Arrow's impossibility theorem

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. 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 Template:Math and Template:Math should not depend on the quality of some third, unrelated option Template:Math.

The result is most often cited in discussions of voting rules. However, Arrow's theorem is substantially broader, and can be applied to methods of social decision-making other than voting. It therefore generalizes Condorcet's voting paradox, and shows similar problems exist for every collective decision-making procedure based on relative comparisons.

Plurality-rule methods like first-past-the-post and ranked-choice (instant-runoff) voting are highly sensitive to spoilers, particularly in situations where they are not forced. By contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections by restricting them to rare situations called cyclic ties. Under some idealized models of voter behavior (e.g. Black's left-right spectrum), spoiler effects can disappear entirely for these methods.

Arrow's theorem does not cover rated voting rules, and thus cannot be used to inform their susceptibility to the spoiler effect. However, Gibbard's theorem shows these methods' susceptibility to strategic voting, and generalizations of Arrow's theorem describe cases where rated methods are susceptible to the spoiler effect.

Background

Template:Main 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. Such a mechanism of study can be a market, voting system, constitution, or even a moral or ethical framework.

Axioms of voting systems

Preferences

Template:FurtherIn the context of Arrow's theorem, citizens are assumed to have ordinal preferences, i.e. orderings of candidates. If Template:Math and Template:Math are different candidates or alternatives, then $A\succ B$ means Template:Math is preferred to Template:Math. Individual preferences (or ballots) are required to satisfy intuitive properties of orderings, e.g. they must be transitive—if $A⪰B$ and $B⪰C$, then $A⪰C$. The social choice function is then a mathematical function that maps the individual orderings to a new ordering that represents the preferences of all of society.

Basic assumptions

Arrow's theorem assumes as background that any non-degenerate social choice rule will satisfy:

• Unrestricted domain – the social choice function is a total function over the domain of all possible orderings of outcomes, not just a partial function.
  • In other words, the system must always make some choice, and cannot simply "give up" when the voters have unusual opinions.
  • Without this assumption, majority rule satisfies Arrow's axioms by "giving up" whenever there is a Condorcet cycle.
• Non-dictatorship – the system does not depend on only one voter's ballot.
  • This weakens anonymity (one vote, one value) to allow rules that treat voters unequally.
  • It essentially defines social choices as those depending on more than one person's input.
• Non-imposition – the system does not ignore the voters entirely when choosing between some pairs of candidates.
  • In other words, it is possible for any candidate to defeat any other candidate, given some combination of votes.
  • This is often replaced with the stronger Pareto efficiency axiom: if every voter prefers Template:Math over Template:Math, then Template:Math should defeat Template:Math. However, the weaker non-imposition condition is sufficient.

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. However, this assumption is not needed or used in his proof (except to derive the weaker condition of Pareto efficiency), and Arrow later corrected his statement of the theorem to remove the inclusion of this condition.

Independence

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

• Independence of irrelevant alternatives (IIA) – the social preference between candidate Template:Math and candidate Template:Math should only depend on the individual preferences between Template:Math and Template:Math.
  • In other words, the social preference should not change from $A\succ B$ to $B\succ A$ if voters change their preference about whether $A\succ C$.
  • This is equivalent to the claim about independence of spoiler candidates when using the standard construction of a placement function.

IIA is sometimes illustrated with a short joke by philosopher Sidney Morgenbesser:

Morgenbesser, ordering dessert, is told by a waitress that he can choose between blueberry or apple pie. He orders apple. Soon the waitress comes back and explains cherry pie is also an option. Morgenbesser replies "In that case, I'll have blueberry."

Arrow's theorem shows that if a society wishes to make decisions while always avoiding such self-contradictions, it cannot use ranked information alone.

Theorem

Intuitive argument

Condorcet's example is already enough to see the impossibility of a fair ranked voting system, given stronger conditions for fairness than Arrow's theorem assumes. Suppose we have three candidates ($A$, $B$, and $C$) and three voters whose preferences are as follows:

Voter	First preference	Second preference	Third preference
Voter 1	A	B	C
Voter 2	B	C	A
Voter 3	C	A	B

If $C$ is chosen as the winner, it can be argued any fair voting system would say $B$ should win instead, since two voters (1 and 2) prefer $B$ to $C$ and only one voter (3) prefers $C$ to $B$. However, by the same argument $A$ is preferred to $B$, and $C$ is preferred to $A$, by a margin of two to one on each occasion. Thus, even though each individual voter has consistent preferences, the preferences of society are contradictory: $A$ is preferred over $B$ which is preferred over $C$ which is preferred over $A$.

