Seat bias is a property describing methods of apportionment.
These are methods used to allocate seats in a parliament among federal states or among political parties.
A method is biased if it systematically favors small parties over large parties, or vice versa.
There are several mathematical measures of bias, which can disagree slightly, but all measures broadly agree that rules based on Droop's quota or Jefferson's method are strongly biased in favor of large parties, while rules based on Webster's method, Hill's method, or Hare's quota have low levels of bias,[1] with the differences being sufficiently small that different definitions of bias produce different results.
(=house size), representing the total number of seats to allocate.
representing the number of parties to which seats should be allocated.
, representing entitlements, that is, the fraction of seats to which some party
This is usually the fraction of votes the party has won in the elections.
The goal is to find an apportionment method is a vector of integers
is the number of seats allocated to party i.
An apportionment method is a multi-valued function
, which takes as input a vector of entitlements and a house-size, and returns as output an apportionment of
[1]: Thm.5.1 This fact can be expressed using the majorization ordering on vectors.
[3] The shifted-quota methods (largest-remainders) with quota
, the set of all possible apportionments yielded by M, for all possible house sizes.
Theoretically, the number of possible house sizes is infinite, but since
are usually rational numbers, it is sufficient to check the house sizes up to the product of their denominators.
Assuming the entitlements are distributed uniformly at random, one can compute the probability that M favors state 1 vs. the probability that it favors state 2.
For example, the probability that a state receiving 2 seats is favored over a state receiving 4 seats is 75% for Adams, 63.5% for Dean, 57% for Hill, 50% for Webster, and 25% for Jefferson.
[1]: Prop.5.2 The unique proportional divisor method for which this probability is always 50% is Webster.
[1]: Thm.5.2 There are other divisor methods yielding a probability of 50%, but they do not satisfy the criterion of proportionality as defined in the "Basic requirements" section above.
The same result holds if, instead of checking pairs of agents, we check pairs of groups of agents.
[1]: Thm.5.3 One can also check, for each vector of entitlements (each point in the standard simplex), what is the seat bias of the agent with the k-th highest entitlement.
Averaging this number over the entire standard simplex gives a seat bias formula.
The formula is applicable when the house size is sufficiently large, particularly, when
Since the mean bias favors large parties when
The seat-bias formula can be extended to settings with such alliances.
, when entitlement vectors are drawn uniformly at random from the standard simplex,
In particular, Hamilton's method is the only unbiased one in this family.
[4] Using United States census data, Balinski and Young argued Webster's method is the least median-biased estimator for comparing pairs of states, followed closely by the Huntington-Hill method.