Questions: Arrow's Impossibility Theorem and Social Choice

Question 1

A committee adopts a voting rule that satisfies unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA). What does Arrow's theorem guarantee about this rule?

• A. It must also satisfy non-dictatorship, since three out of four conditions are already met
• B. It must be a dictatorship — the three conditions together force all decisive power to concentrate in one voter
• C. It is the fairest possible rule for groups of three or more alternatives
• D. It satisfies all four conditions, disproving Arrow's theorem for this particular case

Answer: (B)

Arrow's theorem states that the four conditions are collectively incompatible. If a rule satisfies unrestricted domain, unanimity, and IIA, it is logically forced to be a dictatorship — the fourth condition (non-dictatorship) must be violated. The proof works precisely by showing that these three conditions together generate an escalating concentration of decisive power that terminates in a single individual. There is no 'three out of four is fine' escape hatch; satisfying the first three makes dictatorship inevitable.

Question 2

Ranked-choice voting (instant-runoff) is often described as fairer than plurality voting. What does Arrow's theorem predict about ranked-choice voting?

• A. It satisfies all four of Arrow's conditions and is therefore genuinely fairer than plurality
• B. It satisfies unanimity and non-dictatorship but still violates independence of irrelevant alternatives — a spoiler candidate can still change the outcome between two other candidates
• C. It violates unanimity because voters rank all candidates, making some rankings effectively ignored
• D. It avoids Arrow's theorem because it uses ordinal rather than cardinal preferences

Answer: (B)

Arrow's theorem applies to all possible preference aggregation rules with three or more alternatives. Ranked-choice voting satisfies unanimity (if everyone prefers A over B, A wins) and non-dictatorship, but it violates IIA: eliminating a third candidate can change the relative outcome between the remaining two (the classic spoiler effect). This is not a flaw specific to ranked-choice — every system must violate at least one condition. Arrow's theorem tells us to ask which condition each system sacrifices, not whether a 'perfect' system exists.

Question 3

Arrow's impossibility theorem applies not just to existing voting systems but to any conceivable rule for aggregating individual preference orderings into a collective ranking.

• T. True
• F. False

Answer: True

Arrow proved a mathematical theorem about all possible aggregation functions satisfying his conditions — it is not an empirical survey of known voting methods. The proof starts with an arbitrary rule satisfying the three positive conditions and derives that it must be a dictatorship, without assuming anything about the rule's specific mechanism. This is why the result is so strong: no clever redesign of voting mechanics can escape it, only changing or abandoning one of the four conditions.

Question 4

Arrow's impossibility theorem shows that majority rule is the uniquely flawed voting system, and alternative systems like the Borda count avoid its core problems.

• T. True
• F. False

Answer: False

Arrow's theorem shows that ALL preference aggregation systems (including Borda counts, approval voting, plurality, ranked-choice, and every other conceivable rule) must violate at least one of his four conditions. Borda counts violate IIA: changing how voters rank a third option can alter the relative ranking of two other candidates. Plurality voting violates IIA through the spoiler effect. No system gets a clean bill of health. Arrow's result is about the impossibility itself — no system is uniquely flawed; all are unavoidably compromised in at least one way.

Question 5

Arrow's four conditions each seem like minimal fairness requirements. Explain the logical chain that makes them collectively impossible to satisfy — why do three of them force the fourth (non-dictatorship) to be violated?

Model answer: The proof works by showing that the conditions force decisive power to concentrate. Start with any group of voters that is 'decisive' over some pair (their preferences determine the group outcome for that pair). IIA and unanimity together imply this group must also be decisive over every other pair. Then any decisive group can be split into two sub-groups, and one of them must itself be decisive. Repeating this process shrinks decisive groups down to a single voter — a dictator. The impossibility is that IIA prevents a rule from 'balancing' one pair's outcome against another using third-option information, so decisive power cannot be diluted or shared.

The key conceptual move is that IIA acts as a kind of 'firewall' that prevents the rule from using cross-pair comparisons to distribute power. Without IIA, a rule could respond to the overall preference landscape in ways that give different voters influence over different pairs. With IIA, the decisive set for each pair must be fixed in advance, and the cascading logic of unanimity ensures that fixedness leads inevitably to a single dictator.