Journal of Applied Probability

The probability of the Alabama paradox

Svante Janson and Svante Linusson

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Hamilton's method is a natural and common method to distribute seats proportionally between states (or parties) in a parliament. In the 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 p1,..., pm of relative sizes of the states. From the formula we deduce a number of consequences. For example, the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1 / e and on average approximately 0.123.

Article information

J. Appl. Probab., Volume 49, Number 3 (2012), 773-794.

First available in Project Euclid: 6 September 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 91B12: Voting theory

Alabama paradox election method apportionment proportional allocation Hamilton's method method of largest remainder


Janson, Svante; Linusson, Svante. The probability of the Alabama paradox. J. Appl. Probab. 49 (2012), no. 3, 773--794. doi:10.1239/jap/1346955333.

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