Mathematics of apportionment
Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results In mathematics and fair division, apportionment problems involve dividing (apportioning) a whole number of identical goods fairly across several parties with real-valued entitlements.
The original, and best-known, example of an apportionment problem involves distributing seats in a legislature between different federal states or political parties.
[1] However, apportionment methods can be applied to other situations as well, including bankruptcy problems,[2] inheritance law (e.g. dividing animals),[3][4] manpower planning (e.g. demographic quotas),[5] and rounding percentages.
Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes, as proven by the Balinski–Young theorem.
The mathematical theory of apportionment was studied as early as 1907 by the mathematician Agner Krarup Erlang.
[citation needed] It was later developed to a great detail by the mathematician Michel Balinski and the economist Peyton Young.
[7] The inputs to an apportionment method are: The output is a vector of integers
This is required, since in some cases there is no fair way to distinguish between two possible solutions.
While such ties are extremely rare in practice, the theory must account for them (in practice, when an apportionment method returns multiple outputs, one of them may be chosen by some external priority rules, or by coin flipping, but this is beyond the scope of the mathematical apportionment theory).
When the agents are political parties, these numbers are usually 0, so this vector is omitted.
But when the agents are states or districts, these numbers are often positive in order to ensure that all are represented.
They can be the same for all agents (e.g. 1 for USA states, 2 for France districts), or different (e.g. in Canada or the European parliament).
[7] The proportionality of apportionment can be measured by seats-to-votes ratio and Gallagher index.
The uniqueness of Jefferson and Adams holds even in the much larger class of rank-index methods.
This can be seen as a disadvantage of divisor methods, but it can also be considered a disadvantage of the quota criterion:[7]: 129 "For example, to give D 26 instead of 25 seats in Table 10.1 would mean taking a seat from one of the smaller states A, B, or C. Such a transfer would penalize the per capita representation of the small state much more - in both absolute and relative terms - than state D is penalized by getting one less than its lower quota.
Similar examples can be invented in which some state might reasonably get more than its upper quota.
"In Monte-Carlo simulations, Webster's method satisfies both quotas with a very high probability.
One way to evaluate apportionment methods is by whether they minimize the amount of inequality between pairs of agents.
, then there are instances in which, for any allocation, moving a seat from one agent to another might decrease their pairwise inequality.
State-population monotonicity means that, if the entitlement of an agent increases, its apportionment should not decrease.
The name comes from the setting where the agents are federal states, whose entitlements are determined by their population.
One variant - the pairwise PM - is satisfied exclusively by divisor methods.
[7]: Thm.6.1 Combined with the previous statements, it implies that no divisor method satisfies both quotas.
House monotonicity means that, when the total number of seats
It was considered particularly important in the early days of the USA, when the congress size increased every ten years.
may violate pairwise-PM: there are examples in which one agent gains in population but loses seats.
[7]: Sec.7 Uniformity (also called coherence[17]) means that, if we take some subset of the agents
[7]: Thm.8.4 Moreover, among all anonymous methods:[12] When the agents are political parties, they often split or merge.
Moreover, every method satisfying both quotas is "almost coalitionally-stable" - it gives every coalition between
[12] The following table summarizes uniqueness results for classes of apportionment methods.
