Choosing what unfair means

No rule has all three of the obvious properties, so the question of which method to use cannot be settled by demanding good behaviour. Huntington’s move, in 1921, was to stop asking what a method avoids and ask what it is for.

The question is local and it is very simple. Take an apportionment. Pick two regions. Would moving one seat from the first to the second make those two more equal? If some transfer would, the apportionment is improvable; if none would, it is stable. A method is characterised by the measure of inequality under which its own answer is always stable.

Three ways to be unequal #

Two regions, populations $p$ and $q$, holding $a$ and $b$ seats. What is the inequality between them?

The three that follow are the symmetric ones. Each takes the four numbers $p, a, q, b$ and returns a quantity that does not care which region was named first.

People per seat. Region one has $p/a$ people behind each of its seats and region two has $q/b$. The difference $|p/a - q/b|$ is measured in people, and it asks how much more work each of one region’s representatives is doing.

Seats per person. Turn it over: $|a/p - b/q|$, measured in seats per head. It asks how much representation each individual has.

The relative difference. Take either of the two ratios and compare them proportionally: $|x - y| / \min(x,y)$. Because a relative difference is unchanged by inverting both sides, the relative difference in people per seat and the relative difference in seats per person are the same number — which is a fact worth checking rather than believing, and the figure checks it by producing the same certified allocation for both.

Those are three genuinely different quantities. The first two are absolute differences of reciprocal quantities and disagree about which pairs are worst: a difference of a thousand people per seat is enormous between two small regions and negligible between two large ones, while a difference in seats per person behaves the other way.

There is a fourth candidate a reader will think of and it is worth disposing of. Why not compare $a$ with $q$ — seats against exact quota — and call the inequality $|(a - q_a) - (b - q_b)|$? Because that measure is not about the pair: it needs the quotas, which need the total population and the house size, so it is a comparison of each region against the whole rather than of the two regions against each other. Huntington’s question is deliberately local, and every measure that answers it uses only the four numbers $p$, $a$, $q$ and $b$.

What the table computes #

Nothing in the figure is quoted from Huntington. For each of the five apportionments and each of the three measures, every ordered pair of regions is taken, one seat is moved from the first to the second, and the measure is recomputed for that pair alone. If it falls, the apportionment is improvable and the cell names the transfer.

The result:

• Difference in people per seat certifies the allocation $15, 7, 2, 2, 1$ — which on this instance is Hill’s answer, and which Hamilton’s method happens to produce as well.
• Difference in seats per person certifies $16, 7, 2, 1, 1$, which is Webster’s.
• The relative difference certifies $15, 7, 2, 2, 1$ again.

Every measure certifies exactly one allocation, which is the property Huntington’s programme needed and which the figure asserts rather than observes. And the correspondence between measures and methods comes out of the computation, which is the point of doing it this way: a table of ticks copied from a source is a table of claims.

The two that are stable for none of them #

Jefferson’s and Adams’s apportionments are improvable under all three measures, and that is not an accident of the instance.

Each of the three measures above is symmetric: swap the two regions and the quantity is unchanged. Jefferson’s and Adams’s characterisations are not symmetric. Adams’s is stable for the test is region $i$ short of a seat compared with region $j$’s standard, applied in one direction only, and Jefferson’s for the mirror image. Huntington’s own list has five measures because he allowed the one-sided tests; restricting to the symmetric ones leaves three, and three of the five methods.

That is a real distinction and it is one of the two arguments Huntington used against Jefferson and Adams. A test that gives a different answer depending on which region is named first is a strange thing to call a measure of inequality between them. The other argument is the previous rung’s: those are the two extreme settings of the dial and they are the two with the largest bias.

Local and global #

There is a gap in the argument as stated, and it is the interesting one.

Stability is a local condition: no transfer of one seat between two regions improves things. What a reader wants is a global one: this apportionment minimises the total inequality, over all apportionments of the same house.

The two are not the same in general — a configuration that no single move improves can be far from the best — and for one of the three measures the gap is real. Huntington’s own analysis established the local statement, and the global statement holds for some of the measures and needs care for others: minimising the sum of pairwise inequalities, or the maximum, gives different answers, and neither is what the local condition characterises.

That is worth being blunt about, because it changes what the result is. A method is not the one that minimises unfairness; it is the one whose answer nobody can improve by a single move. As a criterion that is weaker than it sounds, and as a description of what a rule is, it is exactly right — an apportionment is arrived at seat by seat, and a rule that cannot be improved one seat at a time is a rule with no obvious next step. The distinction between local and global optimality is the same one a greedy algorithm lives inside, and the same warning applies: no single move improving things is a weaker statement than nothing improving things, and the gap between them is where the counterexamples live.

When the three measures agree #

The two instances above make a point the theory does not: most of the time the three questions have the same answer.

Webster’s answer and Hill’s differ only when the rounding falls a particular way, and finding an instance where they differ took a search. On tame instances — regions of comparable size, a house comfortably larger than the number of regions — every reasonable method gives the same seats, and the entire subject is invisible.

