Five rules and one dial

Five rules for turning five fractional shares into five whole numbers, five different answers, and a literature that treats each as a proposal with a champion and a history. The rung below had to introduce all five before it could show what any of them breaks. Three of them are the same rule.

A divisor method works like this. Pick a number $d$ — a divisor, a notional number of people per seat — and give region $i$ the whole number $a_i$ obtained by rounding $p_i / d$. Adjust $d$ until the $a_i$ add up to the house size. Everything about the method is in the word rounding.

Where the rounding boundary sits #

Before the dial, the reason there is anything to choose. Five exact shares that are not whole numbers have to become five whole numbers with the same total, and rounding each one to the nearest integer does not work — the rounded values usually do not add to the house size. Something has to give, and every method in the subject is a different account of what.

Hamilton’s answer is to take the floors and hand out the leftovers by remainder, which is the worksheet the rung below starts from and the source of its paradoxes. The divisor methods’ answer is different in kind: do not round the shares — round a rescaled version of them, and choose the scale so that the total comes out right. The shares themselves are never rounded at all. That is why a divisor method can hand a region more seats than its share’s ceiling, which a reader who has only met Hamilton’s method finds impossible on first hearing.

Rounding $x$ to a whole number then means choosing a signpost between each pair of consecutive integers: a value $s(k)$ such that $x$ rounds up to $k+1$ when it passes $s(k)$ and down to $k$ when it does not.

Ordinary rounding puts the signpost halfway: $s(k) = k + \tfrac12$. Rounding down puts it at the top, $s(k) = k+1$; a share of $4.99$ still gives four seats. Rounding up puts it at the bottom, $s(k) = k$; a share of $4.01$ gives five.

Those three are Webster, Jefferson and Adams. Writing $s(k) = k + \delta$ with a single parameter $\delta$ between nought and one covers all three at once, and the whole family in between:

$$\delta = 0 \; \text{Adams} \qquad \delta = \tfrac12 \; \text{Webster} \qquad \delta = 1 \; \text{Jefferson}.$$

The figure runs $\delta$ across that range. Nothing else changes: the populations are the same, the house is the same, the search for the divisor is the same search. What changes is one number, and the answer changes with it in a completely predictable direction.

A region’s seat count is monotone in $\delta$. The largest region never loses a seat as the threshold rises, and the smallest never gains one. Both are checked at every step of the dial.

Why the dial is the honest picture #

The three names carry two hundred years of argument, and the argument is easier to have when they are three proposals than when they are one proposal with a dial on it.

As three rules, the question is which rule is right, and every answer is a claim about rounding. Jefferson’s case is that a region should get every whole seat it has fully paid for; Adams’s is that no region with any population at all should be unrepresented; Webster’s is that ordinary rounding is what rounding means.

As one rule, the question changes shape entirely: where should the threshold be, and what does moving it do? The answer to the second half is on the face of the figure. Moving the threshold up moves seats from small regions to large ones, one seat at a time, in a fixed order. So choosing a method is choosing a position on a scale from favours the small to favours the large, and the three named methods are three points on it with no special claim to be the only ones. The same is true of the many ways a group’s power can be measured: a named index is a choice of formula, and the informative question is what the formulas have in common.

That reframing has a cost worth being honest about. It makes the choice look like a matter of taste, when the next rung shows it is not: one setting of the dial is unbiased and the others are not, and which one is a theorem rather than a preference.

The two that are not on the dial #

Five methods were named and only three are on the dial. Hill and Dean are the other two, and the reason they are off it is the interesting part.

Hill’s signpost is the geometric mean $\sqrt{k(k+1)}$ and Dean’s is the harmonic mean $\frac{2k(k+1)}{2k+1}$. Neither is a fixed distance above $k$. The geometric mean of 4 and 5 is 4.472, so the signpost sits 0.472 above the lower integer; between 0 and 1 it sits at 0, and between 40 and 41 at 0.4969. The signpost creeps up toward a half as the numbers grow and is well below a half when they are small.

That is not a technicality; it is the whole design. A method whose signpost is low at small numbers rounds small shares up, which is a deliberate protection for small regions that a fixed $\delta$ cannot express. Adams achieves the same protection by setting $\delta$ to zero everywhere, which protects small regions at every size and is the reason its bias is so large.

The two of them sit between Adams and Webster in effect, and a reader who wants a single line to remember them by can have this one: the five classical methods are the five classical means. That is a much better organising fact than a list of five names with five champions, and noticing it is what turned the subject from a sequence of proposals into a structure with a shape.

So the family is really signpost functions rather than numbers, and the linear ones are the sub-family a dial can draw. The five classical methods are the five whose signpost is one of the five classical means of $k$ and $k+1$ — minimum, harmonic, geometric, arithmetic, maximum — which is Huntington’s observation and the reason his name is on the analysis rather than on a method. It is the same kind of unification as reading several cost-sharing rules as one formula: the rules were not competing accounts of one question, they were answers to different ones.

What the divisor is doing #

There is a second way to see a divisor method, and it explains why the dial’s monotonicity is not a coincidence.

