Vote-ratio monotonicity

"Population-ratio monotonicity" redirects here and is not to be confused with Population 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 ${\displaystyle A}$ 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. 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}

Population-pair monotonicity

Pairwise monotonicity says that if the ratio 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 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}

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.^{[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.

Relation to other properties

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 ${\displaystyle i}$ to state ${\displaystyle j}$ while the populations of other states do not change, then both ${\displaystyle a_{i}'\geq a_{i}}$ and ${\displaystyle a_{j}'\leq a_{j}}$ must hold.^{[8]: Sub.9.9}

See also

• Apportionment paradox
• Seats-to-votes ratio