Arrow's Impossibility Theorem

Basic Assumptions

Let $N = \{1, \dots, n\}$ be a set of voters and $A = \{a_1, \dots, a_m\}$ with $m \ge 3$ a set of alternatives. A social welfare function $F$ maps each profile of individual preference orderings $(\succeq_1, \dots, \succeq_n)$ to a social ordering $\succeq$ on $A$. Arrow's theorem assumes as background that any non-degenerate social welfare function will satisfy:

U — Unrestricted Domain $F$ is defined for every logically possible profile of strict orderings on $A$. The system must always produce a ranking; it cannot "give up" when voters have unusual preferences.

D — Non-Dictatorship The system does not depend on only one voter's ballot. There is no voter $i$ such that $a \succ_i b \Rightarrow a \succ b$ for all $a, b$ and all profiles.

N — Non-Imposition For every pair $a, b \in A$, there exists some profile under which $a \succ b$ in the social ordering. That is, no social ranking between two candidates is imposed regardless of votes.

Note: Non-imposition is often replaced with the stronger Pareto efficiency (if $a \succ_i b$ for all $i$, then $a \succ b$), which implies non-imposition. The weaker condition is sufficient for the theorem.

Arrow's Theorem

Theorem (Arrow, 1951)

If $|A| \ge 3$ and $|N| \ge 2$, no social welfare function satisfying U, D, and N can also satisfy:

IIA — Independence of Irrelevant Alternatives The social ranking of $a$ vs. $b$ depends only on individual rankings of $a$ vs. $b$. If every voter's relative preference between $a$ and $b$ is unchanged, the social ranking of $a$ vs. $b$ must also be unchanged — regardless of how preferences over other candidates shift.

Equivalently: any social welfare function on $\ge 3$ alternatives satisfying U, N, and IIA must be a dictatorship.

Interactive Exploration

Drag candidates to reorder each voter's preference ranking (top = most preferred). Then observe how common aggregation methods each violate at least one of Arrow's conditions.

Preset Profiles

Aggregation Methods & Arrow's Conditions

Each column reflects a method-level property (whether the method satisfies the condition across all possible profiles), not just the current profile. All four methods satisfy U (unrestricted domain) trivially. The "T" column indicates whether the method always produces a transitive social ordering (required for a valid SWF under U).

Dictator: — the social ordering is simply this voter's ranking

Method	Result	T (Transitive)	N (Non-Impos.)	IIA	D (Non-Dict.)

Pairwise Majority Detail

Head-to-Head Contests

Demonstrating IIA Violation (Borda Count)

The Borda count violates IIA: the social ranking between two candidates can change when a third candidate's position shifts, even if no voter changes their relative preference between the two.