There are ten seats around the table. Three groups must share them in proportion to their size. You work out a rule, check the arithmetic and distribute every seat. Then somebody brings an eleventh chair. Surely no group can be worse off.
Under the largest-remainder method, one can. The surprise is called the Alabama paradox. It is not a story about a population shrinking, a vote being removed or an accountant losing a chair. The population weights stay fixed while the total allocation grows.
Build the smallest useful scene
Use three fictional groups with weights 6, 6 and 2. Their total is 14. With ten seats, their exact proportional quotas are 60/14, 60/14 and 20/14: approximately 4.286, 4.286 and 1.429.
The rule first gives everyone the whole-number part: four seats, four seats and one seat. Nine seats have been assigned. The final seat goes to the largest fractional remainder. Group C’s remainder is 6/14; A and B each have 4/14. C receives the last seat, so the final distribution is 4, 4, 2.
This is the Hamilton, or largest-remainder, approach described in the Census Bureau’s methods account. The numbers here are an original compact illustration, not historic census populations.
Now add the chair
For eleven seats the quotas become 66/14, 66/14 and 22/14: about 4.714, 4.714 and 1.571. The whole-number allocations are again four, four and one. But two seats remain. A and B each have a remainder of 10/14, ahead of C’s 8/14. They receive those seats.
The new distribution is 5, 5, 1. C’s exact quota increased from about 1.429 to about 1.571, yet its actual seat count fell from two to one. Everyone’s quota grew; the ordering of the remainders changed. No arithmetic error is required.
You can audit this result without decimals. At ten seats, compare integer remainders 4, 4 and 6 over the common denominator 14. At eleven, compare 10, 10 and 8. The ordering has genuinely changed. Rounding a displayed quota to three decimal places cannot be blamed for the lost chair.
Add one seat; watch the remainders
A largest-remainder allocation for weights 6,6,2 shows the Alabama paradox between ten and eleven seats.
Use the control to move from ten to eleven, then from eleven to twelve. Watch the fractional parts rather than only the quotas. A remainder climbs until its quota crosses an integer, where the remainder resets. The competition is continually reorganized by those crossings.
A fair-looking rule can miss a different kind of fairness
Largest remainder has an immediately attractive feature: each group receives either the floor or the ceiling of its exact quota. Our C receives its ceiling at ten seats and its floor at eleven. Both allocations respect that quota condition.
The condition that nobody should lose when the total grows is different. It is usually called house monotonicity. The Census Bureau’s historical perspective identifies the Alabama paradox as a difficulty of the Hamilton/Vinton method. The name marks an episode in apportionment history; the mechanism applies more generally wherever whole units are shared proportionally by this rule.
Distinguishing criteria is more useful than declaring the calculation “unfair” and stopping. Would your group rather preserve each exact quota’s neighboring integers, prevent losses when the pool grows, or minimize some measure of population per seat? Those are different questions. A rule should be judged against the properties it was actually chosen to deliver.
Try allocating something besides seats
Imagine three departments sharing ten identical display panels in proportion to request weights 6, 6 and 2. Write the 4–4–2 allocation. Then announce that eleven panels are available and recompute the same rule. The department with two panels now receives one.
If the panels have already been installed, that rule creates a practical problem. You might instead decide that additional panels can only be assigned, never taken back. That is a change to the allocation procedure. State it before using the rule, because a sequence of additions need not match a fresh largest-remainder allocation of the final total.
Svante Janson and Svante Linusson’s research on the paradox studies its probability under specified assumptions. The existence of such research should not be mistaken for a single universal frequency. Real populations, the chosen total and the model of changing house size all matter.
The eleventh chair has exposed a seam between continuous shares and discrete objects. Fractions can all rise smoothly. Chairs must land somewhere, one at a time. Before trusting a proportional rule, it helps to ask what happens when the table gets a little bigger.
Sources and further exploration: Census Bureau: methods of apportionment; Census Bureau: historical perspective; Janson and Linusson: probability of the Alabama paradox.
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