Vote-ratio monotonicity
Vote-ratio,' weight-ratio, or population-ratio monotonicity is a property of some apportionment methods. It says that if the entitlement for grows at a faster rate than, should not lose a seat to.' More formally, if the ratio of votes or populations increases, then should not lose a seat while 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.
Population-pair monotonicity
Pairwise monotonicity says that if the ratio between the entitlements of two states increases, then state should not gain seats at the expense of state. 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.
Strong monotonicity
A stronger variant of population monotonicity, called strong monotonicity requires that, if a state's [Entitlement (fair division)|entitlement] 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. 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.
Relation to other properties
Balinski and Young proved that an apportionment method is VRM if-and-only-if it is a divisor method.Palomares, Pukelsheim and Ramirez proved that very apportionment rule that is anonymous, balanced, concordant, homogenous, and coherent is vote-ratio monotone.
Vote-ratio monotonicity implies that, if population moves from state to state while the populations of other states do not change, then both and must hold.