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Apportionment Method

An apportionment method is a rule that selects an apportionment for each choice of positive populations p_i and house size h. If P is the total population and a_i is the number of seats assigned to state i, then the exact quota of state i is q_i=hp_i/P. A method satisfies the quota rule when

|_q_i_|<=a_i<=q_i.

for every state.

Natural requirements on apportionment methods include the quota rule, house monotonicity, and population monotonicity. These conditions cannot all be guaranteed simultaneously. In particular, the Balinski-Young impossibility theorem makes the quota rule and population monotonicity incompatible.

Failures of the natural monotonicity requirements give the Alabama paradox, population paradox, and new-states paradox.

See also

Alabama Paradox, Balinski-Young Impossibility Theorem, House Monotonicity, New-States Paradox, Population Paradox

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References

Balinski, M. L. and Young, H. P. "The Quota Method of Apportionment." Amer. Math. Monthly 82, 701-730, 1975. https://doi.org/10.1080/00029890.1975.11993911.Balinski, M. L. and Young, H. P. Fair Representation: Meeting the Ideal of One Man, One Vote, 2nd ed. Washington, DC: Brookings Institution Press, 2001.Bogomolny, A. "The Constitution and Paradoxes." https://cut-the-knot.org/ctk/Democracy.shtml.

Cite this as:

Weisstein, Eric W. "Apportionment Method." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ApportionmentMethod.html

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