Independence of irrelevant alternatives: Difference between revisions
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{{Wikipedia}}
In [[voting system]]s, '''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 [[voting system]]s, '''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:
{{definition | 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}}
[[Arrow's impossibility theorem]] states that no voting system can satisfy universal domain, non-imposition, non-dictatorship, unanimity, 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.▼
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 method]]s 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. 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).
:- '''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.▼
=== Ranked methods ===
:- 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. ▼
▲[[Arrow's impossibility theorem]] states that no [[ordinal voting]] system
==== Proof for majoritarian methods (simple case) ====
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|
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 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 will fail IIA.
== Related criteria ==
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)
* '''[[Independence of covered alternatives]]'''
* '''[[Independence of Pareto-dominated alternatives]]''' (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 ==
An anecdote which illustrates a violation of this property has been attributed to Sidney Morgenbesser:
== Implications ==
IIA can be interpreted to imply 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 ballots when the set of alternatives expands or contracts; with something like [[score voting]], this implies voters cannot [[normalization|normalize]] their ballots.
As an example of normalization, 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]] so as to keep the terrible candidate out of the race. The voter thus changes their scale so that the palatable candidates rate higher. An example can be found at the Election Science site.<ref>{{cite web|url=https://www.electionscience.org/wp-content/uploads/2019/09/image03.jpg|publisher=Center for Election Science|title=Approval Voting Ballot}}</ref> This kind of rescaling must not happen if the second implication is to hold.
Empirical results from panel data suggest that judgments are at least in part relative,<ref name="Stadt Kapteyn Geer 1985 pp. 179–187">{{cite journal | last=Stadt | first=Huib van de | last2=Kapteyn | first2=Arie | last3=Geer | first3=Sara van de | title=The Relativity of Utility: Evidence from Panel Data | journal=The Review of Economics and Statistics | publisher=The MIT Press | volume=67 | issue=2 | year=1985 | issn=00346535 | jstor=1924716 | pages=179–187 | url=http://www.jstor.org/stable/1924716 | access-date=2024-04-28}}</ref><ref>{{cite journal |last1=Richard H. |last2=Diener |first2=Ed |last3=Wedell |first3=Douglas H. |title=Intrapersonal and Social Comparison Determinants of Happiness: A Range-Frequency Analysis |journal=Journal of Personality and Social Psychology |date=1989 |volume=56 |issue=3 |pages=317–325 |doi=10.1037/0022-3514.56.3.317 |pmid=2926632 |url=https://psycnet.apa.org/record/1989-18931-001}}</ref> which would weaken the second implication. In addition, [[Warren Smith]] suggested that [[Range voting]] would fail the implication.<ref>{{cite web|url=https://rangevoting.org/WhyNotNormalize.html|title=Why Range Votes should not be Automatically 'Normalized'|quote=[Normalization] also forces a failure of 'independence of irrelevant alternatives.' In practice [Range voting] will largely fail anyway, but I think individuals should be able to vote independently for each candidate.|last=Smith|first=Warren D.|website=The Center for Range Voting}}</ref>
=== Strategic implications ===
Voting systems which are not independent of irrelevant alternatives suffer from [[strategic nomination]] considerations.
== Irrelevant alternatives as context ==
{{section-stub}}
IIA has been criticized as being unduly strong, on the grounds that it would require that the outcome not change when additional information becomes available. <ref name="reddit 2024">{{cite web | title=IIA is A Nonsensical Concept To Apply To Elections and I'm Tired Of Pretending It's Not. : EndFPTP | website=reddit | date=2024-06-14 | author=choco-pi | url=https://www.reddit.com/r/EndFPTP/comments/1dg2lzk/iia_is_a_nonsensical_concept_to_apply_to/ | access-date=2020-02-10}}</ref>
This criticism applies both on an individual and societal level. For instance, a person who doesn't want to be greedy might refrain from taking an apple out of a basket if the only two choices are "take this particular apple" or "don't". But if a seemingly irrelevant alternative, "take another apple" is added, this informs the person that there are actually two apples in the basket, making it safe to pick the first one. In this case, the irrelevant alternative provides context that the person needs to make a decision.<ref name="l108">{{cite journal | last=Sen | first=Amartya | title=Internal Consistency of Choice | journal=Econometrica | publisher=[Wiley, Econometric Society] | volume=61 | issue=3 | year=1993 | issn=00129682 | jstor=2951715 | page=501 | url=http://www.jstor.org/stable/2951715 | access-date=2024-08-17}}</ref>
==Related effects==
I.J. Good and [[Nicolaus Tideman]] argued that voters who can't tell close enough candidates apart may change their ranking or rating of candidates when unrelated candidates enter the race. The unrelated candidates work as reference points, allowing the voter to tell otherwise indiscernible candidates apart.<ref name="m085">{{cite journal | last=Good | first=I. J. | last2=Tideman | first2=T.N. | title=C93. The relevance of imaginary alternatives | journal=Journal of Statistical Computation and Simulation | volume=12 | issue=3-4 | date=1981 | issn=0094-9655 | doi=10.1080/00949658108810466 | pages=313–315}}</ref> As pointed out in the paper, this apparent "IIA failure" can be eliminated if the voters pretend that such reference point candidates exist even when they're not actually running.
