The Balinski-Young impossibility theorem states that, for four or more states, no apportionment method can always satisfy the quota rule while also being free of the population paradox. Thus the quota rule and population monotonicity, although each is a natural requirement for an apportionment method, are incompatible in general.
Balinski-Young Impossibility Theorem
See also
Alabama Paradox, Apportionment Method, New-States Paradox, Population ParadoxExplore with Wolfram|Alpha
References
Balinski, M. L. and Young, H. P. "Apportionment Schemes and the Quota Method." Amer. Math. Monthly 84, 450-455, 1977. https://doi.org/10.1080/00029890.1977.11994382.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. "Balinski-Young Impossibility Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Balinski-YoungImpossibilityTheorem.html