State-population monotonicity

Entitlement-ratio, weight-ratio, vote-ratio, or population-ratio monotonicity is a property of apportionment methods. It says that if the entitlement for A increases proportionally to that of B, then A should not lose any seats to B. Apportionments violating this rule are called population paradoxes; a particularly severe variant, where voting for a party (or moving into a state) causes it to lose seats, is called a no-show paradox.

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. This property is also called vote-ratio monotonicity.

Weak monotonicity

Voter monotonicity or participation criterion is a property weaker than pairwise-PM. It says that, if party i attracts more voters, while all other parties keep the same number of voters, then party i must not lose a seat. Failure of voter monotonicity is called the no-show paradox, since a voter can help their party by not voting. The largest remainders method fails voter monotonicity.

Maskin monotonicity

A stronger variant of population monotonicity, calle 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. 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.