Raynaud or Pairwise-Elimination is a Condorcet method in which the loser of the strongest pairwise defeat is repeatedly eliminated until only one candidate remains. Raynaud can also be described as the sequential loser-elimination method that uses Minmax as its base method. Defeat strength is usually measured as either the absolute number of votes cast for the winning side (winning votes), or the number of votes for the winning side minus those for the losing side (margins). It was devised by Hervé Raynaud.
A variant called Raynaud(Gross Loser) does satisfy the Plurality criterion. It successively eliminates the candidate with the fewest votes for him in any pairwise contest. In this way, it is not possible to eliminate candidate A before candidate B when A has more first preferences than B has any preferences, since this situation means that the minimum number of votes for A in any contest is greater than the maximum number of votes for B in any contest. This variant was devised by Chris Benham.
25 A>B>C 40 B>C>A 35 C>A>B
A beats B (60 to 40) beats C (65 to 35) beats A (75 to 25), so there is no Condorcet winner. If using winning votes as the defeat strength measure, then the C>A defeat is strongest, and so A is eliminated (Raynaud(Gross Loser) does the same). Then, since B beats C, C is eliminated for being the only candidate remaining with a defeat, and B wins.
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Raynaud does satisfy the Smith criterion. This is because if all but one candidate in the Smith set is eliminated at any point in Raynaud, the remaining candidate will pairwise beat all other candidates (since all the other candidates are not members of the Smith set), therefore the final remaining member of the Smith set won't have any pairwise defeats and will be the final remaining candidate overall.
Raynaud passes ISDA. This is because Raynaud can be thought of as hinging a candidate's victory chances on the successive evaluation of the ordering of all defeats from strongest to weakest; since all candidates in the Smith set have no pairwise defeats to any candidates not in the Smith set, there is no way for Smith set candidates' elimination chances to be impacted by the existence or non-existence of andidates not in the Smith set.
Raynaud's calculation can be simplified by only eliminating candidates until there is a Condorcet winner.