Self-referential Smith-efficient Condorcet method

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Self-referential Condorcet methods (SR-Condorcet methods) guarantee the election of a candidate, the SR-Condorcet winner, who would win within the voting method if it was just them and any other candidate, when one exists (related to a Condorcet winner). More broadly, if there are a smallest set of candidates, the SR-Smith set, who would each win if it was just them and any other candidate not in the set, some voting methods always elect someone from this set of candidates (related to the Smith set).

(Note that an underlying assumption is that a voter would cast the same ballot no matter which candidates are added or removed; if voters change their preferences, then most likely no voting method can be an SR-Condorcet method.)

Condorcet methods, Approval voting (which is just a form of Score voting), and Score voting are the most notable such methods. Any voting method that passes the majority criterion in the two-candidate case has to be a Condorcet method to be an SR-Condorcet method. Any voting method that passes the "utility criterion" (i.e. always elects the candidate with the most utility/approval) in the two-candidate case has to be Score voting to be an SR-Condorcet method.

Score voting is SR-Smith-efficient and guarantees there will always be an SR-Smith set of either SR-weak Condorcet winners (candidates who would either tie or win if it was just them and anyone not else) or a single SR-Condorcet winner. The same is not true for Condorcet methods because of Condorcet cycles.

Notes[edit | edit source]

Practically all terms relating to Condorcet methods or pairwise counting can be adapted to the SR context by prefixing them with SR. So, for example, the SR-Schwartz set is the union of all of the smallest sets of candidates that either SR-beat or tie all other candidates using a given voting method.

Pairwise counting can generally be adapted in some way (not necessarily using majority rule) to determine if there is an SR-Condorcet winner or SR-Smith set in a particular voting method.

In some sense, SR methods are the natural result of making majority rule and utilitarianism maximally satisfy Independence of irrelevant alternatives. Indeed, one could define SR methods with reference to 3-candidste matchups, 4-candidate, etc. With majority rule, since there won't necessarily be a majority or a tie when there are more than 2 candidates, defining this is tricky.

It is possible to fuse Score voting and Condorcet methods by allowing voters to cast either rated or ranked ballots (or submit a rated ballot with the option to request it be counted as either a rated or a ranked ballot; see the ballot article for how to do this) and then counting those voters who cast rated ballots as having weak preferences in each runoff (i.e. a voter who scored one candidate a 1 and the other a 0.8 would be counted as giving 0.2 votes to help the first candidate pairwise beat the latter) and counting the voters who cast ranked ballots as having maximally strong preferences in each runoff (i.e. the previously mentioned voter would be counted as giving 1 vote to help the first candidate pairwise beat the latter).