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Scheme for Multiwinner Elections

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Scheme for Multiwinner Elections Smith typeset 19:44 8 Mar 2005 multiwinner “Asset voting” scheme for multiwinner elections Warren D. Smith∗ [email protected] March 8, 2005 Abstract — voter\canddt AB C D #1 .7 .3 0 0 Most democracies involve elections in which some large #2 .3 .5 .2 0 number of voters somehow choose W winners from among #3 00 1 0 N candidates, 0 <W <N. It is desired that those winners #4 0 0 .1 .9 “represent” those voters well, and (simultaneously) that #5 00 0 1 the voters tend to “prefer” them to the nonwinners. Un- total 1 .8 1.3 1.9 fortunately, these requirements can conflict and are vague. We propose “asset voting,” a new voting scheme which Figure 1.1. Voting example. N seems subjectively superior to all previously proposed multiwinner schemes I know. Objectively, it has 12 ad- A perceived disadvantage of the proposed scheme might be vantages (which we state reasonably explicitly), whereas the unconstrained character of the negotiations. Voters might I know no competitor with all 12. An experimental com- become unhappy about the possibility of mysterious, perhaps parison of asset voting with other schemes in the single partially secret, deals. For example, suppose all the candi- winner case shows that it is sometimes superior to, and dates walk into a smoke-filled room, then walk out an hour other times inferior to, the previously undisputed cham- later to announce the W winners, which seem completely un- pion “range voting.” The best available competitors to as- related to the votes, and then everybody refuses to say how set voting in the multiwinner case are forms of Hare’s Sin- those W were chosen. To overcome that, here are four possi- gle Transferable Vote equipped with the “Droop quota.” ble more constrained variant schemes: 1. We could demand the following bottom-up negotiation Keywords — Asset voting, range voting, multiwinner elections, procedure: Bayesian regret, STV, Arrow and Gibbard impossibility theorems. 1. Find the candidate who got the fewest votes. 2. Eliminate him and ask him to distribute his votes among the (not-yet-eliminated) candidate(s) of his choice in any way he pleases. 1 Asset voting (and some variants) 3. Repeat until only W candidates remain. They are the winners. Let there be N candidates from whom the voters have to choose W winners, 0 <W <N. This is sometimes called the 2. Same except that “sure winner” candidates with more than problem of “electing a committee.” 1/(W +1) of the votes are also questioned and asked if they “Asset voting,” our new election scheme: wish to donate any of their votes to anybody else (which they Each voter’s vote is an N-tuple of nonnegative reals summing may well wish to do, since they have “excess” votes). to 1. These vote-vectors are added to get each of the N “vote- 3. We could also further constrain the negotiations by de- totals.” Now, we do not just take the greatest W coordinates manding, in step 2 of the procedure of variant 1, that each of the sum-vector to determine the winners. Instead we enter eliminated candidate award all of his votes to a single not- a “negotiation” stage, in which any subset of the candidates yet-eliminated candidate of his choice. is allowed to agree to re-allocate its votes among themselves. 4. Optionally, we could modify variant 3 by permitting the After all such re-allocations have ceased, then the owners of eliminated candidate to keep all his votes for himself, “taking the top W vote-counts are the winners. them to the grave.” Example: Suppose there are N = 4 candidates A,B,C,D These variants also have the advantage that they definitely and 5 voters, and we wish to select W = 2 winners. Suppose terminate in N or N−W steps. In all of these variant schemes, the votes are as in table 1.1. If no negotiation were to take the full step-by-step description of who donated how much to place, then C and D would be the co-winners. However, sup- whom, would be announced to the public as part of the an- pose instead that B agrees to sacrifice his votes by awarding nouncement of the election results. Also, it would be best them to A. In that case A, now with 1.8 votes, and D (still if the terms of any deals the candidates make among them- with 1.9) would (regardless of what C does) be the co-winners. selves, were made public too. In most cases the candidates ∗21 Shore Oaks Drive, Stony Brook NY 11790. Sep 2004 1 1. 0. 