Vote-ratio monotonicity

There is a resource to allocate, denoted by ${\displaystyle h}$. For example, it can be an integer representing the number of seats in a house of representatives. The resource should be allocated between some ${\displaystyle n}$ agents, such as states or parties. The agents have different entitlements, denoted by a vector ${\displaystyle t_{1},\ldots ,t_{n}}$. For example, t_i can be the fraction of votes won by party i. An allocation is a vector ${\displaystyle a_{1},\ldots ,a_{n}}$ with ${\displaystyle \sum _{i=1}^{n}a_{i}=h}$. An allocation rule is a rule that, for any ${\displaystyle h}$ and entitlement vector ${\displaystyle t_{1},\ldots ,t_{n}}$, returns an allocation vector ${\displaystyle a_{1},\ldots ,a_{n}}$.

To define vote-ratio monotonicity, denote ${\displaystyle M(\mathbf {t} ;h)=\mathbf {a} }$ and ${\displaystyle M(\mathbf {t'} ;h)=\mathbf {a'} }$. An allocation rule M is called vote-ratio monotone[4] if the following holds:

• If ${\displaystyle {\frac {t'_{i}}{t'_{j}}}>{\frac {t_{i}}{t_{j}}}}$, then either ${\displaystyle a_{i}'\geq a_{i}}$ or ${\displaystyle a_{j}'\leq a_{j}}$ or both (note that the apportionments of both states may decrease, or both may increase, but it is not allowed that the apportionment of ${\displaystyle i}$ decreases and simultaneously the apportionment of ${\displaystyle j}$ increases).

The original definition of population monotonicity by Balinski and Young has an additional condition:[2]^: Sec.4

• If ${\displaystyle {\frac {t'_{i}}{t'_{j}}}={\frac {t_{i}}{t_{j}}}}$, then either ${\displaystyle a_{i}'\geq a_{i}}$, or ${\displaystyle a_{j}'\leq a_{j}}$, or ${\displaystyle a_{i}'+a_{j}'=a_{i}+a_{j}}$.

Some of the earlier Congressional apportionment methods, such as Hamilton's, did not satisfy VRM, and thus could exhibit the population paradox. For example, after the 1900 census, Virginia lost a seat to Maine, even though Virginia's population was growing more rapidly.[5]^{: 231–232} See here [3] for a simple numeric example of a this paradox.

Balinski and Young proved the following theorems (note that they call the VRM property "population monotonicity"):

• If ${\displaystyle h\geq n\geq 2,n\neq 3}$, then a partial apportionment method is VRM if-and-only-if it is a partial divisor method.[2]^: Thm.4.2
• An apportionment method is VRM if-and-only-if it is a divisor method.[2]^: Thm.4.3

Palomares, Pukelsheim and Ramirez prove the following theorem:

• Every apportionment rule that is anonymous, balanced, concordant, decent and coherent is vote-ratio monotone.

Vote-ratio monotonicity implies that, if population moves from state ${\displaystyle i}$ to state ${\displaystyle j}$ while the populations of other states do not change, then both ${\displaystyle a_{i}'\geq a_{i}}$ and ${\displaystyle a_{j}'\leq a_{j}}$ must hold.[1]^: Sub.9.9

• Apportionment paradox
• Seats-to-votes ratio