Condorcet Method - Two-method Systems

Two-method Systems

One family of Condorcet methods consists of systems that first conduct a series of pairwise comparisons and then, if there is no Condorcet winner, fall back to an entirely different, non-Condorcet method to determine a winner. The simplest such methods involve entirely disregarding the results of pairwise comparisons. For example, the Black method chooses the Condorcet winner if it exists, but uses the Borda count instead if there is an ambiguity (the method is named for Duncan Black).

A more sophisticated two-stage process is, in the event of an ambiguity, to use a separate voting system to find the winner but to restrict this second stage to a certain subset of candidates found by scrutinizing the results of the pairwise comparisons. Sets used for this purpose are defined so that they will always contain only the Condorcet winner if there is one, and will always, in any case, contain at least one candidate. Such sets include the

• Smith set: The smallest non-empty set of candidates in a particular election such that every candidate in the set can beat all candidates outside the set. It is easily shown that there is only one possible Smith set for each election.
• Schwartz set: This is the innermost unbeaten set, and is usually the same as the Smith set. It is defined as the union of all possible sets of candidates such that for every set:
  1. Every candidate inside the set is pairwise unbeatable by any other candidate outside the set (i.e., ties are allowed).
  2. No proper (smaller) subset of the set fulfills the first property.
• Landau set (or uncovered set or Fishburn set): the set of candidates, such that each member, for every other candidate (including those inside the set), either beats this candidate or beats a third candidate that itself beats the candidate that is unbeaten by the member.

One possible method is to apply instant-runoff voting to the candidates of the Smith set. This method has been described as 'Smith/IRV'.