==See also==
* [[Spoiler effect]]
* [[Independence of clone alternatives]]
* [[Vote splitting]]
* [[w:Gibbard's theorem]]
==References==
<references/>
[[Category:Voting system criteria]]
''Some text of this article is derived with permission from [http://condorcet.org/emr/criteria.shtml Condorcet.org] ([http://miroirs.ironie.org/condorcet/condorcet.org/emr/criteria.shtml mirror])''
{{fromwikipedia}}
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Latest revision as of 11:45, 17 August 2024
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. 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 normalize their ballot. This also requires voters not to vote strategically (which can cause majority failures).
Ranked methods
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.
Proof for majoritarian methods (simple case)
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:
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 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 will fail IIA.
Related criteria
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 (ISDA)
- 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.
- 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 and Schulze satisfy ISDA, and River satisfies IPDA as well. Kemeny-Young and Ranked Pairs satisfy LIIA, but the Schulze method does not.
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 can be interpreted to imply 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 ballots when the set of alternatives expands or contracts; with something like score voting, this implies voters cannot normalize their ballots.
As an example of normalization, 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 so as to keep the terrible candidate out of the race. The voter thus changes their scale so that the palatable candidates rate higher. An example can be found at the Election Science site.[1] This kind of rescaling must not happen if the second implication is to hold.
Empirical results from panel data suggest that judgments are at least in part relative,[2][3] which would weaken the second implication. In addition, Warren Smith suggested that Range voting would fail the implication.[4]
Strategic implications
Voting systems which are not independent of irrelevant alternatives suffer from strategic nomination considerations.
Irrelevant alternatives as context
This section needs expansion. You can help by adding to it. |
IIA has been criticized as being unduly strong, on the grounds that it would require that the outcome not change when additional information becomes available. [5]
This criticism applies both on an individual and societal level. For instance, a person who doesn't want to be greedy might refrain from taking an apple out of a basket if the only two choices are "take this particular apple" or "don't". But if a seemingly irrelevant alternative, "take another apple" is added, this informs the person that there are actually two apples in the basket, making it safe to pick the first one. In this case, the irrelevant alternative provides context that the person needs to make a decision.[6]
Related effects
I.J. Good and Nicolaus Tideman argued that voters who can't tell close enough candidates apart may change their ranking or rating of candidates when unrelated candidates enter the race. The unrelated candidates work as reference points, allowing the voter to tell otherwise indiscernible candidates apart.[7] As pointed out in the paper, this apparent "IIA failure" can be eliminated if the voters pretend that such reference point candidates exist even when they're not actually running.
See also
References
- ↑ "Approval Voting Ballot". Center for Election Science.
- ↑ Stadt, Huib van de; Kapteyn, Arie; Geer, Sara van de (1985). "The Relativity of Utility: Evidence from Panel Data". The Review of Economics and Statistics. The MIT Press. 67 (2): 179–187. ISSN 0034-6535. JSTOR 1924716. Retrieved 2024-04-28.
- ↑ Richard H.; Diener, Ed; Wedell, Douglas H. (1989). "Intrapersonal and Social Comparison Determinants of Happiness: A Range-Frequency Analysis". Journal of Personality and Social Psychology. 56 (3): 317–325. doi:10.1037/0022-3514.56.3.317. PMID 2926632.
- ↑ Smith, Warren D. "Why Range Votes should not be Automatically 'Normalized'". The Center for Range Voting.
[Normalization] also forces a failure of 'independence of irrelevant alternatives.' In practice [Range voting] will largely fail anyway, but I think individuals should be able to vote independently for each candidate.
- ↑ choco-pi (2024-06-14). "IIA is A Nonsensical Concept To Apply To Elections and I'm Tired Of Pretending It's Not. : EndFPTP". reddit. Retrieved 2020-02-10.
- ↑ Sen, Amartya (1993). "Internal Consistency of Choice". Econometrica. [Wiley, Econometric Society]. 61 (3): 501. ISSN 0012-9682. JSTOR 2951715. Retrieved 2024-08-17.
- ↑ Good, I. J.; Tideman, T.N. (1981). "C93. The relevance of imaginary alternatives". Journal of Statistical Computation and Simulation. 12 (3–4): 313–315. doi:10.1080/00949658108810466. ISSN 0094-9655.
Some text of this article is derived with permission from Condorcet.org (mirror)