Vote-ratio Monotonicity
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Vote-ratio monotonicity (VRM) is a property of apportionment methods, which are methods of allocating seats in a
parliament In modern politics, and history, a parliament is a legislative body of government. Generally, a modern parliament has three functions: representing the electorate, making laws, and overseeing the government via hearings and inquiries. Th ...
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 In modern politics, and history, a parliament is a legislative body of government. Generally, a modern parliament has three functions: representing the electorate, making laws, and overseeing the government via hearings and inquiries. Th ...
among
federal states A federation (also known as a federal state) is a political entity characterized by a union of partially self-governing provinces, states, or other regions under a central federal government (federalism). In a federation, the self-governi ...
. In this context, it is called population monotonicity or population-pair monotonicity. 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 the term "population monotonicity" is more commonly used to denote a very different property of resource-allocation rules; see
population monotonicity Population monotonicity (PM) is a principle of consistency in allocation problems. It says that, when the set of agents participating in the allocation changes, the utility of all agents should change in the same direction. For example, if the resou ...
. Therefore, we prefer to use here the term "vote-ratio monotonicity", which is unambiguous.


Definitions

There is a ''resource'' to allocate, denoted by h. For example, it can be an integer representing the number of seats in a ''h''ouse of representatives. The resource should be allocated between some n ''agents'', such as states or
parties A party is a gathering of people who have been invited by a host for the purposes of socializing, conversation, recreation, or as part of a festival or other commemoration or celebration of a special occasion. A party will often feature ...
. The agents have different ''entitlements'', denoted by a vector t_1,\ldots,t_n. For example, ''ti'' can be the fraction of votes won by party ''i''. An ''allocation'' is a vector a_1,\ldots,a_n with \sum_^n a_i = h. An ''allocation rule'' is a rule that, for any h and entitlement vector t_1,\ldots,t_n, returns an allocation vector a_1,\ldots,a_n. To define vote-ratio monotonicity, denote M(\mathbf; h) = \mathbf and M(\mathbf; h) = \mathbf . An allocation rule ''M'' is called vote-ratio monotone if the following holds: * If \frac > \frac, then either a_i' \geq a_i or 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 i decreases and simultaneously the apportionment of j increases). The original definition of population monotonicity by Balinski and
Young Young may refer to: * Offspring, the product of reproduction of a new organism produced by one or more parents * Youth, the time of life when one is young, often meaning the time between childhood and adulthood Music * The Young, an American roc ...
has an additional condition: * If \frac = \frac, then either a_i' \geq a_i, or a_j' \leq a_j, or a_i' +a_j' = a_i + a_j.


Population paradox

Some of the earlier Congressional apportionment methods, such as
Hamilton's Buck Meadows (formerly Hamilton's and Hamilton's Station) is a census-designated place in Mariposa County, California, United States. It is located east-northeast of Smith Peak, at an elevation of . The population was 21 at the 2020 census. Buck ...
, did not satisfy VRM, and thus could exhibit the population paradox. For example, after the 1900 census,
Virginia Virginia, officially the Commonwealth of Virginia, is a state in the Mid-Atlantic and Southeastern regions of the United States, between the Atlantic Coast and the Appalachian Mountains. The geography and climate of the Commonwealth ar ...
lost a seat to
Maine Maine () is a state in the New England and Northeastern regions of the United States. It borders New Hampshire to the west, the Gulf of Maine to the southeast, and the Canadian provinces of New Brunswick and Quebec to the northeast and ...
, even though Virginia's population was growing more rapidly. See here for a simple numeric example of a this paradox.


Relation to other properties

Balinski and
Young Young may refer to: * Offspring, the product of reproduction of a new organism produced by one or more parents * Youth, the time of life when one is young, often meaning the time between childhood and adulthood Music * The Young, an American roc ...
proved the following theorems (note that they call the VRM property "population monotonicity"): * If 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 A highest-averages method, also called a divisor method, is a class of methods for allocating seats in a parliament among agents such as political parties or federal states. A divisor method is an iterative method: at each iteration, the number ...
. * An apportionment method is VRM if-and-only-if it is a
divisor method A highest-averages method, also called a divisor method, is a class of methods for allocating seats in a parliament among agents such as political parties or federal states. A divisor method is an iterative method: at each iteration, the number ...
. Palomares, Pukelsheim and Ramirez prove the following theorem: * Every apportionment rule that is anonymous,
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ci ...
, concordant, decent and
coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deri ...
is vote-ratio monotone. Vote-ratio monotonicity implies that, if population moves from state i to state j while the populations of other states do not change, then both a_i' \geq a_i and a_j' \leq a_j must hold.{{Rp, Sub.9.9


See also

*
Apportionment paradox An apportionment paradox exists when the rules for apportionment in a political system produce results which are unexpected or seem to violate common sense. To apportion is to divide into parts according to some rule, the rule typically being one ...
* Seats-to-votes ratio


References

Apportionment method criteria