Because of this example, some authors credit Condorcet with having given an intuitive argument that presents the core of Arrow's theorem. 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.

Formal statement

Let $A$ be a set of alternatives. A voter's preferences over $A$ are a complete and transitive binary relation on $A$ (sometimes called a total preorder), that is, a subset $R$ of $A\times A$ satisfying:

1. (Transitivity) If $(𝐚,𝐛)$ is in $R$ and $(𝐛,𝐜)$ is in $R$, then $(𝐚,𝐜)$ is in $R$,
2. (Completeness) At least one of $(𝐚,𝐛)$ or $(𝐛,𝐚)$ must be in $R$.

The element $(𝐚,𝐛)$ being in $R$ is interpreted to mean that alternative $𝐚$ is preferred to alternative $𝐛$. This situation is often denoted $𝐚\succ 𝐛$ or $𝐚R𝐛$. Denote the set of all preferences on $A$ by $\mathrm{\Pi }(A)$. Let $N$ be a positive integer. An ordinal (ranked) social welfare function is a function

$\mathrm{F}:\mathrm{\Pi }(A{)}^N\to \mathrm{\Pi }(A)$

which aggregates voters' preferences into a single preference on $A$. An $N$-tuple $(R_1,\dots ,R_N)\in \mathrm{\Pi }(A{)}^N$ of voters' preferences is called a preference profile.

Arrow's impossibility theorem: If there are at least three alternatives, then there is no social welfare function satisfying all three of the conditions listed below:

Pareto efficiency

If alternative $𝐚$ is preferred to $𝐛$ for all orderings $R_1,\dots ,R_N$, then $𝐚$ is preferred to $𝐛$ by $F(R_1,R_2,\dots ,R_N)$.

Non-dictatorship

There is no individual $i$ whose preferences always prevail. That is, there is no $i\in \{1,\dots ,N\}$ such that for all $(R_1,\dots ,R_N)\in \mathrm{\Pi }(A{)}^N$ and all $𝐚$ and $𝐛$, when $𝐚$ is preferred to $𝐛$ by $R_i$ then $𝐚$ is preferred to $𝐛$ by $F(R_1,R_2,\dots ,R_N)$.

Independence of irrelevant alternatives

For two preference profiles $(R_1,\dots ,R_N)$ and $(S_1,\dots ,S_N)$ such that for all individuals $i$, alternatives $𝐚$ and $𝐛$ have the same order in $R_i$ as in $S_i$, alternatives $𝐚$ and $𝐛$ have the same order in $F(R_1,\dots ,R_N)$ as in $F(S_1,\dots ,S_N)$.

Formal proof

Arrow's proof used the concept of decisive coalitions.

Definition:

• A subset of voters is a coalition.
• A coalition is decisive over an ordered pair $(x,y)$ if, when everyone in the coalition ranks $x{\succ }_{i}y$, society overall will always rank $x\succ y$.
• A coalition is decisive if and only if it is decisive over all ordered pairs.

Our goal is to prove that the decisive coalition contains only one voter, who controls the outcome—in other words, a dictator.

The following proof is a simplification taken from Amartya Sen and Ariel Rubinstein. The simplified proof uses an additional concept:

• A coalition is weakly decisive over $(x,y)$ if and only if when every voter $i$ in the coalition ranks $x{\succ }_{i}y$, and every voter $j$ outside the coalition ranks $y{\succ }_{j}x$, then $x\succ y$.

Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes. Template:Math theorem Template:Math proof Template:Math theorem Template:Math proof

By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma, there is a size-one decisive coalition—a dictator. Template:Collapse bottom

Template:Collapse top Proofs using the concept of the pivotal voter originated from Salvador Barberá in 1980. The proof given here is a simplified version based on two proofs published in Economic Theory.

