When Elections Misbehave: Arrow's Impossibility in Practice

Introduce Kenneth Arrow's 1951 impossibility theorem: any voting rule that aggregates ranked preferences for three or more candidates must violate at least one fairness condition (unanimity, IIA, non-dictatorship, transitivity). Frame the driving question with three real voting rules: plurality, Borda count, and instant-runoff voting (IRV).

• Arrow's theorem: no perfect voting rule exists for 3+ candidates
• Three fairness conditions can never all hold simultaneously
• Plurality, Borda, and IRV are the rules we will stress-test
• The question: which condition does each rule actually break?

Ask the learner to commit to which Arrow condition they expect plurality to break before seeing the spoiler evidence.

• Commit to one Arrow condition before the ballot evidence is shown
• The choice sets up the spoiler scenario that follows

Display two ballots under plurality voting. Ballot set A: three candidates A, B, C with A winning on first-preference votes. Ballot set B: identical ballots plus a fourth near-clone candidate X stealing votes from A, so B now wins. Show that the relative ranking of A vs. B did not change, but the winner did.

• Ballot set without X: A wins plurality
• Adding a near-clone X flips the winner to B
• A's position vs. B is unchanged in every voter's ranking
• This is a concrete IIA violation

Let the learner manipulate voter blocs and see Borda points flip a Condorcet winner into a loser. Widget exposes group sizes and ranking preferences for three candidates so the user can watch Borda totals and head-to-head records change.

• Adjust the size of three voter blocs with different rankings
• Watch Borda totals update live
• Compare Borda winner to the Condorcet winner
• Find a configuration where Borda picks the Condorcet loser

Connect the visible failures to Arrow's proof logic: with three or more candidates, the space of preference profiles is rich enough that no aggregation function can simultaneously satisfy unanimity, IIA, non-dictatorship, and transitivity. Plurality violates IIA, Borda can violate transitivity through Condorcet reversal, and IRV fails transitivity outright.

• Arrow's proof uses only the structure of ranked preferences
• It applies to ANY rule, including the three tested here
• Plurality → IIA failure (spoiler)
• Borda → transitivity failure (Condorcet reversal)
• IRV → transitivity failure (cycle across profiles)

Clarify the boundary of the result. Arrow's theorem does not apply to two-candidate elections, to approval or score-based voting with cardinal utilities, or to probabilistic/social-choice functions with restricted domains. Frame these as escapes from the impossibility rather than refutations.

• Two candidates: any reasonable rule works
• Cardinal scoring (approval, range) sidesteps ranked-preference assumptions
• Restricted preference domains (single-peaked) restore consistency
• Arrow's impossibility is about ranked-preference aggregation over 3+ options

Transfer task: the learner manipulates a three-profile IRV election to produce a non-transitive social outcome. Each profile is one election; chained together, profile 1 says A beats B, profile 2 says B beats C, profile 3 says C beats A. Show that IRV inherits this cycle even when individual profile outputs look sensible.

• Run three IRV elections, one per profile
• Each profile gives a clear pairwise winner
• Chain the three winners to reveal a cycle
• Recognize this as IRV's transitivity failure

Directly answer the driving question. Plurality breaks IIA via spoilers. Borda count breaks transitivity by overriding a Condorcet winner. IRV breaks transitivity through cycles across profiles. Arrow's impossibility is not an academic curiosity — it is visible in every real ranked-choice election.

• Plurality fails IIA — a near-clone flips the winner
• Borda fails transitivity — a Condorcet winner can lose on points
• IRV fails transitivity — chaining elections can produce a cycle
• Arrow's theorem is a structural fact about ranked-preference aggregation
• Voting rule choice is therefore a tradeoff, not a search for perfection