For people who insist on ordinal voting methods, should we not recommend Borda Count?
It is clearly the best among them, based on this chart comparing common voting methods according to the criteria of simplicity, fairness (voter satisfaction efficiency), and vulnerability to strategic voting:
I commonly recommend the “top 3” Borda Count: Rank your top 3 choices only and compile by the Borda method. This both simplifies voting and minimizes tactical voting. However in some situations you may need a primary vote by proportional representation to eliminate candidates who’ve been added only for tactical purposes.
Dick Burkhart, Ph.D. mathematics
4802 S Othello St, Seattle, WA 98118
206-721-5672 (home) 206-851-0027 (cell)
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If I recall, Warren analyzed the rank-order methods once and concluded that Schulze beatpath method was his favorite. It is clone-proof, so it should be fairly resistant to strategic nomination.It's important to ask who controls the candidates.There is much that I like about Borda, but it is not clone-proof. (http://en.wikipedia.org/wiki/Independence_of_clones_criterion#Borda_count) Factions have an incentive to nominate as many candidates as possible that are similar to their preferred alternative. So I think Borda is my favorite when the set of candidates arises naturally, but am wary of it when the candidates come from some nomination process.
I strongly disagree with recommending Borda. In my opinion, Borda is the only method which could, under reasonable strategy, give results that are worse than plurality. I think that Borda's famous defensive quote that it's a method "for honest men" about sums it up. That is, with honest voters, it's nearly as good as score; but with strategy, it could be unboundedly bad.
In particular, it's the only method where I think the "dark horse 3" (Warren's DH3) pathology would be a real threat. In my VSE simulations (my redo of Warren's BR), this was evident: several of the "top/bottom" strategic scenarios had negative VSE for Borda (that is, worse than random ballot); and Borda was worse than anything but Plurality using one-sided strategy (where the supporters of the honest winner are honest, but anyone who prefers the honest second-place is strategic). In the latter case, the VSE's in a representative voter utility model are:
scenarios | honBallot0 | stratBallot1 | stratBallot2 | OssChooser3 | OssChooser4 | ProbChooser5 | ProbChooser6 | ProbChooser7 | ProbChooser8 | ProbChooser9 | LazyChooser10 | ProbChooser11 | DeltaChooser12 |
Score0 | 0.9278496244 | 0.8224658875 | 0.7971869977 | 0.6831684184 | 0.8333245178 | 0.9160040478 | 0.9008207469 | 0.8717989204 | 0.9237152553 | 0.9318729627 | 0.8183036043 | 0.8288883968 | 0.8125142468 |
Score1 | 0.5747001944 | 0.7830457092 | 0.7568044558 | 0.6043617289 | 0.6686293517 | 0.8148670394 | 0.8247867322 | 0.8139234301 | 0.7829571812 | 0.7410734687 | 0.7767793987 | 0.7786840125 | 0.799286268 |
Approval2 | 0.9133709997 | 0.792490559 | 0.7468809598 | 0.6755451388 | 0.8336253242 | 0.9194441106 | 0.8801824049 | 0.8483380453 | 0.9086492086 | 0.9232029493 | 0.801553656 | 0.7838536842 | 0.8006959849 |
Plurality3 | 0.7042032473 | 0.381966361 | 0.3016107936 | 0.3864870779 | 0.632611169 | 0.6823754192 | 0.4839470264 | 0.3712716361 | 0.6180618543 | 0.706190903 | 0.3788714488 | 0.3693226161 | 0.3679137522 |
Irv4 | 0.8656768772 | 0.6739399915 | 0.6765919629 | 0.7627178557 | 0.8144248604 | 0.8692821464 | 0.7730744812 | 0.7105123224 | 0.8046867946 | 0.8421858271 | 0.673209407 | 0.6678520634 | 0.6742136402 |
Minimax5 | 0.8753775113 | 0.4038590253 | 0.3713852954 | 0.5871970602 | 0.789271416 | 0.8487545051 | 0.6997755328 | 0.5473404414 | 0.7614101834 | 0.8321969256 | 0.4031689164 | 0.4081938674 | 0.4050216189 |
