Independence of irrelevant alternatives: Difference between revisions

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(Change links to "independence of" criteria as these now exist. In addition, rephrase reference to CES example of min-max voting.)
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=== Cardinal methods ===
=== 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 [[Normalization|normalize]] their ballot, and so in a two-candidate election the majority can't vote strategically to make their preferred candidate win.
[[Range voting]], [[approval voting]], and [[majority judgment]] satisfy the criterion. This implies that if voters grade or rate the candidates on an absolute scale that doesn't depend on who is in the running, these methods will never suffer from spoiler effects.

Note, however, that this means no voter can [[Normalization|normalize]] their ballot. This also requires voters not to vote strategically (which can cause majority failures).


=== Ranked methods ===
=== 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.
[[Arrow's impossibility theorem]] states that no [[ordinal voting]] system (a function from ranked ballots to a ranking of candidates) can satisfy non-dictatorship and independence of irrelevant alternatives. 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 ====
==== Proof for majoritarian methods (simple case) ====


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:
Let's say that we have a majoritarian ranked ballot method, i.e. one that elects the candidate with a majority of the vote (if there are only 2 candidates). With an election that's a Condorcet cycle (rock-paper-scissors situation), like this:


{{ballots|
{{ballots|
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35: C>A>B}}
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:
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 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 1: ''A'' is elected. IIA is violated because the 75% who prefer ''C'' over ''A'' would elect ''C'' if ''B'' were not a candidate.
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*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.
*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.
No matter who wins, the method will fail IIA.


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


* '''[[Independence of Smith-dominated Alternatives|Independence of Smith-dominated alternatives]]''' (ISDA) and '''[[Independence of covered alternatives]]'''
* '''[[Independence of Smith-dominated Alternatives|Independence of Smith-dominated alternatives]]''' (ISDA)
* '''[[Independence of covered alternatives]]'''
* '''[[Independence of Pareto-dominated alternatives]]''' (IPDA)
* '''[[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.
* '''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.
* 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.
* Woodall's '''Weak IIA''': If we add a new alternative y (who is ahead of x on some 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. [[Ranked Pairs]] does satisfy ISDA, and [[River]] satisfies IPDA.
Neither the [[Borda count]], [[Coombs' method]] nor [[Instant-runoff voting]] satisfies the less strict criteria above. [[Ranked Pairs]] and [[Schulze method|Schulze]] satisfy ISDA, and [[River]] satisfies IPDA as well. [[Kemeny-Young]] and [[Ranked Pairs]] satisfy LIIA, but the [[Schulze method]] does not.


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


<blockquote>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."</blockquote>
<blockquote>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."</blockquote>


== Implications ==
== Implications ==
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* A candidate can enter or drop out of the election without changing the result (unless they win in one of the cases).
* 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.<ref>https://www.electionscience.org/wp-content/uploads/2019/09/image03.jpg</ref>
The second implication is controversial for methods that pass IIA. It requires assuming voters won't change their ballots when the set of alternatives expands or contracts; with something like [[score voting]], this implies voters cannot [[normalization|normalize]] their ballots. Another common example is 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.<ref>https://www.electionscience.org/wp-content/uploads/2019/09/image03.jpg</ref>


=== Strategic implications ===
=== Strategic implications ===
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