Vote-ratio monotonicity

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 ${\displaystyle A}${\displaystyle A} grows at a faster rate than ${\displaystyle B}${\displaystyle B} (i.e. ${\displaystyle A}${\displaystyle A} grows proportionally more than ${\displaystyle B}${\displaystyle B}), ${\displaystyle A}${\displaystyle A} should not lose a seat to ${\displaystyle B}${\displaystyle B}.^{[1] : Sub.9.6} More formally, if the ratio of votes or populations ${\displaystyle A/B}${\displaystyle A/B} increases, then ${\displaystyle A}${\displaystyle A} should not lose a seat while ${\displaystyle B}${\displaystyle B} gains a seat. An apportionment method violating this rule may encounter 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}

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

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

A stronger variant of population monotonicity, called strong monotonicity requires that, if a state's entitlement (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 a fixed divisor and allowing the house size to vary, satisfies strong monotonicity in this sense.

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

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

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

• Apportionment paradox
• Seats-to-votes ratio

• Apportionment method criteria

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