Independence of irrelevant alternatives

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.

Any voting method which passes the majority criterion in the two-candidate case will fail IIA, because of the Condorcet paradox. Certain voting methods only do so when all voters are strategic (i.e. Approval voting, Score voting, and Majority Judgment); they are guaranteed to fail IIA under those particular circumstances (i.e. when all voters are strategic).

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. Note that this means no voter can normalize their ballot.

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) and Independence of covered alternatives
• Independence of Pareto-dominated alternatives (IPDA)
• 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.

See also

• Spoiler effect
• Independence of clone alternatives
• Vote splitting
• w:Gibbard's theorem

Some text of this article is derived with permission from Condorcet.org (mirror)

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