Congrats to Rob and the list on 25 years. I've greatly appreciated this list over the 18 years I've been here.
I've mentioned a while ago some concepts of identifying the median voter, or the alternative supported by that voter. My main idea was to ask each voter to propose a majority of the voters (including themselves) as though proposing a majority coalition to govern. Then we might identify the median voter as that voter who was included in the proposals of the most voters.
Transforming this into an election method is difficult, since we can't normally vote on voters. Even if candidates can be "weighted" according to the number of voters that would elect them, it would still be difficult on a ballot to enforce the rule that voters must propose (basically approve) candidates accounting for a majority of the voters.
In the three-candidate case, I believe I noted, the principle can be achieved using Majority Favorite//Antiplurality. That is to say, with three candidates, as long as no candidate is a majority favorite, each voter would need to approve two of the three candidates (equivalent to antiplurality), and it would not matter exactly how many voters are represented by each candidate.
A downside is that this method MF//Antiplurality is probably not very satisfying to the voter. For example if there is no prospect of a majority favorite, and the "worst" candidate is generally agreed on, then the outcome will be determined by those (few?) who don't agree on who the worst candidate is (either sincerely disagreeing, or strategically disagreeing).
I thought of a different approach. Suppose that each voter may "exclude" one other voter, from being the voter who will determine the outcome. Have exclusions selected in some order, so that each voter is excluded by at most one other voter. Theoretically they will exclude the voter most distant from themselves in issue space. Then the last voter to be excluded is a good guess at the median voter.
In the three-candidate case this can be simulated pretty well, resulting in a method I might call "Adaptive Antiplurality." Have each voter fully rank the three candidates. Call these "for," "middle," and "against" votes. There are only six possible votes. There is no need for a special majority favorite rule.
Suppose that you vote A>B>C. Your "against-C" votes are used to exclude "for-C" votes. If C received more "for" than "against" votes, the latter number of "for" voters are excluded (doesn't matter which ballots exactly), and an A>B>C ballot needs no further processing. But if against-C votes exceed for-C votes, determine what percentage of the against-C votes were unused. Say the surplus was only 20%. Then an A>B>C ballot is further allowed to exclude 20% of a for-B vote (i.e. the middle preference of an A>B>C ballot). We don't need to actually get down to a single voter (meaning, a for-A ballot need not be called on to exclude other for-A ballots, even if for-A votes are the only non-excluded votes).
Running random scenarios it seems that this normally leaves a single candidate with positive "for" votes. It's possible for all "for" tallies to end up at zero, but this seems like a tie of the sort all methods could have.
This method should be more satisfying than MF/Antiplurality since surplus "against" votes aren't wasted as obviously. However, there is still potential for strategy in the form of burial. And it is not a Condorcet method. Consider:
The against-B votes fully exclude for-B votes with 13 votes surplus, consuming 63% of against-B. For-A is reduced to 4, for-C to 9.
Next the ACB and CAB voters can use the remainder of their against-B power against C and A respectively. This fully excludes for-A, and leaves for-C with 1.2 votes. C wins, being the only candidate with non-excluded "for" votes.
However, the Condorcet winner is A. If 5 ACB voters had instead voted ABC, A would have won. So there was incentive to vote against the viable competition rather than the actual perceived worst candidate.
In any case I think it is interesting and worthwhile to consider how to identify the median voter(s) and what we can do with that information. From a theoretical standpoint, it should be better if a governing majority is centered around the median voter, rather than merely, and barely, including that voter's position off to one side.
The idea could also be applied in an assembly. A procedure could be used to identify a subset (of some size) of the assembly closest to the median, and that subset could control general policy. A monolithic majority in the assembly will still get whatever it wants. But if the majority is split between a moderate and extreme wing, the moderate wing could make certain decisions alone, without the extreme wing and without needing explicit supporting votes from the opposition minority.
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