Where they pull apart is exactly where the populations are skewed and the house is tight, which is where a rounding rule has the most to decide. That has a practical consequence worth stating: the choice of method matters in proportion to how badly the problem needs one. A body dividing a large house between similar regions can use any of them; a body dividing a small house between wildly unequal ones will find that the choice moves seats, and will therefore find that everyone has a preference about it.

What this settles, and what it does not #

The reframing does one thing very well: it converts an argument about methods into an argument about measures, and an argument about measures is one that a person can have an opinion about without knowing any of the arithmetic.

Should two regions be compared by how many people are behind each seat, or by how much of a seat each person has? That is not a mathematical question. It is a question about what representation is for, and the two answers give Hill’s method and Webster’s — which are the two methods that assemblies actually argue between. The same substitution of a question for a rule is what makes the neighbouring impossibility bearable: once the conditions are separated, giving one up is a decision about priorities rather than an admission of defeat.

What it does not do is escape the previous rung. Every method in the table breaks the quota rule on some instance, stability or no stability, and no measure of inequality changes that. The impossibility is about the demands and this rung is about the objectives, and a rule can be the unique optimum for a perfectly sensible objective and still do something a reader will call absurd on some census.

Why a characterisation is worth having #

A rule that is merely defined is a procedure; a rule that is characterised is an answer. The difference shows up in three places.

It makes the rule portable. A method defined as “rank the priorities $p/\sqrt{k(k+1)}$ and stop at the house size” says nothing about why anyone should want it, and cannot be transferred to a problem where the priorities look different. A method defined as “the apportionment no transfer improves for the relative difference in representation” transfers to any problem where that measure makes sense.

It makes disagreement productive. Two people who disagree about which method to use, with only the procedures in front of them, can only trade instances. The same two with the characterisations in front of them are disagreeing about something specific, and one of them may be persuaded.

And it tells a reader when to stop looking. A method that is the unique optimum for a stated objective is finished as a design problem. Any improvement has to change the objective, which is a different kind of work — and this is the same service an impossibility theorem performs from the other direction.

What Huntington was arguing against #

The programme had a target, and knowing it explains why the analysis takes the form it does.

Before 1921 the argument between methods was conducted on examples. Somebody would produce a census on which their preferred method gave the sensible answer and another gave an odd one, and somebody else would produce a different census. Nothing settled, because every method looks bad on some instance and good on most.

Huntington’s response was to stop comparing outputs and start comparing objectives. If a method is the unique answer to a stated question, then producing a census on which it looks odd is not an argument against it — it is an argument against the question, and the person producing the census has to say which other question they prefer. That is a debate with somewhere to go.

The same manoeuvre is what a set of axioms does for a solution concept: it moves the disagreement from the answers to the assumptions, where it can be examined. Whether the manoeuvre succeeds depends entirely on whether the objectives are things a reader can have an opinion about, and people per seat versus seats per person passes that test more comfortably than most.

What the pictures cannot show #

Two instances are two instances. The correspondence between measures and methods is a theorem about all instances; the figures compute it on two, which is evidence and not a proof.

The transfers are between pairs, and only one seat at a time. An apportionment that no single transfer improves might be improved by moving two seats at once, or by a three-way rotation. Nothing here searches for those, and the local nature of the condition is exactly what the section above is about.

A region with one seat cannot give it up. Every search skips transfers that would leave a region with none, because an apportionment doing that is a different object and three of the five methods refuse to produce one. Allowing it would change some cells.

Hamilton’s row is a coincidence of this instance. Hamilton’s method is not a divisor method and has no Huntington characterisation; the ticks in its row are there because its answer happens to coincide with Hill’s on both instances drawn, and on another instance it would not.

And the two one-sided measures are described and not computed. Jefferson’s and Adams’s own characterisations use tests that are not symmetric in the two regions; the figures test three symmetric measures and report, correctly, that neither method is stable for any of them.

Where the ladder goes next #

This closes the ladder. It began with five regions, twenty-seven seats and three paradoxes on one instance; it ends with the observation that the five rules are five answers to five different questions, three of which can be stated in a line.

What remains unsettled is what was unsettled at the start: which question to ask. The subject has an impossibility theorem saying no rule has everything, a bias theorem saying one rule has no favourites, and a characterisation saying each rule is the unique answer to one objective. Those three results do not point at the same method, and there is no fourth result that adjudicates between them. That is a stable state for a subject rather than an unfinished one: the mathematics has produced everything it can, and what is left is the kind of choice that has to be made by whoever will live with it.

Sideways: the same shape appears wherever a rule for choosing is characterised by what it optimises rather than by what it avoids — a cost is split by the unique rule satisfying four properties, and a matching is stable exactly when no pair wants to defect, which is a local condition of precisely this kind.

What is worth carrying away #

When a choice between rules cannot be settled by asking what each avoids, the move is to ask what each optimises — and the answers are usually more distinguishable than the rules.

Three measures of inequality, none of them exotic, produce three different apportionments, and each of the classical methods turns out to be the unique stable answer for one of them. The disagreement between the methods was never about arithmetic; it was about the units. People per seat and seats per person are reciprocal quantities, they cannot both be equalised, and choosing which to equalise is choosing the method — which is a decision an assembly can have an argument about, and a decision no theorem will make for it.