Setup

Assume there are n voters. We assign all of these voters an arbitrary ID number, ranging from 1 through n, which we can use to keep track of each voter's identity as we consider what happens when they change their votes. Without loss of generality, we can say there are three candidates who we call A, B, and C. (Because of IIA, including more than 3 candidates does not affect the proof.)

We will prove that any social choice rule respecting unanimity and independence of irrelevant alternatives (IIA) is a dictatorship. The proof is in three parts:

1. We identify a pivotal voter for each individual contest (A vs. B, B vs. C, and A vs. C). Their ballot swings the societal outcome.
2. We prove this voter is a partial dictator. In other words, they get to decide whether A or B is ranked higher in the outcome.
3. We prove this voter is the same person, hence this voter is a dictator.

Part one: There is a pivotal voter for A vs. B

Part one: Successively move B from the bottom to the top of voters' ballots. The voter whose change results in B being ranked over A is the pivotal voter for B over A.

Consider the situation where everyone prefers A to B, and everyone also prefers C to B. By unanimity, society must also prefer both A and C to B. Call this situation profile[0, x].

On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each i let profile i be the same as profile 0, but move B to the top of the ballots for voters 1 through i. So profile 1 has B at the top of the ballot for voter 1, but not for any of the others. Profile 2 has B at the top for voters 1 and 2, but no others, and so on.

Since B eventually moves to the top of the societal preference as the profile number increases, there must be some profile, number k, for which B first moves above A in the societal rank. We call the voter k whose ballot change causes this to happen the pivotal voter for B over A. Note that the pivotal voter for B over A is not, a priori, the same as the pivotal voter for A over B. In part three of the proof we will show that these do turn out to be the same.

Also note that by IIA the same argument applies if profile 0 is any profile in which A is ranked above B by every voter, and the pivotal voter for B over A will still be voter k. We will use this observation below.

Part two: The pivotal voter for B over A is a dictator for B over C

In this part of the argument we refer to voter k, the pivotal voter for B over A, as the pivotal voter for simplicity. We will show that the pivotal voter dictates society's decision for B over C. That is, we show that no matter how the rest of society votes, if pivotal voter ranks B over C, then that is the societal outcome. Note again that the dictator for B over C is not a priori the same as that for C over B. In part three of the proof we will see that these turn out to be the same too.

Part two: Switching A and B on the ballot of voter k causes the same switch to the societal outcome, by part one of the argument. Making any or all of the indicated switches to the other ballots has no effect on the outcome.

In the following, we call voters 1 through k − 1, segment one, and voters k + 1 through N, segment two. To begin, suppose that the ballots are as follows:

• Every voter in segment one ranks B above C and C above A.
• Pivotal voter ranks A above B and B above C.
• Every voter in segment two ranks A above B and B above C.

Then by the argument in part one (and the last observation in that part), the societal outcome must rank A above B. This is because, except for a repositioning of C, this profile is the same as profile k − 1 from part one. Furthermore, by unanimity the societal outcome must rank B above C. Therefore, we know the outcome in this case completely.

Now suppose that pivotal voter moves B above A, but keeps C in the same position and imagine that any number (even all!) of the other voters change their ballots to move B below C, without changing the position of A. Then aside from a repositioning of C this is the same as profile k from part one and hence the societal outcome ranks B above A. Furthermore, by IIA the societal outcome must rank A above C, as in the previous case. In particular, the societal outcome ranks B above C, even though Pivotal Voter may have been the only voter to rank B above C. By IIA, this conclusion holds independently of how A is positioned on the ballots, so pivotal voter is a dictator for B over C.

Part three: There exists a dictator

Part three: Since voter k is the dictator for B over C, the pivotal voter for B over C must appear among the first k voters. That is, outside of segment two. Likewise, the pivotal voter for C over B must appear among voters k through N. That is, outside of Segment One.

In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying parts one and two to the other pairs of candidates). First, the pivotal voter for B over C must appear earlier (or at the same position) in the line than the dictator for B over C: As we consider the argument of part one applied to B and C, successively moving B to the top of voters' ballots, the pivot point where society ranks B above C must come at or before we reach the dictator for B over C. Likewise, reversing the roles of B and C, the pivotal voter for C over B must be at or later in line than the dictator for B over C. In short, if k_X/Y denotes the position of the pivotal voter for X over Y (for any two candidates X and Y), then we have shown

k_B/C ≤ k_B/A ≤ k_C/B.