Borda6 | 0.8790139305 | -0.8500994391 | -0.6600492104 | 0.521018883 | 0.7685387008 | 0.8412409559 | 0.6326718822 | -0.1535071888 | 0.6860257393 | 0.8582263523 | -0.8529629248 | -0.8562396573 | -0.8697301604 |
MavBy7 | 0.8705056185 | 0.881986234 | 0.8627542754 | 0.8676974096 | 0.870591752 | 0.8762386843 | 0.8952695276 | 0.8812859608 | 0.8778834885 | 0.8850518009 | 0.8200808431 | 0.8808748973 | 0.8395890512 |
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I've wondered about the Borda results on Warren's Bayesian Regret charts before, given that BR has been used to promote score, but not used to promote Borda over the other ranked methods to anywhere near the same extent.However, even if we take the graph at face value, it doesn't mean that Borda is better than Condorcet. It means it's slightly better with honest voting, and slightly better with full strategic voting. It doesn't say anything about how much voters would feel encouraged to vote strategically by each system. So it doesn't say where on each bar the most likely BR level is for Borda or Condorcet, and so doesn't give a conclusive answer as to which is better.
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It doesn't say anything about how much voters would feel encouraged to vote strategically by each system.
This depends on the voting method. For the Borda Count the most common method is the “Modified Borda Count” of Peter Emerson. If ‘n’ candidates are elected, then the top candidate gets n points, the next n-1 points, etc., whereas a full vote for m > n candidates would be (m,m-1,m-2,…,1,0,…). The undervote sums to n(n+1)/2 points whereas the full vote sums to m(m+1)/2 points, so the undervote is penalized by the factor n(n+1)/[m(m+1)]. This is a very heavy penalty if the voter selects only his or her top candidates (n = 1).
For a less heavy penalty, I specify a minimum and a maximum number of candidates to be ranked for a full vote. For example, if 3 is the minimum and 6 is the maximum, out of 9 candidates, then 6 rated candidates would be (6,5,4,3,2,1,0,0,0) summing to 21 points, 3 rated candidates would be (6,5,4,1,1,1,1,1,1) also summing to 21 points, 2 rated candidates would be (6,5,10/7,10/7,10/7,10/7,10/7,10/7,10/7) scaled by 2/3 so that it sums to 14 points instead of 21, and 1 rated candidate would be (6,15/8,15/8,15/8,15/8,15/8,15/8,15/8,15/8) scaled by 1/3 so that it sums to 7 points instead of 21.
Dick Burkhart
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This depends on the voting method. ...
I permit equal rankings in my Borda type voting algorithms. I assign points (possibly fractional) equal to the average of a strong ranking of the equally ranked candidates. Thus if a ranking of (1,2,2,2,3,4) would convert to points of (5,3,3,3,1,0) instead of (5,4,3,2,1,0), and (1,1,2,3) would become (2.5,2.5,1,0) instead of (3,2,1,0). Thus the point sum remains the same.
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>I would be particularly interested to know whether Steve Cobb would consider a given
>system to be "ordinal" if it allows to equal-rank the unnamed candidates on a particular ballot.
I’m not one of the theorists, but it seems to me that an ordinal system could allow equal-ranking. I can imagine that the question is the source of some debate. However, such a system certainly wouldn’t be cardinal.
The motivation for the original question arose from the chart summarizing Warren’s simulation, which we often refer to. The original post included a link to it. According to that chart, Borda is clearly superior to Condorcet and IRV in both fairness and simplicity, and it bears more resemblance to cardinal systems than Condorcet and IRV, so we should recommend it as the best ordinal option. Jameson questioned the simulation that produced the results.
Personally, my favorite ranking system is randomly choosing one among several ranking systems after all the ballots are cast.