Vote-ratio monotonicity explained

Vote-ratio, weight-ratio,^[1] or population-ratio monotonicity^[2] is a property of some apportionment methods. It says that if the entitlement for

A grows at a faster rate than B (i.e. A grows proportionally more than B ), A should not lose a seat to B . More formally, if the ratio of votes or populations A / B increases, then A should not lose a seat while 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.

Population-pair monotonicity

Pairwise monotonicity says that if the ratio between the entitlements of two states

i, j increases, then state j should not gain seats at the expense of state 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.^[3]

Strong monotonicity

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.^[4] 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.^[5]

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

i to state j while the populations of other states do not change, then both a _{i'} \geq a _{i} and a _{j'} \leq a _{j} must hold.

See also

• Apportionment paradox
• Seats-to-votes ratio

References

1. Chakraborty . Mithun . Schmidt-Kraepelin . Ulrike . Suksompong . Warut . 2021-04-29 . Picking sequences and monotonicity in weighted fair division . Artificial Intelligence . 301 . 103578 . 2104.14347 . 10.1016/j.artint.2021.103578 . 233443832.
2. Book: Balinski . Michel L. . Fair Representation: Meeting the Ideal of One Man, One Vote . Young . H. Peyton . Yale University Press . 1982 . 0-300-02724-9 . New Haven . registration.
3. Book: Stein, James D. . How Math Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics . Smithsonian Books . 2008 . 9780061241765 . New York.
4. Book: Balinski . Michel L. . Fair Representation: Meeting the Ideal of One Man, One Vote . Young . H. Peyton . Yale University Press . 1982 . 0-300-02724-9 . New Haven . registration.
5. Book: Balinski . Michel L. . Fair Representation: Meeting the Ideal of One Man, One Vote . Young . H. Peyton . Yale University Press . 1982 . 0-300-02724-9 . New Haven . registration.