Now repeating the entire argument above with B and C switched, we also have

k_C/B ≤ k_B/C.

Therefore, we have

k_B/C = k_B/A = k_C/B

and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election. Template:Collapse bottom

Stronger versions

Arrow's impossibility theorem still holds if Pareto efficiency is weakened to the following condition:

Non-imposition

For any two alternatives a and b, there exists some preference profile Template:Math such that Template:Math is preferred to Template:Math by Template:Math.

Interpretation and practical solutions

Arrow's theorem establishes that no ranked voting rule can always satisfy independence of irrelevant alternatives, but it says nothing about the frequency of spoilers. This led Arrow to remark that "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."

Attempts at dealing with the effects of Arrow's theorem take one of two approaches: either accepting his rule and searching for the least spoiler-prone methods, or dropping one or more of his assumptions, such as by focusing on rated voting rules.

Template:AnchorMinimizing IIA failures: Majority-rule methods

An example of a Condorcet cycle, where some candidate must cause a spoiler effect

The first set of methods studied by economists are the majority-rule, or Condorcet, methods. These rules limit spoilers to situations where majority rule is self-contradictory, called Condorcet cycles, and as a result uniquely minimize the possibility of a spoiler effect among ranked rules. (Indeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proven, however, that under any such restriction, if there exists any social welfare function that adheres to Arrow's criteria, then Condorcet method will adhere to Arrow's criteria.) Condorcet believed voting rules should satisfy both independence of irrelevant alternatives and the majority rule principle, i.e. if most voters rank Alice ahead of Bob, Alice should defeat Bob in the election.

Unfortunately, as Condorcet proved, this rule can be intransitive on some preference profiles. Thus, Condorcet proved a weaker form of Arrow's impossibility theorem long before Arrow, under the stronger assumption that a voting system in the two-candidate case will agree with a simple majority vote.

Unlike pluralitarian rules such as ranked-choice runoff (RCV) or first-preference plurality, Condorcet methods avoid the spoiler effect in non-cyclic elections, where candidates can be chosen by majority rule. Political scientists have found such cycles to be fairly rare, suggesting they may be of limited practical concern. Spatial voting models also suggest such paradoxes are likely to be infrequent or even non-existent.

Template:AnchorLeft-right spectrum

Template:Main Soon after Arrow published his theorem, Duncan Black showed his own remarkable result, the median voter theorem. 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.

More formally, Black's theorem assumes preferences are single-peaked: a voter's happiness with a candidate goes up and then down as the candidate moves along some spectrum. For example, in a group of friends choosing a volume setting for music, each friend would likely have their own ideal volume; as the volume gets progressively too loud or too quiet, they would be increasingly dissatisfied. If the domain is restricted to profiles where every individual has a single-peaked preference with respect to the linear ordering, then social preferences are acyclic. In this situation, Condorcet methods satisfy a wide variety of highly-desirable properties, including being fully spoilerproof.

The rule does not fully generalize from the political spectrum to the political compass, a result related to the McKelvey-Schofield chaos theorem. However, a well-defined Condorcet winner does exist if the distribution of voters is rotationally symmetric or otherwise has a uniquely-defined median. 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).

Generalized stability theorems

The Campbell-Kelly theorem shows that Condorcet methods are the most spoiler-resistant class of ranked voting systems: whenever it is possible for some ranked voting system to avoid a spoiler effect, a Condorcet method will do so. In other words, replacing a ranked method with its Condorcet variant (i.e. elect a Condorcet winner if they exist, and otherwise run the method) will sometimes prevent a spoiler effect, but can never create a new one.

In 1977, Ehud Kalai and Eitan Muller gave a full characterization of domain restrictions admitting a nondictatorial and strategyproof social welfare function. These correspond to preferences for which there is a Condorcet winner.

