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.<br>
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<br>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:<br>
<br>1. Attainable (simple to explain, simple to vote, close to systems which have already been used in government, runs easily on existing equipment)<br>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]<br>
3. Good honest results (by Bayesian Regret [2], or by Condorcet Criterion [3], or by Monotonicity, or other measures.<br>4. Summable (thus easily recountable, sampleable, etc. - this is important for confidence and legitimacy.)<br>
<br>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.<br>
<br>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).<br>
<br>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.<br>
<br>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).<br>
<br>This system is proportional: a group of N droop quotas who approve only the same M>N 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.<br>
<br>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.<br>
<br>Note that that does not mean that it obeys the participation criterion. By participating you can cause one of your preferred candidates to be elected in an earlier round than otherwise, and this could cause another of your preferred candidates not to be elected. Still, one should remember that in a proportional system, you do not have any right to determine who wins once "your" representative is elected. I believe that it does obey a weaker "proportional participation" criterion: your participation cannot cause a given candidate X to lose instead of win, unless there is some candidate Y, whom you rank equal or higher, who wins when you participate (whether or not they won without you). Personally, I find it implausible that anyone could find a practical, safe strategy involving this flaw; especially since one would have to be strategizing against "allies" who share relatively strong support (or, support that is at least as strong as your motivation to strategize) for at least one candidate.<br>
<br>Let's work out an example. Say you have the following approvals in the first round of a 5-seat election:<br><br>A<br>AB<br>AC<br>ACD<br>BDFG<br>
CE<br>DE<br>DEFG<br>F<br>G<br><br>The approval matrix is: (Only the lower-left half; the upper-right is just symmetrical)<br><font face="courier new,monospace"><b> </b> A B C D E F G<br>A 4<br>B 1 2<br>C 2 3<br>
D 1 1 1 4 <br>E 1 2 3 <br>F 1 2 1 3 <br>G 1 2 1 2 3</font><br><br>A and D have the most approvals: 4 each. That's more than the Droop quota of 2, so we'll elect one. We'll break the tie towards the candidate with a lower average number of approvals on the ballots that approve them (this uses the "scarcer per ballot, more valuable" approvals first, and leaves the "more flexible" ones for later). In this case, that's A. So we have to take half (Droop quota/total approvals - 2/4) of each A vote away.<br>
<font face="courier new,monospace"><b> </b> A B C D E F G<br>A 2<br>B .5 1.5<br>C 1 2<br>D .5 1 1 3.5 <br>E 1 2 3 <br>F 1 2 1 3 <br>G 1 2 1 2 3</font><br><br>
Next comes D, using 4/7 of the votes<br>
<font face="courier new,monospace"><b> </b> A B C D E F G<br>A 1.79<br>B .5 1.5<br>C 1 1.57<br>D .29 .57 1.5 <br>E 1 1.14 2.14 <br>F 1 1.14 1 2.14 <br>
G 1 1.14 1 2 2.14</font><br><br>Next, either E, F, or G, using 14/15 of their total. E wins the tiebreaker.<br>
<font face="courier new,monospace"><b> </b> A B C D E F G<br>A 1.79<br>B .5 1.5<br>C 1 .64<br>D .29 .57 .46 <br>E .07 .1 .14 <br>F 1 1.14 .07 .21 <br>
G 1 1.14 .07 2 .21</font><br><br>We've elected A, D, and E; and there is no Droop quota left, so we'd proceed to the next round.<br><br>There's a couple of things that would make this system more user-friendly.<br>
<br>This system reduces to Bucklin in the one-candidate case. However, to get it to reduce to MCV - remember, my motivation is that classing into "preferred", "approved", and "disapproved" is easier and involves less strategic thinking for the voter than more rounds; and voters don't really want to be forced to approve, even minimally, something over half of the candidates for a large assembly - you need some way to "round out" the legislature after there are no Droop quotas left. There are 3 ways to do this: with some kind of a runoff (as in MCV); approval-style, where the lowest-approval candidates are forced to share their votes which aren't already shared; or simply continuing to pick the highest diagonal elements, and deducting a full Droop quota (leaving negative numbers). (This last option can be used with Hare quotas from the beginning to yield a more Saint-Lague-style method).<br>
<br>Finally, to reduce the burden of individually marking candidates, there could be (overlapping) party slates which could be marked in either preference category wholesale.<br><br>I believe that this system's advantages speak for itself. I am not aware of another system which is easy, monotonic, preferential, summable, and almost strategy-free.[5]<br>
<br>This email is long enough for now, but I'd be happy to answer questions if I've left something(s) unclear.<br><br>[0] MCV asks voters only to class candidates into "preferred", "approved", and "disapproved". To me, this is always a natural division. If there's a known CW, that candidate will almost certainly be salient to most voters, and thus anchoring effects will tend to naturally put the approved/disapproved breakpoint next to that candidate (perhaps with an empty set of approved); that's why MCV will tend to elect the CW. In Approval, one is forced to choose whether to put one's cutoff between "preferred" and "approved", or between "approved" and "disapproved"; this is fundamentally a strategic, not a natural, choice. In Range, a [nearly] rational voter is forced to do the same, and then to exaggerate to a [nearly] approval-style vote; this feels even more strategic. And for me, MCV feels less strategic than Bucklin versions with more ranks. For instance, in 3-rank, equality allowed Bucklin, it could often be rational to "hold out to the last minute" by leaving the middle rank blank (or voting Mickey Mouse there); yet that's purely due to strategic considerations, and not at all about the inherent relative qualities of the given candidates.<br>
[1] OK, if there are only a few 100% rational voters and all of them have exact knowledge of each others' preferences, you can construct scenarios where the game-theoretic optimum for one voter would not be semi-honest. It happens because their preferences are only orthogonal, maximizing the differences between candidates that all other voters regard as near-clones. Warren Smith showed this for Approval in a paper somewhere (I suspect he can provide the reference if someone's curious) and MCV is alike enough to Approval for the result to apply. But it's a contrived and fragile case, inhuman even when it happens and improbable exponentially in the number of voters - that is, vastly less probable than exact ties.<br>
[2] In WDS's simulations, it's the second-best system, after Range, if I'm not mistaken.<br>[3] The method does not strictly satisfy the Condorcet Criterion, but in practice, violations are rare for monte-carlo voters, and I think that human nature would tend to make them rarer. If there's a known CW, they will win in a cabal equilibrium.<br>
[4] I've also explored the idea of a system like asset voting, with
your preferred candidate(s) transferring your vote to make Droop
quotas, but with guarantees to make sure your vote is never transferred
to a disapproved candidate. However, you have to go through crazy
contortions to make a system like that summable, and even with those
contortions, I can't find a "natural" way to make it a well-determined
function of the votes and the candidate's preference orders - that is,
strategy would matter too much at the "candidate's convention".<br>[5] Actually, this system is related to RRV. Perhaps a similar trick, using a "correlation matrix", would make RRV summable. But RRV would still call for an exaggeration strategy - one that's significantly more complex than simple Range, by the way.<br>
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