State-population monotonicity is a property of apportionment methods, which are methods of allocating seats in a parliament among federal states or political parties. The property says that, if the population of a state increases faster than that of other states, then it should not lose a seat. Apportionments violating this rule are called population paradoxes.

In the apportionment literature, this property is simply called population monotonicity.^[1]: Sec.4 However, the term "population monotonicity" is more commonly used to denote a very different property of resource-allocation rules:

• In resource allocation, the property relates to the set of agents participating in the division process. A population increase means that the previously-present agents are entitled to fewer items, as there are more mouths to feed. See population monotonicity for more information.
• In apportionment, the property relates to the population of an individual state, which determines the state's entitlement. A population-increase means that a state is entitled to more seats. The parallel property in fair division is called weight monotonicity:^[2] when an agent's entitlement increases, their utility should not decrease.

Population-pair monotonicity

Pairwise monotonicity says that if the ratio between the entitlements of two states ${\displaystyle i,j}$ increases, then state ${\displaystyle i}$ should not receive less seats while state ${\displaystyle j}$ receives more seats. In other words, a shrinking state should not "steal" a seat from a growing state. This property is also called vote-ratio monotonicity; see that page for more information.

Weak monotonicity

Weak monotonicity, also called voter monotonicity, 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-remainder method with the Droop quota fails voter monotonicity.^{[3]: Sub.9.14}

Strong Monotonicity

A stronger variant of population 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 too 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.^{[1]: 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.

Static population-monotonicity ("concordance")

Static population-monotonicity^[4]: 147, also called concordance^[5]: 75, says that a state with a larger population should not receive a smaller allocation. Formally, if ${\displaystyle t_{i}>t_{j))$ then ${\displaystyle a_{i}\geq a_{j))$.

All apportionment methods must be concordant (by definition, to be considered an apportionment method); occassionally this requires using a "tiebreaking" rule, such as assigning ties to the largest state.