Holliday and Pacuit devised a voting system that provably minimizes the number of candidates who are capable of spoiling an election, albeit at the cost of occasionally failing vote positivity (though at a much lower rate than seen in instant-runoff voting).Template:Clarify

Going beyond Arrow's theorem: Rated voting

Template:Main As shown above, the proof of Arrow's theorem relies crucially on the assumption of ranked voting, and is not applicable to rated voting systems. This opens up the possibility of passing all of the criteria given by Arrow. These systems ask voters to rate candidates on a numerical scale (e.g. from 0–10), and then elect the candidate with the highest average (for score voting) or median (graduated majority judgment).Template:Rp

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. Intuitively, cardinal information can only lead to such immunity if it's meaningful; simply providing cardinal data is not enough.

Some rated systems, such as range voting and majority judgment, pass independence of irrelevant alternatives when the voters rate the candidates on an absolute scale. However, when they use relative scales, more general impossibility theorems show that the methods (within that context) still fail IIA. As Arrow later suggested, relative ratings may provide more information than pure rankings, but this information does not suffice to render the methods immune to spoilers.

While Arrow's theorem does not apply to graded systems, Gibbard's theorem still does: no voting game can be straightforward (i.e. have a single, clear, always-best strategy).

Template:AnchorMeaningfulness of cardinal information

Template:Main Arrow's framework assumed individual and social preferences are orderings or rankings, i.e. statements about which outcomes are better or worse than others. Taking inspiration from the strict behaviorism popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of well-being. 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.

Arrow originally agreed with these positions and rejected cardinal utility, leading him to focus his theorem on preference rankings. However, he later stated that cardinal methods can provide additional useful information, and that his theorem is not applicable to them.

John Harsanyi noted Arrow's theorem could be considered a weaker version of his own theoremTemplate:Failed verification and other utility representation theorems like the VNM theorem, which generally show that rational behavior requires consistent cardinal utilities.

Nonstandard spoilers

Behavioral economists have shown individual irrationality involves violations of IIA (e.g. with decoy effects), suggesting human behavior can cause IIA failures even if the voting method itself does not. However, past research has typically found such effects to be fairly small, 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.Template:Page needed Similar techniques are often discussed in the context of contingent valuation.

Esoteric solutions

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

Supermajority rules can avoid Arrow's theorem at the cost of being poorly-decisive (i.e. frequently failing to return a result). In this case, a threshold that requires a $2/3$ majority for ordering 3 outcomes, $3/4$ for 4, etc. does not produce voting paradoxes.

In spatial (n-dimensional ideology) models of voting, this can be relaxed to require only $1-e^{-1}$ (roughly 64%) of the vote to prevent cycles, so long as the distribution of voters is well-behaved (quasiconcave). 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.

Infinite populations

Fishburn shows all of Arrow's conditions can be satisfied for uncountably infinite sets of voters given the axiom of choice; 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".

Common misconceptions

Arrow's theorem is not related to strategic voting, which does not appear in his framework, though the theorem does have important implications for strategic voting (being used as a lemma to prove Gibbard's theorem). The Arrovian framework of social welfare assumes all voter preferences are known and the only issue is in aggregating them.

Monotonicity (called positive association by Arrow) is not a condition of Arrow's theorem. This misconception is caused by a mistake by Arrow himself, who included the axiom in his original statement of the theorem but did not use it. Dropping the assumption does not allow for constructing a social welfare function that meets his other conditions.

Contrary to a common misconception, Arrow's theorem deals with the limited class of ranked-choice voting systems, rather than voting systems as a whole.

See also

• Comparison of electoral systems
• Condorcet paradox
• Doctrinal paradox
• Gibbard–Satterthwaite theorem
• Gibbard's theorem
• Holmström's theorem
• May's theorem
• Market failure

Further reading

• Template:Cite book Surveys many of approaches discussed in #Alternatives based on functions of preference profilesTemplate:Broken anchor.
• Template:Cite journal preprint.
• Template:Cite journal
• Template:Cite book. The chapter "Defining Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof.
• Template:Cite book Gives explicit examples of preference rankings and apparently anomalous results under different electoral system. States but does not prove Arrow's theorem.
• Template:Cite book
• Template:Cite book
• Template:Cite journal

External links

• Template:SEP
• A proof by Terence Tao, assuming a much stronger version of non-dictatorship