## A difficult proportional representation election example

Here is a 102-voter 6-candidate election. (The candidates are named A,B,M,X,Y,Z and the votes are rank-orderings with unranked candidates to be regarded as co-equal last.)

#Voters their vote
16 X>A>M>B
15 X>B>M>A
17 Y>A>M>B
16 Y>B>M>A
16 Z>A>M>B
18 Z>B>M>A
4 M>A>B

If this is a 1-winner election, then it seems fairly clear that M should win. That is because M is a "Condorcet winner," i.e. beats every rival in a head-to-head contest (M beats A by 53:49, B by 53:49, X by 71:31, Y by 69:33, and Z by 68:34). Instant Runoff Voting, would, however, declare Z the winner – probably a bad decision.

If this is a 2-winner election, then it seems fairly clear that A & B should be the winners: Every voter ranks A or B as their second choice, and no other candidate-pair can say that. (STV as used in Ireland and Australia, would, however, not see it that way. It would make Z win, with Y as the second winner.)

If this is a 3-winner election, then Bram Cohen has argued that the winners ought to be X, Y, and Z. (And that is what STV would do.) That is because 96% of the voters have either X,Y, or Z as top choice. No other candidate-triple can say that. On the other hand, one could also argue that the winners ought to be A, B, and M. That is because every voter has one of them as their second or top choice, and no other candidate-triple can say that. Cohen would presumably riposte that only 4% of the voters would then be switching the winner-set from {X,Y,Z} to {A,B,M} and it is not right to pay attention to only 4% of the voters in such a situation because they'd be overruling too many top-choices.

This example is interesting because the "correct" winner-sets are totally disjoint if there are to be 1, 2, or 3 winners. No vote-handling method (including STV and RRV) which chooses the winners one-by-one without knowing how many there are to be in all, can do a good job.

### How would "Asset Voting" handle this?

If we were using Simmons's (or Lewis Carroll's) simplified version of asset voting in which each vote just names one candidate, then the assets would be X=31, Y=33, Z=34, M=4, rest=0. X and Y, realizing they could not win, might agree to give their assets to A. If so, A would win. If X and Y regarded M as a better compromise than either A or B, then they could make M win. All in all, it seems unlikely that M would be the winner if this were a single-winner election. If so, this is an example of failure for simplified-Asset voting.

With full asset voting, it seems more likely that M would win. For example, if each voter gave 3, 2, and 1 points to her top, 2nd, and 3rd choices, then M would have the most asset-points. If the first 98 voters gave 4,3,2,1 points to their top, 2nd, 3rd, 4th choices while the last two voters each gave 5,3,2 to their top, 2nd, 3rd choices, then also M would (more clearly) have the most asset-points. So M would be in the strongest negotiating position.

Now let's instead suppose this were a 2-winner election. Presumably X,Y,Z would be willing to donate to A and B but not to anybody else in which case A & B would indeed win. However, if even a single one among {X,Y,Z} wanted to give it all to M, he could could force M to be a winner. It seems likely to me that not both of {A,B} would win, with simplified asset voting. But with full asset voting, I think it more likely A and B could both win.

We also should note that some of our original claims about who "should" be the winners might change if strengths of preferences were taken into account. Full asset voting and RRV can do that but STV and simplified asset voting cannot.

Thus this election illustrates how full asset voting might be superior both to its simplified version, and also to other forms of proportional representation.