Smith set: Difference between revisions

The Smith set represents one approach to identifying a subset of candidates from which an election method should choose a winner. The idea is to choose the "best compromise." (In the example above, B is the first or second choice of every voter.) The most widely used voting method, the sequential pairwise elimination method used when voting on motions and amendments under Robert's Rules of Order, always chooses an alternative in the Smith set.

 

 

Also of note is the '''sincere Smith set''': the smallest non-empty set of candidates such that every candidate in the set is ''sincerely preferred'' by a majority of the voters over every candidate not in the set. (That isn't correct. It's only necessary that each candidate in the set be preferred to each candidate outside the set by more voters than vice-versa. It needn't be a majority Maybe some voters are indifferent between some candidate-pairs] Since the definition depends on voters' preferences, it does not matter what voting method is used, assuming voters' preferences do not depend on the voting method. However, that assumption is naive, because candidates who want to win choose positions on issues that they hope will help them win, and winning positions depend in part on the voting method, and candidates' positions affect voters' preferences. For example, Plurality Rule, Top Two Runoff, Instant Runoff and many other methods can cause candidates to avoid centrist positions due to the risk of being sandwiched between a candidate "on the left" and a candidate "on the right" whereas voting methods that satisfy the Smith criterion can cause candidates to take more centrist positions. In this important sense, the sincere Smith set (and the voted Smith set) depend on the voting method. However, this is beyond the scope of this article. (Unfortunately, many comparisons of voting methods naively oversimplify their analysis by assuming candidates' positions and voters' preferences are constant, and neglect the possibility that the most important criteria for comparing voting methods may involve the effects that voting methods have on candidates' positions.)

 

==Properties==

 

The Smith set is the maximal cycle equivalence class of the [[Beatpath#Beat-or-tie path|beat-or-tie order]].