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Vote-ratio, weight-ratio, or population-ratio monotonicity is a property of some apportionment methods. It says that if the entitlement for ${\displaystyle A}$ grows at a faster rate than ${\displaystyle B}$ (i.e. ${\displaystyle A}$ grows proportionally more than ${\displaystyle B}$), ${\displaystyle A}$ should not lose a seat to ${\displaystyle B}$. More formally, if the ratio of votes or populations ${\displaystyle A/B}$ increases, then ${\displaystyle A}$ should not lose a seat while ${\displaystyle B}$ gains a seat. An apportionment method violating this rule may encounter population paradoxes.

A particularly severe variant, where voting for a party causes it to lose seats, is called a no-show paradox. The largest remainders method exhibits both population and no-show paradoxes.

Population-pair monotonicity

Pairwise monotonicity says that if the ratio between the entitlements of two states ${\displaystyle i,j}$ increases, then state ${\displaystyle j}$ should not gain seats at the expense of state ${\displaystyle i}$. In other words, a shrinking state should not "steal" a seat from a growing state.

Some earlier apportionment rules, such as Hamilton's method, do not satisfy VRM, and thus 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.

Strong monotonicity

A stronger variant of population monotonicity, called strong 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 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.

Relation to other properties

Balinski and Young proved that an apportionment method is VRM if-and-only-if it is a divisor method.

Palomares, Pukelsheim and Ramirez proved that very apportionment rule that is anonymous, balanced, concordant, homogenous, 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.

See also

• Apportionment paradox
• Seats-to-votes ratio

References

1. ^ 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
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. 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.
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
5. Stein, James D. (2008). How Math Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics. New York: Smithsonian Books. ISBN 9780061241765.
6. 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.
7. 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.
8. 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

• Apportionment method criteria