Arrow's Impossibility Theorem

Arrow's Impossibility Theorem

Recap: Where We Are

In Lecture 2, we introduced four aggregation rules — majority, unanimity, Borda count, and dictatorship — and evaluated them against three axioms: Completeness (C), Pareto (P), and No Dictator (ND). We saw that each rule has a flaw: unanimity violates C, dictatorship violates ND, and majority rule violates transitivity (which we observed implicitly but did not yet formalize). Borda count seemed to pass all three, but we flagged that a fourth axiom would cause trouble. That fourth axiom is Independence of Irrelevant Alternatives, and the resulting impossibility result is Arrow's theorem.

The IIA Axiom

Independence of Irrelevant Alternatives (IIA): The social ranking of any two alternatives $x$ and $y$ depends only on how individual voters rank $x$ against $y$ — not on how they rank $x$ or $y$ against any third alternative $z$.

More formally: suppose we have two preference profiles $(\succsim_1, \ldots, \succsim_N)$ and $(\succsim_1', \ldots, \succsim_N')$ such that for every voter $i$ and every pair $\{x,y\}$, voter $i$'s ranking of $x$ vs. $y$ is the same in both profiles (that is, $x \succsim_i y \iff x \succsim_i' y$). Then the social ranking of $x$ vs. $y$ must also be the same under both profiles.

The intuition is that the social ranking of two options should be determined by how people compare those two options, full stop. Changing how voters rank a third option $z$ — which does not affect their relative views on $x$ vs. $y$ — should not change the social ranking of $x$ vs. $y$. This sounds eminently reasonable, but it rules out Borda count.

Borda Count Violates IIA: The Russian Election Example

Consider four candidates: Navalny (N), Ivanov (I), Putin (P), and Zyuganov (Z). There are 100 voters: 60 of type 1 and 40 of type 2, with the following rankings:

• Type 1 (60 voters): Navalny $\succ$ Ivanov $\succ$ Putin $\succ$ Zyuganov
• Type 2 (40 voters): Ivanov $\succ$ Zyuganov $\succ$ Putin $\succ$ Navalny

With four candidates, the Borda scores are 4, 3, 2, 1 from first to last. The total Borda scores are:

$$\begin{aligned} \text{Navalny:} & \quad 4 \cdot 60 + 1 \cdot 40 = 240 + 40 = 280 \\ \text{Ivanov:} & \quad 3 \cdot 60 + 4 \cdot 40 = 180 + 160 = 340 \\ \text{Putin:} & \quad 2 \cdot 60 + 2 \cdot 40 = 120 + 80 = 200 \\ \text{Zyuganov:} & \quad 1 \cdot 60 + 3 \cdot 40 = 60 + 120 = 180 \end{aligned}$$

The social ranking is: Ivanov $\succ_S$ Navalny $\succ_S$ Putin $\succ_S$ Zyuganov. In particular, Navalny ranks above Putin.

Now suppose Zyuganov drops out of the race. We re-run Borda with only three candidates, using scores 3, 2, 1. The rankings of the remaining three candidates are unchanged for every voter. But the Borda scores change:

$$\begin{aligned} \text{Navalny:} & \quad 3 \cdot 60 + 1 \cdot 40 = 180 + 40 = 220 \\ \text{Ivanov:} & \quad 2 \cdot 60 + 3 \cdot 40 = 120 + 120 = 240 \\ \text{Putin:} & \quad 1 \cdot 60 + 2 \cdot 40 = 60 + 80 = 140 \end{aligned}$$

The social ranking is now: Ivanov $\succ_S$ Navalny $\succ_S$ Putin. That is the same relative ranking of Navalny vs. Putin — still Navalny above Putin. So far no violation.

But let us reconsider with a slightly different type-2 ranking. Suppose instead:

• Type 1 (60 voters): Navalny $\succ$ Ivanov $\succ$ Putin $\succ$ Zyuganov
• Type 2 (40 voters): Zyuganov $\succ$ Ivanov $\succ$ Putin $\succ$ Navalny

Now with all four candidates:

$$\begin{aligned} \text{Navalny:} & \quad 4 \cdot 60 + 1 \cdot 40 = 280 \\ \text{Ivanov:} & \quad 3 \cdot 60 + 3 \cdot 40 = 300 \\ \text{Putin:} & \quad 2 \cdot 60 + 2 \cdot 40 = 200 \\ \text{Zyuganov:} & \quad 1 \cdot 60 + 4 \cdot 40 = 220 \end{aligned}$$

Social ranking: Ivanov $\succ_S$ Navalny $\succ_S$ Zyuganov $\succ_S$ Putin. Navalny ranks above Putin.

