Arrow's impossibility theorem is a key result in social choice theory, proved by American economist Kenneth Arrow. It shows that no group decision-making procedure based on ordinal utilities (citizens' orderings of different alternatives) can satisfy the requirements of rational choice. Specifically, no such rule can satisfy independence of irrelevant alternatives, the principle that a choice between two alternatives and should not depend on the quality of some third, unrelated option, .
The result is often cited in discussions of voting rules, where it shows no ranked voting rule can eliminate the spoiler effect. This result was first shown by the Marquis de Condorcet, whose voting paradox showed that majority rule violates; Arrow's theorem generalizes Condorcet's findings to include non-majoritarian rules like oligarchies, collective leadership, or consensus decision-making.
While the impossibility theorem shows all ranked voting rules must have spoilers, the frequency of spoilers differs dramatically by rule. Plurality-rule methods like choose-one and ranked-choice (instant-runoff) voting are highly sensitive to spoilers, creating them even in some situations where they are not mathematically necessary (e.g. in center squeezes). In contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections by restricting them to voting cycles, which are rare in ideologically-driven elections. Under some models of voter preferences (like the left-right spectrum assumed in the median voter theorem), spoilers disappear entirely for these methods.
Rated voting rules, where voters assign a separate grade to each candidate, are not affected by Arrow's theorem. Arrow initially asserted the information provided by these systems was meaningless and therefore could not be used to prevent paradoxes, leading him to overlook them. However, Arrow would later describe this as a mistake, admitting rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.Dr. Arrow: Well, I'm a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.
Background
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 In the context of Arrow's theorem, citizens are assumed to have ordinal preferences, i.e. orderings of candidates. If and are different candidates or alternatives, then means is preferred to . Individual preferences (or ballots) are required to satisfy intuitive properties of orderings, e.g. they must be transitive—if and , then . 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 over , then should defeat . 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 standard principle of rational choice is independence of irrelevant alternatives (IIA), which says that when deciding between and , one's opinion about a third option should not affect their decision. Independence of irrelevant alternatives (IIA) – the social preference between candidate and candidate should only depend on the individual preferences between and . In other words, the social preference should not change from to if voters change their preference about whether . 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 attributed, perhaps apocryphally, to philosopher Sidney Morgenbesser:Variations can be found in, e.g., 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 wants to avoid such self-contradictions in its decision-making, 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 (, , and ) 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 is chosen as the winner, it can be argued any fair voting system would say should win instead, since two voters (1 and 2) prefer to and only one voter (3) prefers to . However, by the same argument is preferred to , and is preferred to , 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: is preferred over which is preferred over which is preferred over .
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-person-one-vote elections, such as markets or weighted voting, based on ranked ballots.
Formal statement Let be a set of alternatives. A voter's preferences over are a complete and transitive binary relation on (sometimes called a total preorder), that is, a subset of satisfying: (Transitivity) If is in and is in , then is in , (Completeness) At least one of or must be in . The element being in is interpreted to mean that alternative is preferred to or indifferent to alternative . This situation is often denoted or . The symmetric part of yields the indifference relation . This is written as or if and only if and are both in . The asymmetric part of yields the (strict) preference relation . This is written as or if and only if is in and is not in . In the following, preference of one alternative over another denotes strict preference.
Denote the set of all preferences on by . Equivalently, is the set of rankings of the alternatives in from top to bottom, with ties allowed. Let be a positive integer. An ordinal (ranked) social welfare function is a function
which aggregates voters' preferences into a single preference on . An -tuple 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 is preferred to for all orderings , then is preferred to by . Non-dictatorship There is no individual whose preferences always prevail. That is, there is no such that for all and all and , when is preferred to by then is preferred to by . Independence of irrelevant alternatives For two preference profiles and such that for all individuals , alternatives and have the same order in as in , alternatives and have the same order in as in .
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 if, when everyone in the coalition ranks , society overall will always rank . 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 if and only if when every voter in the coalition ranks , and every voter outside the coalition ranks , then .
Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes.
By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma, there is a size-one decisive coalition—a dictator.
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: 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. 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. 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
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.