Alabama Paradox, explained.
The Alabama paradox is a loss of allocation for a group when the total number of seats rises under Hamilton’s largest-remainder method, with populations unchanged.
Why it happens
Each larger house size changes fractional quotas. The ranking of their remainders can change, moving the final whole seats between groups even while every underlying quota increases.
Hamilton’s largest-remainder method can reduce a group’s allocation when the total number of seats increases, even with populations unchanged.
Read the result
Compare four and five total seats for the fixed populations. Inspect both quotas and actual allocations. The smallest group’s quota increases while its allocated seat disappears, separating proportional amounts from indivisible assignments.
A worked example
Four seats become five
Populations 5, 3 and 1 yield four-seat quotas about 2.22, 1.33 and 0.44, allocating 2, 1 and 1.
At five seats, quotas are about 2.78, 1.67 and 0.56; the two remainder seats go to A and B, allocating 3, 2 and 0.
Group C’s proportional quota rises but its whole-seat allocation falls, illustrating the Alabama paradox with unchanged populations.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Quotas equal population share times seats. Floors are assigned first; remaining seats go to the largest fractional remainders, with group order resolving exact ties.
Where this idea is useful
A practical use
Allocating indivisible representatives or resources requires rules whose behavior can differ from proportional intuition.
A common misconception
“Proportional quotas guarantee every group’s allocation rises.”
Quotas are fractions; a rounding rule can distribute indivisible seats differently. House-size monotonicity is a separate property.
What this explanation leaves out
- This three-group example isolates house-size monotonicity. It does not imply that every apportionment rule has the same failure.
Does the paradox happen at every increase?
No. It requires particular population shares and house sizes; the experiment should display a verified transition.
What monotonicity or fairness property does your allocation method need to preserve?
Further reading
Explore the original research or the teaching reference behind this experiment.