Five rules and five winners

The rung below left the pairwise majority in pieces: a majority preferring A to B, another preferring B to C, another preferring C to A, and no ranking at all to be had out of the three. That failure is loud, and it is also uncommon. The trouble on this page is quieter, needs no breakdown of any kind, and is harder to shrug off — because in it every rule works perfectly.

Nothing in that picture is a trick. The ballots are a single fixed list. Nobody changed their mind between one column of the table and the next. Each rule reads the same twenty-seven rankings and returns a different name, and every one of those names is correct under the rule that produced it.

The five rules, stated exactly #

Each rule takes a preference profile — a list of voters, each of whom has ranked all the candidates from first to last — and returns a name. All five have been proposed in earnest, all five are in use somewhere, and none is anybody’s joke.

• Plurality. Count first places only. Whoever has the most is the winner.
• Borda. Each ballot awards $c-1$ points to its first choice, $c-2$ to its second, down to zero for its last. Whoever has the most points wins. With five candidates and twenty-seven voters the largest score anybody could reach is $27 \times 4 = 108$.
• Instant runoff. Count first places. If nobody holds a majority of the whole electorate, delete whoever has fewest, promote each deleted name’s ballots to their next surviving choice, and count again.
• Condorcet. For each pair of candidates ask which of the two a majority ranks higher. Whoever wins every one of those contests is the winner.
• Coombs. As instant runoff, except that the candidate deleted each round is the one with the most last places rather than the fewest firsts.

The figure’s generator implements all five separately, and refuses to compute one from another even where it easily could. That is not tidiness. If instant runoff were computed by reusing the plurality tally, a single shared mistake would pull two winners together and the picture would understate the very thing it exists to show.

The count of distinct winners is then taken twice over, once through a set and once by walking the list and keeping first occurrences. Two routes to a number a caption is going to quote is the habit this collection runs on, and here the headline is a single digit.

Each rule also refuses to guess. When a deciding count comes out level — two candidates tied for elimination, two tied at the top — the generator stops and says which step was level rather than breaking the tie. A tie-break is another rule, and an unstated rule doing work inside a picture is the one thing this collection will not have.

Where this profile came from #

The provenance matters, and hiding it would be dishonest, so: this profile was found by search. Profiles were generated and counted until one turned up on which all five rules disagreed. It was then shrunk — from ninety-three voters to twenty-seven, which is the smallest electorate the search reached while keeping every deciding count strict — and finally the candidates were relabelled so that the five winners read A, B, C, D, E in the order the rules are listed.

That last step costs nothing whatever, and it is worth saying why. The candidates are letters, with nothing behind them that a permutation of the alphabet could disturb, so relabelling is a change of notation rather than of content and every count in the table is identical before and after. The tidiness of the answer column is presentational; the disagreement is not.

What the search does not establish is how common such profiles are. A successful search proves that the thing exists and says nothing at all about density. What has been produced is a counterexample to the idea that the ballots settle the winner, and it is worth setting beside its opposite: an argument that proves some object exists without producing one leaves the other half of the question open, and both gaps are easy to overlook once an object is on the page. The six groups are of sizes seven, six, five, five, two and two, and $7 + 6 + 5 + 5 + 2 + 2 = 27$.

The majorities underneath the table #

The same twenty-seven ballots carry a second structure that the rules table does not show: for each pair of candidates, which of the two a majority puts higher.

Two rows repay a long look. Row D is positive across the board: D wins all four of its head-to-head contests, and by margins as small as $+1$. And row A is negative across the board — a majority prefers B to A, C to A, D to A and E to A. Yet A is plurality’s winner, with the most first places of anybody.

So the profile contains four candidates whom a majority would rather have than the plurality winner. That is not an artefact of the search; it is what happens whenever support is concentrated in one bloc and opposition is spread across several. Plurality sees the concentration and has no instrument for seeing the spread, because it never looks below the top line of a ballot.

This matrix of pairwise majority verdicts is the whole of what the Condorcet rule reads, and Condorcet is the only one of the five that can return nobody at all — the rung below is entirely about the profiles on which it does. Here it returns D, cleanly, with 10 and 17 splitting the electorate on the very first pair and $10 + 17 = 27$.