0 Smith typeset 19:44 8 Mar 2005 multiwinner themselves would wish to do that, but in cases when they do a11. It seems immune to manipulation by artificial introduc- not, the publication requirement seems unenforceable. tion of “cloned” candidates. That is, if all voters split their Of these, I presently prefer variant #2, with the candidate votes for a candidate among his clones in some fashion, and chosen each negotation round for questioning being the one if all mutual clones agree to collaborate and“act as one”dur- with > 1/(W + 1) of the votes (but the least votes subject ing the negotiation stage, then the committee output by asset to this) if one exists, otherwise the one with the least votes, voting will be the same (up to replacements of members by both of these subject to the overriding rule that no candidate clones) as it was before. This had been a well known defect of is ever questioned for a second time. However, I do not have plurality voting: introducing a clone of a candidate causes his any very convincing argument that this variant is the best, “vote to be split” often causing both clones to lose an election and there are other possible variants (e.g. restrictions could one otherwise would have won. Some multiwinner schemes be placed on the“total flow”of votes transferred in and out of have the opposite property – cloning a candidate causes that each candidate, or we could stop the negotiation once certain “team” to be more likely to win. It seems best for neither conditions are satisfied). splitting nor teaming to work, i.e. cloning should have little or no effect on the composition of the output committee. a12. It seems to minimize the probability of near-tied elec- 2 Asset advantages tions (which have a bad habit of plunging the USA into crisis). STV voting schemes involve a large number (up to N) of a1. Asset voting tends to cause more-popular candidates to “rounds” in which either candidates are eliminated, or are de- win, as opposed to simply attempting to get a representative clared to be winners because their vote counts exceed the cross section of the electorate without any regard for quality “Droop quota”2. Every round offers the terrifying possibility of the candidates as perceived by the voters. of a vote near-tie and consequent crisis. In contrast, systems a2. It tends to cause the winners to represent the electorate like multiwinner plurality (v14 and v15 of §5) involve only as a whole – as opposed to zeroing out minority groups.1 one possible tie, and at least in that respect are far less likely a3. It encourages peaceful harmony among candidates (since to induce a crisis than, i.e. are far superior to, STV. they must negotiate). It is debatable how true a12 really is for asset voting. In some sense asset voting completely ignores issues of who got a4. In the single-winner special case W = 1 it will always more votes than whom, and hence ties are irrelevant to it elect a true-majority winner (i.e. one with more than 50% and we have superiority even over multiwinner plurality. But of the total votes) if one exists. (But if there is none, then it of course, in reality undoubtably very small shifts in certain remains possible that the other candidates can combine forces vote counts could completely alter the nature of the negotia- to defeat the top vote-getter.) More generally, any candidate tion stage. getting more than 1/(W +1) of the votes can guarantee to be among the W winners. a5. The well known defect of plurality voting in 3-way single- 3 Possible disadvantages winner elections – the logic that voting for the apparently least popular candidate is a “wasted vote” – seems absent in asset d1. A “unanimous Condorcet winner” U (whom every voter voting, because the third candidate is free to tip the election awards a greater vote to, than to any other candidate) can among the top 2 in any way he likes. Mike Ossipoff called fail to be among the winners, even in a single-winner elec- this defect the “favorite betrayal” property – voters in 3-way tion. For example, make every vote be (.4, .3, .3) and then plurality elections often can feel tactically forced not to vote the two lowest-ranked candidates join forces. (However, see for their favorite candidate. It appears that in N = 3, W =1 advantage a4, and this pathology cannot occur if the candi- asset voting elections, favorite betrayal does not happen. dates themselves feel the same way about U as do the voters.) a6. Asset voting neither assumes nor requires political parties Personally, I do not regard this as a “disadvantage” – at least to exist. not in this example. a7. It offers voters very high levels of expressivity.
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