Arrow's impossibility theorem

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results Arrow's impossibility theorem is a key result in social choice theory, showing that no ranking-based decision rule can satisfy the requirements of rational choice theory.

[1] Most notably, Arrow showed that no such rule can satisfy all of a certain set of seemingly simple and reasonable conditions that include independence of irrelevant alternatives, the principle that a choice between two alternatives A and B should not depend on the quality of some third, unrelated option C.[2][3][4] The result is most often cited in discussions of voting rules.

It therefore generalizes Condorcet's voting paradox, and shows similar problems exist for every collective decision-making procedure based on relative comparisons.

[1] Plurality-rule methods like first-past-the-post and ranked-choice (instant-runoff) voting are highly sensitive to spoilers,[6][7] particularly in situations where they are not forced.

[8][9] By contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections[9] by restricting them to rare[10][11] situations called cyclic ties.

[8] Under some idealized models of voter behavior (e.g. Black's left-right spectrum), spoiler effects can disappear entirely for these methods.

[12][13] Arrow's theorem does not cover rated voting rules, and thus cannot be used to inform their susceptibility to the spoiler effect.

When Kenneth Arrow proved his theorem in 1950, it inaugurated the modern field of social choice theory, a branch of welfare economics studying mechanisms to aggregate preferences and beliefs across a society.

Arrow's theorem assumes as background that any non-degenerate social choice rule will satisfy:[15] Arrow's original statement of the theorem included non-negative responsiveness as a condition, i.e., that increasing the rank of an outcome should not make them lose—in other words, that a voting rule shouldn't penalize a candidate for being more popular.

[3][18] A commonly-considered axiom of rational choice is independence of irrelevant alternatives (IIA), which says that when deciding between A and B, one's opinion about a third option C should not affect their decision.

[2] IIA is sometimes illustrated with a short joke by philosopher Sidney Morgenbesser:[19] Arrow's theorem shows that if a society wishes to make decisions while always avoiding such self-contradictions, it cannot use ranked information alone.

[20] However, Arrow's theorem is substantially more general; it applies to methods of making decisions other than one-man-one-vote elections, such as markets or weighted voting, based on ranked ballots.

[23] The simplified proof uses an additional concept: Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA.

We will prove that any social choice rule respecting unanimity and independence of irrelevant alternatives (IIA) is a dictatorship.

The theorem proves that if voters and candidates are arranged on a left-right spectrum, Arrow's conditions are all fully compatible, and all will be met by any rule satisfying Condorcet's majority-rule principle.

In this situation, Condorcet methods satisfy a wide variety of highly-desirable properties, including being fully spoilerproof.

[31][32] In most realistic situations, where voters' opinions follow a roughly-normal distribution or can be accurately summarized by one or two dimensions, Condorcet cycles are rare (though not unheard of).

[9] In 1977, Ehud Kalai and Eitan Muller gave a full characterization of domain restrictions admitting a nondictatorial and strategyproof social welfare function.

[34]: 4–5 Because Arrow's theorem no longer applies, other results are required to determine whether rated methods are immune to the spoiler effect, and under what circumstances.

However, when they use relative scales, more general impossibility theorems show that the methods (within that context) still fail IIA.

[41] Arrow's framework assumed individual and social preferences are orderings or rankings, i.e. statements about which outcomes are better or worse than others.

[42] Taking inspiration from the strict behaviorism popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of well-being.

[43][19] Such philosophers claimed it was impossible to compare the strength of preferences across people who disagreed; Sen gives as an example that it would be impossible to know whether the Great Fire of Rome was good or bad, because despite killing thousands of Romans, it had the positive effect of letting Nero expand his palace.

[39] Arrow originally agreed with these positions and rejected cardinal utility, leading him to focus his theorem on preference rankings.

[47] However, past research has typically found such effects to be fairly small,[48] and such psychological spoilers can appear regardless of electoral system.

Balinski and Laraki discuss techniques of ballot design derived from psychometrics that minimize these psychological effects, such as asking voters to give each candidate a verbal grade (e.g. "bad", "neutral", "good", "excellent") and issuing instructions to voters that refer to their ballots as judgments of individual candidates.

[40] In addition to the above practical resolutions, there exist unusual (less-than-practical) situations where Arrow's requirement of IIA can be satisfied.

Supermajority rules can avoid Arrow's theorem at the cost of being poorly-decisive (i.e. frequently failing to return a result).

[50] These results provide some justification for the common requirement of a two-thirds majority for constitutional amendments, which is sufficient to prevent cyclic preferences in most situations.

[50] Fishburn shows all of Arrow's conditions can be satisfied for uncountably infinite sets of voters given the axiom of choice;[51] however, Kirman and Sondermann demonstrated this requires disenfranchising almost all members of a society (eligible voters form a set of measure 0), leading them to refer to such societies as "invisible dictatorships".