Vote-ratio monotonicity[1]: Sub.9.6 (VRM) is a property of apportionment methods, which are methods of allocating seats in a parliament among political parties. The property says that, if the ratio between the number of votes won by party A to the number of votes won by party B increases, then it should NOT happen that party A loses a seat while party B gains a seat.
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The property was first presented in the context of apportionment of seats in a parliament among federal states. In this context, it is called population monotonicity.[2]: Sec.4 The property says that, if the population of state A increases faster than that of state B, then state A should not lose a seat while state B gains a seat. An apportionment method that fails to satisfy this property is said to have a population paradox. Note that the term population monotonicity is more commonly used to denote a very different property of resource-allocation rules. Therefore, the term "vote-ratio monotonicity" is sometimes used instead.
There is a resource to allocate, denoted by . For example, it can be an integer representing the number of seats in a house of representatives. The resource should be allocated between some agents, such as states or parties. The agents have different entitlements, denoted by a vector . For example, ti can be the fraction of votes won by party i. An allocation is a vector with . An allocation rule is a rule that, for any and entitlement vector , returns an allocation vector .
To define vote-ratio monotonicity, denote and . An allocation rule M is called vote-ratio monotone[3] if the following holds:
- If , then either or or both (note that the apportionments of both states may decrease, or both may increase, but it is not allowed that the apportionment of decreases and simultaneously the apportionment of increases).
The original definition of population monotonicity by Balinski and Young has an additional condition:[2]: Sec.4
- If , then either , or , or .
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.[4]: 231–232
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 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.