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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]]: Considerfor aeach voter, toassume havethey approved"approve" oneall of their 1st choicechoices. Find the ballot which approves the most candidates,; thenfor another,each etc.other ballot, goinguntil downit theapproves ranksas many candidates as necessary,this until"most-approvals" ballot, the firstmost-approvals roundballot whereshould abe candidateprevented orfrom candidate(s)approving areany approvedmore bycandidates. Once a majority.ballot Thenapproves checkas ifmany anor absolutemore majoritycandidates ofthan votersthe rankmost-approvals theseballot, approvedit candidatesshould abovebe allconsidered others.the Itmost-approvals mayballot beinstead, necessaryand tolikewise, whenit someshould stop approving additional candidates. reachFor aeach majority,ballot considerthat votersis tonot approvea allmost-approvals ofballot, theapprove all candidates at the next consecutive rank lastwhere considered,candidates ratherhaven't thanbeen onlyapproved someyet iffor that wasballot. theDo casethis i.e.until ifsome acandidate(s) voterare wasapproved consideredby toa approve onemajority of theirvoters, 3and 2ndthen choicescheck inif theall roundballots someoneapproving got aeach majority,-approved nowcandidate considerdo themnot toapprove haveanyone else. If so, then the majority-approved allcandidates 3are 2ndthe choicessmallest andmutual seemajority ifset, thisbut helpsif anynot, ofthen themthere beis approvedno bysmallest amutual majority as wellset. For example: <blockquote>17 A>B>C
17 A>B>C
17 A=B>C
17 C>A>B
49 D>E>F </blockquote>34 voters approve A as their 1st choice, 17 B, 17 C, and 49 D. The 17 A=B voters approve both A and B, two candidates, making them the most-approvals voters currently, so they are not allowed to approve any more candidates for now. Adding in the next rank, 17 voters now approve B as their 2nd choice, 17 A, and 49 E. Now 51 voters approve A, so check whether they are a mutual majority. In this case, the only candidates any of the 51 voters prefer more than or equally to A are B and C; it is seen that all 51 voters prefer any of A, B, or C over all other candidates (D, E, and F), so ABC is the smallest mutual majority set.
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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