State-population monotonicity

State-population monotonicity is a property of apportionment methods, which are methods of allocating seats in a parliament among federal states. The property says that, if the population of a state increases faster than that of other states, then it should not lose a seat. An apportionment method that fails to satisfy this property is said to have a population paradox.

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 the "weight" (- entitlement) of an agent increases, his utility should not decrease.

There are several variants of the state-population monotonicity (PM); see mathematics of apportionment for definitions and notation.

Erlang PM

The simplest definition of PM is that, if the population of one state ${\displaystyle i}$ increases and the populations of the other states remain the same (such that the entitlement of ${\displaystyle i}$ increases and the entitlements of the other states decrease), then the apportionment of ${\displaystyle i}$ weakly increases. This notion was proposed by Agner Krarup Erlang in 1907 and studied by Aanund Hylland in 1978.[3]

The problem with this notion is that, in practice, populations of states do not remain the same, they increase simultaneously.

Strong PM

A stronger variant of PM requires that, if the entitlement of state ${\displaystyle i}$ increases (that is: its population divided by the sum of all populations), then the apportionment of ${\displaystyle i}$ weakly increases. This variant is too strong: whenever there are at least 3 states, and the number of seats is not exactly equal to the number of states, no partial apportionment method (= apportionment method for a fixed number of states and seats) is strongly-PM.[1]: Thm.4.1  Proof: Suppose by contradiction that the partial apportionment ${\displaystyle M^{*}}$ is strongly-PM. Consider several cases:

• ${\displaystyle h=1}$ - there is only one seat. Consider two entitlement vectors:
• All entitlements are equal to ${\displaystyle 1/n}$. Then by symmetry, ${\displaystyle M^{*}}$ must return all ${\displaystyle n}$ apportionments in which some state ${\displaystyle i}$ receives 1 and the others receive 0.
• The entitlements of some two states are larger than ${\displaystyle 1/n}$. Then by strong PM both must receive at least 1 seat, but this is impossible.[clarification needed]
• ${\displaystyle 1 - there are fewer seats than states. Consider three entitlement vectors:
• An arbitrary vector with ${\displaystyle t_{1}>\cdots >t_{n}}$, where the entitlements of the first ${\displaystyle h-1}$ states satisfy ${\displaystyle 1/n, and the entitlement of state ${\displaystyle h}$ is less than ${\displaystyle 1/n}$. Denote the apportionment for such a vector by ${\displaystyle a_{1}\geq \cdots \geq a_{n}}$.
• The entitlements of the first ${\displaystyle n-1}$ states are all equal to ${\displaystyle t_{1}+\epsilon }$, where ${\displaystyle \epsilon }$ is sufficiently small such that ${\displaystyle t_{1}+\epsilon <1/(n-1)}$, and the entitlement of state ${\displaystyle n}$ is less than ${\displaystyle 1/n}$. Then by strong PM and symmetry the first ${\displaystyle n-1}$ states must receive at least ${\displaystyle a_{1}}$ seats in any apportionment. This requires at least ${\displaystyle (n-1)a_{1}}$ seats. This is impossible if ${\displaystyle h. Even if ${\displaystyle h=n-1}$, it is possible only if ${\displaystyle a_{1}=1}$. This means that any state with entitlement less than ${\displaystyle 1/(n-1)}$ must get at most 1 seat, and any state with entitlement less than ${\displaystyle 1/n}$ must get 0 seats.[why?]
• [for ${\displaystyle h=n-1}$]: The entitlement of state 1 is ${\displaystyle 1/(n-1)-\epsilon }$ and the entitlements of the other states are ${\displaystyle (1-1/(n-1)+\epsilon )/(n-1)}$, which are smaller than ${\displaystyle 1/n}$. Then, the apportionment of state 1 must be ${\displaystyle a_{1}=1}$ and of the other states 0. But then the sum of apportionments is smaller than the number of seats - a contradiction.
• ${\displaystyle h>n}$ - the usual situation - there are more seats than states.

In the special case in which ${\displaystyle h=n}$, there are strongly-PM rules.[1]: Prop.4.1

Population-pair monotonicity

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. This property is also called vote-ratio monotonicity; see that page for more information.

Voter monotonicity

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 his party by not voting. The largest-remainder method with the Droop quota fails voter monotonicity.[4]: Sub.9.14

Weak PM

Weak PM is a static property: it says that a state with a larger population should not receive a smaller allocation. Formally, if ${\displaystyle t_{i}>t_{j}}$ then ${\displaystyle a_{i}\geq a_{j}}$. This property is also called concordance.

References

1. ^ a b c Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
2. ^ Chakraborty, Mithun; Schmidt-Kraepelin, Ulrike; Suksompong, Warut (2021-04-29). "Picking sequences and monotonicity in weighted fair division". Artificial Intelligence. 301: 103578. arXiv:2104.14347. doi:10.1016/j.artint.2021.103578. S2CID 233443832.
3. ^ A. Hylland (1978), "Allotment methods: procedures for proportional distribution of indivisible entities", Ph.D. thesis, [[Harvard University]].
4. ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Securing System Consistency: Coherence and Paradoxes", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 159–183, doi:10.1007/978-3-319-64707-4_9, ISBN 978-3-319-64707-4, retrieved 2021-09-02