Vote-ratio monotonicity

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Vote-ratio,^{[1]: Sub.9.6} weight-ratio,^[2] or population-ratio monotonicity^[3]: Sec.4 is a property of some apportionment methods. It says that if the entitlement for ${\displaystyle A}$ grows at a faster rate than ${\displaystyle B}$ (i.e. ${\displaystyle A}$ grows proportionally more than ${\displaystyle B}$), ${\displaystyle A}$ should not lose a seat to ${\displaystyle B}$.^{[1]: Sub.9.6} More formally, if the ratio of votes or populations ${\displaystyle A/B}$ increases, then ${\displaystyle A}$ should not lose a seat while ${\displaystyle B}$ gains a seat. Apportionments violating this rule are called population paradoxes.

A particularly severe variant, where voting for a party causes it to lose seats, is called a no-show paradox. The largest remainders method exhibits both population and no-show paradoxes.^{[4]: Sub.9.14}