State-population monotonicity

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 another states, then it should not lose a seat to that state. Apportionments violating this rule are called population paradoxes.

In the apportionment literature, this property can sometimes simply be called population monotonicity. However, the term "population monotonicity" is more commonly however used to denote a very different property of resource-allocation rules within that realm. Specifically, as it relates to the concept of population monotonicity, the term "population" refers 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. Conversely, in the domain of legislative seat apportionment, the term "population" refers 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: 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 $i,j$ 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

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.

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. 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, also called concordance, says that a state with a larger population should not receive a smaller allocation. Formally, if $t_{i}>t_{j}$ then $a_{i}\geq a_{j}$.

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.