Voting rules
A Socratic walk-through of voting rules — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #How can the same set of votes elect different winners under different counting rules?
We speak as though an election discovers something: the will of the electorate, sitting in the ballots, waiting to be counted correctly. If that were true, the counting rule would be a technicality. So here is the awkward test — take one fixed set of ballots, with not a single voter changing their mind, and count them three defensible ways. If the winner changes, what exactly was the count discovering?
Reasoning it through
REASONING #Let us build the ballots ourselves, so nothing is hidden. A hundred voters, three candidates, and each voter ranks all three:
- 40 voters rank A first, then C, then B
- 35 voters rank B first, then C, then A
- 25 voters rank C first, then B, then A
Count by plurality first — read only each ballot's top line. A has 40, B has 35, C has 25. A wins. But notice what plurality discarded: every second and third line. Sixty voters put A last. A wins a race that most of the electorate would rather A lost.
So add a second round: eliminate the lowest and let their voters transfer. C, on 25, drops out, and all 25 ranked B above A, so B has 60 to A's 40. B wins — same ballots, and this time the rule did read the second line, but only of the losers' ballots.
Now ask a third question, the one the Marquis de Condorcet posed in the 1780s: is there a candidate who beats every other in a straight head-to-head? Check all three pairs. A against B: 40 prefer A, 60 prefer B, so B wins. A against C: 40 prefer A, 60 prefer C, so C wins. B against C: only the 35 B-voters put B above C, while the other 65 put C above B, so C wins. C beats both rivals one-on-one — yet C was eliminated first, for the crime of having the fewest enthusiasts.
One more count, to see whether that is a fluke. Borda's method awards 2 points for a first place, 1 for a second, 0 for a third. A scores 40x2 = 80. B scores 35x2 + 25x1 = 95. C scores 25x2 + 40x1 + 35x1 = 125. (The three total 300, as they must: 100 ballots x 3 points each.) C wins, comfortably.
Three winners, three rules, one set of ballots — because each rule answers a genuinely different question: who has the most first-choice support, who survives elimination, who would win every duel, who ranks highest on average. Is the profile a contrivance? Partly; ones this clean are constructed. But the pattern is real, and reversals of this shape have occurred in actual multi-candidate elections.
The analogy
THE ANALOGY #Think of a sports league arguing over a champion. The season's results are fixed, but the trophy depends on the rule: most wins outright, a knockout playoff among the top four, or the best head-to-head record against everyone else. Nobody thinks one of these is "the true champion" hiding in the fixtures list; the trophy is defined by the rule, not found by it.
teams actually play differently under different formats, whereas our voters' rankings were held fixed — and more importantly, nobody claims a league table expresses the will of a people, which is exactly the claim voting rules are asked to carry.
Clarifying the model
THE MODEL #The tempting conclusion — "no voting system is fair, so it is all arbitrary" — is both too strong and too vague. The precise result is Kenneth Arrow's 1951 theorem, and it says something narrower and sharper: for three or more alternatives, no rule that turns individual rankings into a complete, transitive social ranking can satisfy all of these at once — it works for every possible set of preferences, it ranks X above Y whenever every voter does, the social ranking of X and Y depends only on how voters rank X against Y (independence of irrelevant alternatives), and no single voter dictates the outcome. One of those must give.
Note what that does and does not say. It is a statement about ranked methods and about that specific list of conditions, not a proof that all systems are equally bad — systems still differ enormously in how often and how badly they misfire. Methods that ask for ratings rather than rankings sidestep Arrow's exact framing, but they do not escape a second result, Gibbard's theorem, which says essentially no non-dictatorial deterministic rule can make honest voting always the best strategy.
And Condorcet's criterion is not a rescue: a Condorcet winner need not exist. Rotate the preferences — A>B>C, B>C>A, C>A>B in equal thirds — and the majority prefers A to B, B to C, and C to A. The collective preference goes round in a circle even though every individual's is perfectly orderly.
A picture of it
THE PICTURE #How to readRead left to right: the three voter blocs on the left feed their first-choice tallies in the middle, and the middle column feeds the runoff on the right. The plurality winner is simply the fattest middle band, A with 40. The runoff winner is the fattest right-hand band, B with 60, because C's 25 all flow into B. What the picture cannot show is the thing that matters most: C, whose band vanishes at the first cut, is the candidate who beats both A and B head-to-head.
What became clearer
WHAT CLEARED #An election does not read off a pre-existing collective preference; it constructs one, using a rule that has already decided which information on the ballot counts. Plurality hears only enthusiasm, runoffs hear the losers' second thoughts, Condorcet hears every duel, Borda hears the whole ranking. Arrow's theorem then tells us this is not a defect awaiting a clever fix: the demands we would want a rule to meet are jointly unsatisfiable. Choosing a voting rule is therefore not an administrative decision downstream of democracy. It is part of what the democracy is.
Where to go next
ONWARD #- What each real system trades away in practice — and how often reversals actually occur.
- Strategic voting: how Gibbard's theorem shows up as tactical squeezing of third candidates.
- Proportional systems, which change the question from "who wins" to "who is represented".
Key terms
TERMS #| Term | What it means |
|---|---|
| Plurality | the candidate with the most first preferences wins, regardless of majority. |
| Runoff (instant-runoff) | lowest-placed candidates are eliminated in turn and their ballots transferred until someone has a majority. |
| Condorcet winner | a candidate who would beat every other in a one-on-one majority contest; may not exist. |
| Borda count | points awarded by rank position, summed across all ballots. |
| Arrow's impossibility theorem | the proof that no ranked voting rule can satisfy unrestricted domain, unanimity, independence of irrelevant alternatives and non-dictatorship at once for three or more options. |
Every term the collection defines is gathered in the glossary.