Independence of irrelevant alternatives

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In voting systems, independence of irrelevant alternatives is the property some voting systems have that, if one option (X) wins the election, and a new alternative (Y) is added, only X or Y will win the election. In single winner systems it is formally defined as:

If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged with the addition of a new candidate

Adding irrelevant (non-winning) candidates should not be able to change the election results. The only time when adding candidates can change the election results is when they change the election result to one in which they are one of the winners.

Condorcet methods necessarily fail this criterion, although some, such as Ranked Pairs, satisfy a related but weaker criterion known as local independence of irrelevant alternatives. Borda count, Coombs' method, and Instant-runoff voting fail. Range voting, approval voting, and majority judgment satisfy the criterion if the voters grade or rate the candidates on an absolute scale that doesn't depend on who is in the running.

Arrow's impossibility theorem states that no voting system can satisfy universal domain, non-imposition, non-dictatorship, unanimity, and independence of irrelevant alternatives. Since universal domain implies that the method is an ordinal method, the impossibility theorem only applies to ordinal voting. In practice, this means that no deterministic ranked ballot system can satisfy independence of irrelevant alternatives without either having a dictator (whose ballot decides who wins no matter the other ballots), failing to elect a candidate that the whole electorate ranks first, or rendering one or more outcomes impossible no matter the ballots.

Therefore, less strict properties have been proposed (some of which are incompatible with IIA):

  • Independence of Smith-dominated alternatives (ISDA), also sometimes called Smith-IIA (Smith-Independence of Irrelevant Alternatives), which says that if one option (X) wins an election, and a new alternative (Y) is added, X will 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. Any Condorcet 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. 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."
  • Independence of covered alternatives which says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the uncovered set. Independence of covered alternatives also implies Condorcet. If a method is independent of covered alternatives, then the method fails monotonicity if perfect ties can always be broken in favor of a choice W by using ballots ranking W first.
  • Independence of Pareto-dominated alternatives (IPDA), which says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is Pareto-dominated. An alternative W is Pareto-dominated if there exists some other alternative Z so that no voter ranks W ahead of Z and at least one voter ranks Z ahead of W.
  • Local independence of irrelevant alternatives (LIIA), which says that if the alternative ranked first or last in the outcome is removed, the relative ordering of the other alternatives in the outcome must not change. Kemeny-Young and Ranked Pairs satisfies this criterion, but the Schulze method does not.
  • Woodall's Weak IIA: If x is elected, and one adds a new calternative y ahead of x on some of the ballots on which x was first preference (and nowhere else), then either x or y should be elected.

Neither the Borda count, Coombs' method nor Instant-runoff voting satisfies the less strict criteria above.

An anecdote which illustrates a violation of this property has been attributed to Sidney Morgenbesser:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."

Voting systems which are not independent of irrelevant alternatives suffer from strategic nomination considerations.

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