The probability of the Alabama paradox

TL;DR

This paper derives a formula for the asymptotic probability of the Alabama paradox occurring in Hamilton's seat distribution method, revealing bounds and expected values for the number of affected states.

Contribution

It provides a closed-form asymptotic probability formula for the Alabama paradox under certain assumptions, advancing understanding of this phenomenon in apportionment methods.

Findings

Expected number of states affected is asymptotically at most 1/e.

For uniform sizes, the expected affected states converge to about 0.123.

The formula's generalization to all relative sizes remains open.

Abstract

Hamilton's method (also called method of largest remainder) is a natural and common method to distribute seats proportionally between states (or parties) in a parliament. In USA it has been abandoned due to some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries. In this paper we give, under certain assumptions, a closed formula for the asymptotic probability, as the number of seats tends to infinity, that the Alabama paradox occurs given the vector p_1,...,p_m of relative sizes of the states. From the theorem we deduce a number of consequences. For example it is shown that the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1/e. For random (uniformly distributed) relative sizes p_1,...,p_m the expected number of states to suffer from the Alabama paradox converges to slightly more than a third of this, or approximately 0.335/e=0.123, as m tends to infinity. We leave open the generalization of our formula to all possible (in particular rational) p_1,...,p_m.