Instead of searching for $d$, list every priority $p_i / s(k)$ for every region $i$ and every $k$ from 0 upward, sort them downward, and hand out seats in that order. The house size decides where to stop. That is the same method — the divisor that awards exactly $h$ seats is any number between the $h$-th and $(h{+}1)$-th priorities — and it is how every implementation actually works, including the ones behind these figures.

In that form the dial’s effect is transparent. Raising $\delta$ divides every priority by a larger number, but not by the same factor: a region’s first seat has priority $p/\delta$ and its tenth has $p/(9+\delta)$, so raising $\delta$ depresses first seats far more than tenth seats. Regions with few seats are the ones whose priorities are dominated by their early seats, so they are the ones that suffer. That is the monotonicity, argued rather than measured.

What the dial cannot settle #

A parameter that runs continuously between two named methods invites a question it cannot answer: which value is right?

Nothing in this rung answers it. That is not evasion: the question has three different answers depending on what is being asked for, and each of them is a rung of this ladder. The dial shows that the choice exists, that it is one-dimensional among the linear methods, and that moving it has a monotone effect. It does not say where to stand.

Three kinds of argument have been offered and each belongs to a later rung. Bias: one setting gives large and small regions their quotas equally often on average, and the others do not. Stability: each method is the unique one no transfer of a seat can improve, for a particular measure of unfairness. Impossibility: whichever setting is chosen, the method breaks the quota rule on some instance, and that is not a fact about the dial but about every method there is. The pattern is the field’s — a list of properties, a proof that no rule has all of them, and a choice about which to give up.

Only the first of those is about the dial at all. The other two are the reasons the subject did not settle down once someone noticed the parameter.

The same move elsewhere #

Collapsing several named rules into one parameterised family is a move worth recognising, because it is available far more often than it is taken.

Five voting rules on one profile give five different winners, and those five are also a family: scoring rules assign points to positions, and plurality, Borda and antiplurality are three settings of the scoring vector. The reframing is the same, and so is what it buys — the question stops being which rule and becomes which point, and the impossibility results stop looking like accidents about particular rules. The circle a majority can walk into is not repaired by any point of that family, which is what makes it a fact about the space rather than about a choice within it.

Where the move is not available is worth noticing too. Arrow’s theorem is about every rule, parameterised or not, and no amount of collapsing helps: the conclusion is that the whole space is empty, and finding a nice coordinate system on an empty space achieves nothing.

What the pictures cannot show #

The dial is drawn at twenty-one settings. The parameter is continuous, and what happens between two adjacent settings is that nothing happens: the seat counts are step functions of $\delta$ with finitely many jumps, and twenty-one samples catch the jumps on this instance and might miss one on another.

One instance is one instance. The monotonicity in $\delta$ is checked here and is a theorem; the sizes of the changes — three seats for region A across the dial — are facts about these five populations and this house.

Hill and Dean are named and not drawn on the dial, because they are not on it. Their worksheets appear separately, and no figure here shows the geometric mean creeping toward a half, which is the property that puts them between Adams and Webster in effect while sitting outside the family in form.

The priorities are not drawn either. Every seat in every figure here was awarded by ranking priorities and stopping at the house size, and what is shown is the result of that ranking. A picture of a hundred and thirty-five priorities in descending order would show where the seats change hands, and it would be a picture of an implementation.

And no figure says which setting is right. Every one of them draws what a rule does. The rest of the ladder is about what could make one of them the answer.

Where the ladder goes next #

The rung above measures the bias the dial makes visible. Four hundred instances, five methods, and the average of seats minus exact quota for the largest region and the smallest; the numbers come out at about a third of a seat in each direction for Jefferson and Adams, and at a hundredth for Webster. That Webster is the unbiased one is a theorem about where its signpost sits rather than a property of the instances.

Above that: the impossibility that says no choice on the dial or off it can have everything, and then Huntington’s account of what each of the five methods is separately optimising — which turns “five names” into “five questions”, and is the sharpest answer available to which one is right.

What a parameter is worth knowing about #

There is a practical reason to find the parameter, beyond tidiness, and it is about what can then be proved.

A statement about five named rules has to be proved five times, once for each, and a sixth rule needs a sixth proof. A statement about a family indexed by $\delta$ is proved once, with $\delta$ in it, and it covers every method in the family including the ones nobody has named. The monotonicity above is exactly such a statement: raising $\delta$ never costs the largest region a seat is one argument about priorities, and it would have been five observations about five tables.

It also makes the edges of the family visible. Every claim about the dial holds at $\delta = \tfrac12$ and at $\delta = 1$, and the case $\delta = 0$ needs its own sentence, because a signpost of zero makes a priority infinite. A family with a degenerate end is a family whose statements need a hypothesis, and the hypothesis is easy to find once the coordinate exists and nearly impossible to notice when the methods are five separate procedures with five separate proofs.

What is worth carrying away #

When several rules are presented as alternatives, the first thing to try is writing them as one rule with a parameter, because the shape of the disagreement is usually a coordinate.

Three of the five apportionment methods are the same rule at $\delta = 0$, $\tfrac12$ and $1$. Their two-century argument was, in that coordinate, an argument about where to stand on a line — and the two methods that are not on the line are not on it for a reason worth stating, which is that their signposts depend on how many seats a region already has. A family that does not contain everything is more informative than one that does: knowing which of the five are points on the dial says exactly what distinguishes the other two.