34 Arrow’s Theorem
Arrow’s impossibility theorem is one of the most important results in social choice theory. As Alan Taylor notes:
“For an area of study to become a recognized field, or even a recognized subfield, two things are required: It must be seen to have coherence, and it must be seen to have depth. The former often comes gradually, but the latter can arise in a single flash of brilliance….With social choice theory, there is little doubt as to the seminal result that made it a recognized field of study: Arrow’s impossibility theorem.”
34.1 Arrow’s Axioms
Arrow identified several basic properties that we might want a Social Welfare Function \(f\) to satisfy:
34.1.1 Universal Domain
The Social Welfare Function \(f\) should output a social ordering for any possible combination of voter preferences. More precisely, it is assumed that:
- voters are free to choose any ranking, and
- voters’ choices are independent.
This axiom requires that the domain of \(f\) includes all possible profiles - we make no special assumptions about voters’ preferences. That is, the Social Welfare Function must produce a coherent social ranking regardless of what rankings voters submit.
This is a minimal assumption satisfied by all the Social Welfare Functions we have discussed. As Arrow notes, “If we do not wish to require any prior knowledge of the tastes of individuals before specifying our social welfare function, that function will have to be defined for every logically possible set of individual orderings.”
34.1.2 Rationality
The Social Welfare Function \(f\) should output a social ordering that is a rational preference relation (Definition 6.1). This means that the social ordering must satisfy the following properties: 1. For any profile, the social ordering must be complete—al candidates are ranked or tied. 2. For any profile, the social ordering must be complete transitive (if \(a\) is ranked above \(b\) and \(b\) above \(c\), then \(a\) must be ranked above \(c\)).
Not all Social Welfare Functions satisfy rationality. For example, while Plurality and Borda always produce rational rankings, the Majority ordering can produce rankings that violate transitivity (Chapter 25).
34.1.3 Pareto (Unanimity)
If every voter prefers candidate \(a\) to candidate \(b\), then so should the social ranking. More formally:
For any profile \(\mathbf{P}\), if every voter ranks \(a\) above \(b\), then \(f(\mathbf{P})\) must rank \(a\) strictly above \(b\)
Most common Social Welfare Functions, including Borda, Majority, Instant Runoff, and Unanimity, satisfy the Pareto property. However, Plurality does not. Consider this simple example:
\[\begin{array}{cc} 2 & 1 \\\hline a & b \\ b & a \\ c & c\\ d & d \end{array}\]In this profile, every voter ranks \(c\) above \(d\). The Pareto property requires that the social ranking must then rank \(c\) strictly above \(d\). However, according to the Plurality ranking:
- \(a\) is ranked above \(b\) and both are ranked at the top (since \(a\) has 2 first-place votes and \(b\) has 1).
- \(c\) and \(d\) are tied for last place (since neither has any first-place votes).
This violates Pareto: even though everyone agrees that \(c\) is better than \(d\), Plurality fails to rank \(c\) above \(d\). The problem arises because Plurality only looks at first-place votes, ignoring the rest of voters’ rankings.
34.1.4 Independence of Irrelevant Alternatives
The relative ranking of two alternatives should depend only on how voters rank those two alternatives. If we change how voters rank other alternatives, it shouldn’t affect the relative ranking of the original pair.
Formally, if two profiles \(\mathbf{P}\) and \(\mathbf{P}'\) have identical relative rankings of \(a\) and \(b\) by all voters, then \(f(\mathbf{P})\) and \(f(\mathbf{P}')\) must rank \(a\) and \(b\) the same way relative to each other.
The Borda Count violates Independence of Irrelevant Alternatives. Here are two examples showing different ways this can happen:
Example 1: Consider the following two profiles with 100 voters:
- Proifle \(\mathbf{P}\): \[\begin{array}{cc} 45 & 55 \\\hline a & b \\ c & a \\ b & c \\ \end{array}\]
- Profile \(\mathbf{P}'\): \[\begin{array}{cc} 45 & 55 \\\hline a & b \\ b & a \\ c & c \\ \end{array}\]
Notice that in both profiles, voters rank \(a\) and \(b\) the same way relative to each other: 45 voters prefer \(a\) to \(b\) and 55 voters prefer \(b\) to \(a\). However:
- In profile \(\mathbf{P}\), the Borda ranking puts \(a\) above \(b\)
- In profile \(\mathbf{P}'\), the Borda ranking puts \(b\) above \(a\)
This violates IIA because the relative ranking of \(a\) and \(b\) changed even though voters’ preferences between \(a\) and \(b\) remained the same.
Example 2: Consider the following two profiles with 100 voters:
- Proifle \(\mathbf{P}\):
- Profile \(\mathbf{P}'\):
In both profiles, voters rank \(b\) and \(c\) the same way relative to each other. However:
- In \(\mathbf{P}\), the Borda ranking has \(b\) and \(c\) tied
- In \(\mathbf{P}'\), the Borda ranking puts \(b\) above \(c\)
Again, this violates IIA: changing the position of \(d\) affected the relative ranking of \(b\) and \(c\), even though voters’ preferences between \(b\) and \(c\) remained unchanged.
34.1.5 Non-Dictatorship
A basic principle of fairness is Anonymity (Definition 32.1). Arrow’s last axiom strengthens Anonymity: it requires not just that the identity of voters should not matter, but that no voter should be able to impose their strict preferences on the group.
A Social Welfare Function \(f\) satisfies Non-Dictatorship provided that there should not exist a voter \(d\) such that for any profile \(\mathbf{P}\) and any pair of candidates \(a\) and \(b\), if \(d\) ranks \(a\) above \(b\) in \(\mathbf{P}\), then \(f(\mathbf{P})\) ranks \(a\) above \(b\).
Every Social Welfare Function we have discussed satisfies Non-Dictatorship.
34.2 Arrow’s Impossibility Theorem
Now we can state Arrow’s remarkable impossibility result:
Theorem 34.1 (Arrow’s Theorem) Suppose there are at least three candidates and a finite number of voters. Then there is no Social Welfare Function that satisfies Universal Domain, Rationality, Pareto, Independence of Irrelevant Alternatives, and Non-Dictatorship.
An equivalent way to state the theorem is: Any Social Welfare Function that satisfies Universal Domain, Rationality, Pareto, and Independence of Irrelevant Alternatives must be dictatorial - that is, some voter must be a dictator.
This profound result shows that it is impossible to design a method for ranking candidates that satisfies all these seemingly reasonable properties simultaneously. Each property appears minimal and desirable on its own, yet together they lead to an impossibility.