Now remove Zyuganov. Three-candidate Borda with type 2 ranking Ivanov $\succ$ Putin $\succ$ Navalny:

$$\begin{aligned} \text{Navalny:} & \quad 3 \cdot 60 + 1 \cdot 40 = 220 \\ \text{Ivanov:} & \quad 2 \cdot 60 + 3 \cdot 40 = 240 \\ \text{Putin:} & \quad 1 \cdot 60 + 2 \cdot 40 = 140 \end{aligned}$$

Still Navalny above Putin. Let us try a version where the violation is stark. Consider:

• Type 1 (60 voters): Navalny $\succ$ Putin $\succ$ Ivanov $\succ$ Zyuganov
• Type 2 (40 voters): Ivanov $\succ$ Zyuganov $\succ$ Putin $\succ$ Navalny

With all four candidates (scores 4,3,2,1):

$$\begin{aligned} \text{Navalny:} & \quad 4 \cdot 60 + 1 \cdot 40 = 280 \\ \text{Ivanov:} & \quad 2 \cdot 60 + 4 \cdot 40 = 280 \\ \text{Putin:} & \quad 3 \cdot 60 + 2 \cdot 40 = 260 \\ \text{Zyuganov:} & \quad 1 \cdot 60 + 3 \cdot 40 = 180 \end{aligned}$$

Social ranking: Navalny $\sim_S$ Ivanov $\succ_S$ Putin $\succ_S$ Zyuganov. Both Navalny and Ivanov rank above Putin.

Now remove Zyuganov. Three-candidate Borda (scores 3,2,1):

$$\begin{aligned} \text{Navalny:} & \quad 3 \cdot 60 + 1 \cdot 40 = 220 \\ \text{Ivanov:} & \quad 2 \cdot 60 + 3 \cdot 40 = 240 \\ \text{Putin:} & \quad 1 \cdot 60 + 2 \cdot 40 = 140 \end{aligned}$$

Now Ivanov $\succ_S$ Navalny. No voter changed their ranking of Navalny vs. Ivanov — yet the social ranking of those two flipped when Zyuganov was removed. This is a clear IIA violation: the social ranking of Navalny vs. Ivanov depends on where Zyuganov fits in everyone's ranking, even though Zyuganov is "irrelevant" to the Navalny-Ivanov comparison.

Summary: Which Rules Satisfy Which Axioms

We now have a complete picture of the four rules against five criteria (C, P, ND, IIA, and Transitivity of the social preference):

Majority rule satisfies C, P, ND, and IIA, but can violate transitivity of $\succsim_S$ (the Condorcet paradox: if voter 1 prefers $x \succ y \succ z$, voter 2 prefers $y \succ z \succ x$, and voter 3 prefers $z \succ x \succ y$, then majority gives $x \succ_S y$, $y \succ_S z$, and $z \succ_S x$ — a cycle).

Unanimity rule satisfies P, ND, IIA, and transitivity, but violates C (incomparability whenever voters disagree).

Borda count satisfies C, P, ND, and transitivity, but violates IIA as shown above.

Dictatorship satisfies C, P, IIA, and transitivity, but violates ND by construction.

No rule in our list satisfies all five properties simultaneously. Arrow's theorem says this is not a coincidence — it is impossible.

Arrow's Impossibility Theorem

Arrow's Impossibility Theorem (Formulation 1): Suppose $|X| \geq 3$. There is no social choice function $F$ that simultaneously satisfies Completeness (C), Pareto (P), No Dictator (ND), and Independence of Irrelevant Alternatives (IIA).

Equivalently:

Arrow's Impossibility Theorem (Formulation 2): Suppose $|X| \geq 3$. Any social choice function $F$ that satisfies C, P, and IIA must be a dictatorship.

The proof is non-trivial and involves showing that IIA forces the social choice function to have a very rigid structure that must ultimately track one individual's preferences entirely. We do not prove this in full, but the key step is showing that IIA implies the existence of a "decisive" voter — one whose strict preferences always determine the social strict preference — which is precisely a dictatorship.

Interpretation: What Arrow's Theorem Does and Does Not Say

Arrow's theorem does not say that democracy is bad, nor that there is no good way to make collective decisions. It says that certain desiderata are mutually incompatible when the set of alternatives has at least three elements. Something has to give.

The three main "exits" from the impossibility are:

Drop IIA: This allows Borda count and similar scoring methods. The cost is that the social ranking between two alternatives can depend on how people rank irrelevant third alternatives, which creates scope for agenda manipulation (strategically adding or removing candidates to change the outcome). Whether this cost is acceptable is a substantive question.

Drop ND: This gives dictatorship. Not very attractive as a democratic ideal.

Restrict the preference domain: If we restrict attention to preference profiles where all voters have single-peaked preferences over a one-dimensional policy space, then majority rule satisfies all the other axioms and produces a transitive social preference. This is the route taken by the Median Voter Theorem, which we develop in Lecture 4.

The theorem is a fundamental result in political economy because it shows that there is irreducible tension among the things we want from a collective decision procedure. It does not counsel despair — it counsels realism about the limits of aggregation and careful thought about which properties matter most in a given context.