Independence of Smith-dominated Alternatives

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Independence of Smith-dominated alternatives (ISDA), also sometimes called Smith-IIA (Smith-Independence of Irrelevant Alternatives), says that if one option (X) wins an election, and a new alternative (Y) is added, X will still win the election if Y is not in the Smith set. ISDA implies Smith and thus Condorcet, since logically speaking, if an ISDA-passing method's winner were not in the Smith set, eliminating everyone outside of the Smith set would have to change the winner. Some Condorcet methods (e.g. Schulze) satisfy ISDA.

Notes[edit | edit source]

Any voting method that starts by eliminating everyone outside the Smith set passes ISDA. Satisfaction of ISDA can sometimes make understanding a voting method or finding the winner easier; see the Schwartz set heuristic for Schulze for an example.

ISDA is a natural extension of the Smith criterion because it can be phrased as analogous to the following property implied by the Condorcet criterion: "if there is a Condorcet winner, and candidates are added or removed to the election who are pairwise beaten by the Condorcet winner, then the winner does not change". (ISDA's analogous phrasing is "if there is a Smith set, and candidates are added or removed to the election who are pairwise defeated by everyone in the Smith set, then the winner does not change." Note that unlike the Condorcet winner, the Smith set always exists.)

ISDA is incompatible with IIA, since ISDA implies majority and majority is incompatible with IIA.

Note that one implication of ISDA is that you can separate the candidates into two groups of candidates, and so long as every candidate in the first group pairwise beats every candidate in the second group, all candidates in the second group can be eliminated; this is because the Smith set logically must be a subset of any set of candidates that pairwise beat all candidates not in the set. This means, for example, you can eliminate all candidates not in the mutual majority-preferred set of candidates, when one exists. This is because the Smith set is guaranteed to be a subset of the mutual majority set. Further, you can eliminate any subset of candidates not in the Smith set, which means that sometimes the computation or demonstration of a Condorcet method can be simplified. One example is the standard Tennessee capital election example:

42% of voters

(close to Memphis)

26% of voters

(close to Nashville)

15% of voters

(close to Chattanooga)

17% of voters

(close to Knoxville)

  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

A mutual majority of voters prefer all other cities over Memphis (the 2nd, 3rd, and 4th columns of voters, which together amount to 58% of the voters), so Memphis can be eliminated, resulting in:

42% of voters

(close to Memphis)

26% of voters

(close to Nashville)

15% of voters

(close to Chattanooga)

17% of voters

(close to Knoxville)

  1. Nashville
  2. Chattanooga
  3. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  1. Chattanooga
  2. Knoxville
  3. Nashville
  1. Knoxville
  2. Chattanooga
  3. Nashville

The first two columns can be combined (because they're now identical), resulting in Nashville being (42%+26%)=68% of voters' 1st choice, a majority, and thus Nashville becomes the only member of the Smith set.

Given Schulze's multi-winner generalization of the Smith set (see the "Multi-winner generalizations" section of the Smith criterion article), an analogous extension of ISDA for the multi-winner case might be "if candidates not in any groups of candidates guaranteed seats by Schulze's multi winner Smith criterion drop out or enter the race, this shouldn't change the seat guarantees given to those same groups."