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Vote-ratio,^{[1]: Sub.9.6}weight-ratio,^[2] orpopulation-ratio monotonicity^[3]: Sec.4 is a property of someapportionment methods. It says that if theentitlement for${\displaystyle A}$ grows at a faster rate than${\displaystyle B}$ (i.e.${\displaystyle A}$ growsproportionally 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. An apportionment method violating this rule may encounterpopulation paradoxes.

A particularly severe variant, where votingfor a party causes it tolose seats, is called ano-show paradox. Thelargest remainders method exhibits both population and no-show paradoxes.^{[4]: Sub.9.14}

Pairwise monotonicity says that if theratio between the entitlements of two states${\displaystyle i,j}$ increases, then state${\displaystyle j}$ should not gain seats at the expense of state${\displaystyle i}$. In other words, a shrinking state should not "steal" a seat from a growing state.

Some earlier apportionment rules, such asHamilton's method, do not satisfy VRM, and thus exhibit the population paradox. For example, after the 1900 census,Virginia lost a seat toMaine, even though Virginia's population was growing more rapidly.^{[5]: 231–232}

A stronger variant of population monotonicity, calledstrong monotonicity requires that, if a state'sentitlement (share of the population) increases, then its apportionment should not decrease, regardless of what happens to any other state's entitlement. This variant is extremely strong, however: whenever there are at least 3 states, and the house size is not exactly equal to the number of states, no apportionment method is strongly monotone for a fixed house size.^{[6]: Thm.4.1} Strong monotonicity failures in divisor methods happen when one state's entitlement increases, causing it to "steal" a seat from another state whose entitlement is unchanged.

However, it is worth noting that the traditional form of the divisor method, which involves using afixed divisor and allowing the house size to vary, satisfies strong monotonicity in this sense.

Balinski andYoung proved that an apportionment method is VRM if-and-only-if it is adivisor method.^{[7]: Thm.4.3}

Palomares, Pukelsheim and Ramirez proved that very apportionment rule that isanonymous,balanced,concordant,homogenous, andcoherent is vote-ratio monotone.^{[citation needed]}

Vote-ratio monotonicity implies that, if population moves from state${\displaystyle i}$ to state${\displaystyle j}$ while the populations of other states do not change, then both${\displaystyle a_i^{\prime }\ge a_i}$ and${\displaystyle a_j^{\prime }\le a_j}$ must hold.^{[8]: Sub.9.9}

• Apportionment paradox
• Seats-to-votes ratio

• Apportionment method criteria

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