# 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.

## Complying methods

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 (see the below Implications section).

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.

### Cardinal methods

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, and so in a two-candidate election the majority can't vote strategically to make their preferred candidate win.

### Ranked methods

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.

#### A simple example

Let's say that we have a majoritarian ranked ballot method. With an election that's a Condorcet cycle (rock-paper-scissors situation), like this:

```25: A>B>C
40: B>C>A
35: C>A>B```

at least one of A, B or C must be elected (or have a chance of winning the election if the method is nondeterministic). There are thus three cases:

• Case 1: A is elected. IIA is violated because the 75% who prefer C over A would elect C if B were not a candidate.
• Case 2: B is elected. IIA is violated because the 60% who prefer A over B would elect A if C were not a candidate.
• Case 3: C is elected. IIA is violated because the 65% who prefer B over C would elect B if A were not a candidate.

No matter who wins, the method can be made to fail IIA.

## Related criteria

To mitigate the reach of IIA failures, less strict properties have been proposed (some of which are incompatible with IIA):

Neither the Borda count, Coombs' method nor Instant-runoff voting satisfies the less strict criteria above. Ranked Pairs does satisfy ISDA, and River satisfies IPDA.

## Anecdote

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."

## Implications

IIA implies two things:

• A voter may change their preference between A and B without impacting the race between B and C.
• A candidate can enter or drop out of the election without changing the result (unless they win in one of the cases).

The second implication is strongly disputed for voting methods that pass IIA. It requires assuming voters won't change their preferences when the set of alternatives expands or contracts; with something like Score voting, this means no voters can do normalization. A commonly used example is that if a candidate that a voter finds terrible enters the race, and is likely to win, then the voter has an incentive to do Min-max voting. An example can be found at the Election Science site.[1]

### Strategic implications

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