Vote-ratio monotonicity

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results Vote-ratio,[1]: Sub.9.6  weight-ratio,[2] or population-ratio monotonicity[3]: Sec.4  is a property of some apportionment methods.

[1]: Sub.9.6  More formally, if the ratio of votes or populations

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

Some earlier apportionment rules, such as Hamilton's method, do not satisfy VRM, and thus exhibit the population paradox.

[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.

[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