted from the approval of the candidate set. It's monotonic, in the sense that raising a winning candidate to a higher approval (earlier round) cannot cause that candidate to lose. It affects only the round where you added their approval, and if it does not cause their election that round, it does not affect the results of that round, because all ballots are treated proportionally, and only a different candidate being elected can affect later results. Note that that does not mean that it obeys the participation criterion.