What each rule refuses to read #

The disagreement becomes much less mysterious once each rule is described by what it discards.

Plurality reads the first name on each ballot and throws the rest away. Borda reads every position and throws away nothing, which is why its verdict on two candidates can be shifted by a third who wins nothing at all — a section further down does exactly that. Instant runoff reads the top of each ballot repeatedly, so it sees a little more of each ballot the longer that ballot’s favourites survive. Coombs reads the bottom, and therefore measures opposition where instant runoff measures support. Condorcet reads every pair and nothing else — it never forms a score for a candidate at all.

Deliberate discarding is how the good ideas in this collection tend to work: Euler solved a problem about a city by deleting the city, and what survived was worth more than the map. Each of these five rules is a deletion of that kind — keep this much of each ballot, discard the rest, count what is left.

The difficulty is that the deletions are incomparable. There is no sense in which reading only the top is less faithful than reading every pair; reading every pair is what produces the cycle of the rung below, and reading the whole ballot is what makes Borda sensitive to candidates who cannot possibly win. Each rule keeps something real and drops something real, and the five choices of what to drop are exactly the five winners.

The control, and why it is the important figure #

A picture of five rules disagreeing is worth very little on its own, because a reader with no other information cannot tell whether it is a curiosity or the normal state of affairs. So the same machinery is run on a different profile of the same size.

On these ballots the five rules are unanimous, and the unanimity is not bought cheaply: no candidate has a majority of first places, so nothing forces the answer. Every rule simply arrives at A, by five different routes and five different counts. The group sizes are ten, seven, four, three, two and one, and $10 + 7 + 4 + 3 + 2 + 1 = 27$.

That is the ordinary case, and it is what makes the first figure a finding rather than a scandal. Rules that disagreed on most profiles would be useless and would have been abandoned long ago; rules that never disagreed would be one rule with five names. The interesting situation is the actual one — broad agreement, punctuated.

Two rules parting company #

Between total agreement and total disagreement sits the case a reader is most likely to meet, in which the rules split into two camps.

Plurality gives it to A on eleven first places out of twenty-seven. The other four rules, each reading a different part of the ballots, all arrive at B — which is the Condorcet winner, has the most Borda points, survives both eliminations, and still has three fewer first places than A. The group sizes here are eight, seven, four, three, three and two, and $8 + 7 + 4 + 3 + 3 + 2 = 27$.

One rule against the rest is the common form of disagreement, and it is where the choice of rule does its real work. A five-way split is arresting; a two-way split is the one that decides things.

Three candidates, and a ceiling on the disagreement #

The disagreement has a hard upper bound, and the bound is a counting argument of the simplest kind.

Five rules return five names, but the names are drawn from the candidates on the ballot. With only three candidates there are three names to go round and five rules asking for one, so two of the rules must return the same candidate — not because they are similar, but because there is nowhere else for the fifth answer to go. “Five rules and five winners” therefore needs at least five candidates, which is why the profile at the top of this page has exactly five.

Three is the ceiling and the ceiling is reached: plurality takes A, Borda takes B, instant runoff takes C, and the remaining two rules are obliged to duplicate. The four groups are of sizes three, three, two and one, and $3 + 3 + 2 + 1 = 9$ — nine voters, which is small enough that the whole thing can be checked by hand in a few minutes, and worth doing.

The same trouble inside a single rule #

It would be comfortable to conclude that the problem is having five rules, and that any one of them, adopted and left alone, would behave. It would not.

Every voter in the second profile ranks A against B exactly as they did in the first. Not one opinion about that pair has changed. All that moved is a third candidate — one who wins nothing, and whose presence ought to be irrelevant to a question about A and B — and Borda’s verdict on A against B reverses.

The pair was not typed in. It was searched for, by fixing each voter’s ranking of A against B, partitioning the profiles by that pattern, and counting the ordered pairs inside a class whose verdicts oppose. The sweep starts at three voters, finds nothing, and finds the first flip at four — which is a genuine fact about Borda count rather than a fact about how hard somebody looked.

