Why Two Voting Rules Choose Different Winners

Present the driving question and frame the paradox: two reasonable voting rules, applied to the same set of ranked ballots, can crown different candidates. Show a stylized election preview — five candidates (A, B, C, D, E) and 100 voters — to set up the concrete example used throughout the investigation.

• Same electorate, same ranked ballots, different winners
• This is not fraud; both results are mathematically correct
• The investigation will build one specific 100-voter profile that triggers the split

Ask the learner to pick the most likely reason two voting systems would disagree on a winner, before any evidence is shown.

• Forces an explicit commitment to a hypothesis
• Surfaces the common assumption that one rule must simply be 'wrong'

Interactive table where the learner can switch the scoring rule (Plurality, Borda count, head-to-head / Condorcet) and see the winner change for one fixed electorate of 100 ranked ballots across candidates A–E.

• Same ballots in every column; only the rule changes
• Plurality: count of first-place ranks
• Borda: weighted sum of rank positions
• Condorcet: pairwise head-to-head victories across all pairs

Display the numeric result of the interaction: e.g., Plurality picks B (28 first-place), Borda picks C (highest average rank), Condorcet picks D (beats every other candidate head-to-head). Make the split undeniable.

• Three different winners from identical ballots
• Each winner is defensible by the rule that produced it
• The Condorcet winner D is often the one voters intuitively judge 'best'

Explain that Plurality rewards being 'someone's top choice,' Borda rewards 'broadly liked,' and Condorcet rewards 'universally preferred.' These are three distinct mathematical objectives that are not guaranteed to peak on the same candidate.

• Plurality optimizes: maximize first-place votes
• Borda optimizes: maximize average rank across all voters
• Condorcet optimizes: maximize pairwise win count
• With three or more options, these maxima need not coincide

Show the boundary condition: with only two candidates, every reasonable ranked rule produces the same winner. The split is a phenomenon specific to three or more alternatives.

• Two-candidate elections: Plurality, Borda, and Condorcet all agree
• Disagreement appears the moment a third option is added
• This is why every real voting paradox is about three or more choices

Interactive explorer: the learner adjusts three fairness properties (Unrestricted Domain, Unanimity/Pareto, Independence of Irrelevant Alternatives) and sees whether any combination can be satisfied by a single rule on 3+ alternatives. The point is to feel the impossibility rather than prove it formally.

• Property 1: Unrestricted Domain — any set of rankings is allowed
• Property 2: Unanimity — if everyone ranks X first, X must win
• Property 3: IIA — adding a spoiler candidate should not flip the winner
• Try to keep all three on at once; the system will push back

Hand the learner a new tiny ranked-ballot profile (e.g., 15 voters, 4 candidates) and ask them to predict which rule will produce which winner, then verify by tabulating. This transfers the explanatory principle to a novel, simpler situation.

• Predict the Plurality, Borda, and Condorcet winners before computing
• Identify whether this profile has a 'Condorcet loser' — the structural trigger for splits
• Confirm: whichever candidate is the Condorcet loser is exactly whose standing the rules disagree on

Close the investigation by directly answering: two voting rules disagree because they optimize different definitions of 'best,' and Arrow's impossibility theorem proves these definitions cannot be unified into a single rule once there are three or more alternatives.

• The disagreement is structural, not an error
• It is triggered specifically by the presence of a Condorcet loser
• Arrow's theorem: no rule on 3+ options satisfies all three fairness properties
• Choosing a voting rule is therefore choosing which definition of 'best' to honor