The Strange Mathematics of Voting Power

In 1880 the United States Census Bureau noticed something disturbing. They had been allocating House seats among the states using a method invented by Alexander Hamilton: give each state its proportional fair share, round down, then hand out the remaining seats to whoever had the largest fractional remainders. It seemed obviously correct. It also produced a result that any reasonable person would call broken.

Alabama had 8 seats when the House had 299 members. When the House was expanded to 300 — adding one seat — Alabama dropped to 7. Adding a seat had taken a seat away. The Alabama Paradox, as it came to be known, exposed the first fault line in what looked like a settled question of arithmetic. The question of how to convert fractions of votes into whole representatives is, it turns out, mathematically impossible to solve fairly. And the choice of which unfairness to accept tells you a great deal about a society.

The problem looks trivial

You have 100 seats and 4 states with populations 49,000, 49,000, 1,000, and 1,000 — total 100,000. Each person counts as 1/1000 of a seat. The two large states should each get 49 seats, the two small states 1 seat each. The arithmetic works exactly.

Now change the numbers. Populations 47,800, 47,800, 2,200, and 2,200. The fair shares become 47.8, 47.8, 2.2, 2.2. You cannot give a state 0.2 of a seat. Round down: 47, 47, 2, 2 — total 98. You have two seats left. Who gets them?

Hamilton said: the two states with the largest fractional remainders. In this case, all four are tied, so it doesn't matter. Pick any two. Try other examples and the rule produces the Alabama Paradox, where adding a seat to the total changes who gets what in non-monotonic ways.

Webster, Jefferson, and the divisor methods

The mathematicians of the early Republic were not stupid. They tried other methods. Thomas Jefferson proposed picking a "divisor" — a target ratio of population to seats — and giving each state the floor of population / divisor. Adjust the divisor until the totals match. This eliminates the Alabama Paradox but introduces a different bias: it systematically over-represents large states because flooring a large number rounds away less proportionally.

Daniel Webster suggested rounding to nearest instead of flooring. This reduces the bias toward large states but introduces a new pathology: a state's seats can decrease when its own population grows, if other states grew faster. The Population Paradox.

By the late 19th century the patterns were clear: every method has a paradox somewhere. Hamilton's method violates monotonicity in seat count (Alabama). Divisor methods violate monotonicity in population. There is no method that simultaneously respects all three of: proportionality, monotonicity in total seats, and monotonicity in state population.

The Balinski–Young theorem

In 1980 Michel Balinski and H. Peyton Young proved this rigorously. Their theorem states: no apportionment method can simultaneously satisfy quota (every state gets either the floor or the ceiling of its fair share), house monotonicity (no Alabama Paradox), and population monotonicity (no Population Paradox). You can have any two but never all three.

This is a deep result, philosophically. It means the question "what is the fair way to allocate seats" has no answer — only trade-offs. Every nation that uses proportional representation has chosen a trade-off, often without realizing they were choosing.

The methods nations actually use

The United States House uses the Huntington–Hill method (since 1941), which uses a geometric mean as its rounding rule. It's a divisor method that minimizes a particular notion of relative inequality. It can violate quota in extreme cases.

Most European parliamentary systems use D'Hondt or Sainte-Laguë. D'Hondt (Jefferson's method by another name) is biased toward larger parties — useful if you want stable governments. Sainte-Laguë (Webster's method) treats parties more evenly — useful if you want fair representation of small movements. Sweden uses a modified Sainte-Laguë specifically to suppress the smallest parties, requiring a 4% threshold before they get any seats at all.

Each system encodes a value judgment: do you want stable two-party rule (D'Hondt), proportional reflection of opinion (Sainte-Laguë), or hybrid systems that try to balance both (Germany's mixed-member proportional)?

Voting power is not the same as votes

A separate strangeness shows up in coalition voting. Suppose three parties hold 40, 40, and 20 seats in a parliament that needs 51 to pass a law. Naively, the small party has 20% of the seats and 20% of the power. Mathematically, that is wrong.

The Banzhaf power index counts how often a party's vote is decisive — meaning, how often switching their vote changes the outcome. In the 40-40-20 example, every coalition that passes requires any two parties. Each party is a swing voter equally often. All three have identical Banzhaf power, despite holding very different seat counts.

Reverse the example: 49-48-3. The 3-seat party now holds the same Banzhaf power as the 49-seat party — both are necessary in some coalitions and unnecessary in others. The vote share differs by a factor of 16; the actual power is identical. This is why kingmaker minor parties in coalition governments wield influence far beyond their seat counts.

The shadow of arithmetic on democratic legitimacy

Mathematics rarely intrudes on politics this directly, but apportionment is one of the few places where it does. The 2020 US Census reapportionment shifted seats from California, New York, and Michigan to Texas, Florida, and Colorado — partly because of population shifts, partly because of the precise rounding rule used. Different rounding rules would have produced different shifts. None of the rules are objectively correct. All of them are defensible.

This is uncomfortable. We expect mathematics to provide certainty: the right answer, knowable in principle. Apportionment is a domain where mathematics provides the opposite — it tells us, with proof, that no right answer exists. We must choose which kind of unfairness to accept.

Why this matters beyond apportionment

The Balinski–Young result is part of a larger family of impossibility theorems — Arrow's Impossibility, the Gibbard–Satterthwaite theorem on strategy-proof voting, Sen's liberal paradox. They share a structure: democratic intuitions that feel obviously compatible turn out to be mathematically inconsistent. Fairness, monotonicity, independence of irrelevant alternatives — pick any two.

What this teaches, in the end, is humility. The voting and apportionment systems we have are not optimal solutions to a clean problem. They are negotiated trade-offs among incompatible goals. Knowing which trade-offs your system has made is not academic — it is the difference between accepting a paradox you understand and being surprised by one you didn't see coming.