That condition is called independence of irrelevant alternatives, and the rung above takes it up properly, as one of four conditions no rule can satisfy at once. What matters here is narrower: the sensitivity is not a symptom of having several rules. It lives inside each of them.

When the ballots settle nothing at all #

One of the five can fail in a way the other four cannot. With every count strict and no tie anywhere, Condorcet can still return nobody at all.

The right-hand panel is what the rung below is about, and the number is worth carrying forward: every one of the $6^3 = 216$ profiles of three voters over three candidates was constructed and decided, and twelve of them have no Condorcet winner. That is a count over a completely exhausted space, and nothing about it is estimated.

Notice what the two halves of this page do to each other. The condorcet cycle the rung below exhibits is a failure of the ballots to determine anything; the five-winner profile is a failure of a different sort, because there the ballots determine five perfectly good answers. The second is the more corrosive, since it comes with no warning light. A rule that returns nobody at least announces its own difficulty.

What these figures cannot show #

Every number on this page is a count over a stated finite object, performed rather than quoted — the same standing of evidence as a theorem settled by a machine walking a list of cases — and the boundary of what that licenses needs saying plainly.

The five winners at the top are exact, and they are exact about this profile. Twenty-seven ballots were counted five ways and the five answers differ; each answer was checked against every other candidate on that rule’s own scale before the picture was drawn. That much is settled and no reader need take it on trust.

The essay’s larger claim — that this is a normal feature of voting rules rather than a contrived specimen — is precisely what a single profile cannot establish. The profile was found by search. A successful search is evidence that such profiles exist, and it is no evidence at all about how common they are; the two questions are not related, and answering the first tells nothing about the second. Neither is the control figure a frequency claim: it is one profile on which the rules agree, and one profile is one profile whichever way it comes out. Nothing here counts how large the disagreeing region is, and it would be easy and wrong to leave a reader with the impression that something had.

The one count here that is a frequency claim is the exhausted one in the cycle figure — twelve of two hundred and sixteen — and it is worth noticing how narrow it is: three voters, three candidates, one property. The syllogism figures can say fifteen of two hundred and fifty-six because all two hundred and fifty-six were built; nothing of that kind has been built for profiles here, and the honest move is to name the searches that were run and stop.

Where the choice actually sits #

The five rules disagree, each is defensible, and each is a different answer to the question of which part of a ballot to read. Adopting one of them is therefore choosing an answer, and the choice is made before a single ballot is filled in.

That is a familiar structure elsewhere in this collection, in a place nobody expects it. The Monty Hall answer is not about the doors; it is about what the host was permitted to do, fixed in advance, invisible in the evidence. Here the answer is not about the ballots; it is about which rule was adopted, fixed in advance, invisible in the table of rankings. In both cases the data is not enough, and the missing ingredient is a procedure chosen earlier by somebody else.

The sharpest version of the same idea is in analysis. A conditionally convergent series has no sum until an order of summation is fixed — rearrange the terms and it converges to whatever number is asked for, adding nothing and removing nothing. A profile of ballots is the same kind of object. It has no winner until a rule is fixed, and the rule, like the ordering, supplies the answer while appearing merely to organise the counting. That two subjects so far apart should share the structure is the most surprising thing on this page.

Two more things follow, and neither is a recommendation. The first is that a rule, once chosen, is not even safe from the voters it counts.

The second is that arguing from these figures to a recommendation is not available. The figures decide which rule returns which winner on the profiles shown, and they decide nothing whatever about which rule anybody should use. Choosing between rules means weighing what each one discards, which is a judgement rather than a computation, and this page holds no such judgement and no materials for one. Two structures can satisfy the same axioms and disagree about everything else — that is how independence is demonstrated in logic, and the five rules stand in close to that relation.

The instant runoff and Coombs columns make the point in miniature: one rule deletes on fewest firsts, the other on most lasts, and on the first profile of this page they eliminate almost opposite sets and end with different names. Neither is reading the ballots wrongly. They are reading different ballots, assembled out of the same marks, and the fourth rung of this anchor asks what happens once a voter notices which one is being read.