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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.
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:
The original definition of population monotonicity by Balinski and Young has an additional condition: [2]: Sec.4
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
Balinski and Young proved the following theorems (note that they call the VRM property "population monotonicity"):
Palomares, Pukelsheim and Ramirez prove the following theorem:
Vote-ratio monotonicity implies that, if population moves from state to state while the populations of other states do not change, then both and must hold. [1]: Sub.9.9
This article needs additional citations for
verification. (December 2021) |
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.
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:
The original definition of population monotonicity by Balinski and Young has an additional condition: [2]: Sec.4
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
Balinski and Young proved the following theorems (note that they call the VRM property "population monotonicity"):
Palomares, Pukelsheim and Ramirez prove the following theorem:
Vote-ratio monotonicity implies that, if population moves from state to state while the populations of other states do not change, then both and must hold. [1]: Sub.9.9