Mutual majority criterion: Difference between revisions
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The Smith set is just B here. When looking at the 70 voters who helped B beat C and the 65 for B>A, it's clear that a majority of them prefer A over C, and that an absolute majority of voters prefer either A or B over C. So the smallest mutual majority set is A and B.
An alternative way to find the smallest mutual majority set would be to, in part, use a modified version of [[Bucklin voting]]: Consider a voter to have approved one of their 1st choice candidates, then another, etc., going down the ranks as necessary, until the first round where a candidate or candidate(s) are approved by a majority. Then check if an absolute majority of voters rank these approved candidates above all others. It may be necessary to, when some candidates reach a majority, consider voters to approve all of the candidates at the rank last considered, rather than only some if that was the case i.e. if a voter was considered to approve one of their 3 2nd choices in the round someone got a majority, now consider them to have approved all 3 2nd choices and see if this helps any of them be approved by a majority as well. For example:
17 A>B>C
17 A=B>C
17 C>A>B
49 D>E>F
In the first round, 17 voters approve A, 17 C, 49 D, and the 17 A=B voters are randomly considered to approve A. In the second round, 34 voters also approve B, 17 A, and 49 E. Finally, in the third round, A, B, and C all get 51 approvals, with D, E, and F at 49. The check indicates at least an absolute majority of voters rank all of A, B and C above all other candidates, so they are the smallest mutual majority set.
Voting methods which pass the majority criterion but not the mutual majority criterion (some ranked methods fall under this category, notably [[FPTP]]) possess a spoiler effect, since if all but one candidate in the mutual majority drops out, the remaining candidate in the mutual majority is guaranteed to win, whereas if nobody had dropped out, a candidate not in the mutual majority might have won. This is also why [[:Category:Sequential loser-elimination methods|Sequential loser-elimination methods]] whose base methods pass the majority criterion pass the mutual majority criterion.
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