Seat bias

There is a positive integer ${\displaystyle h}$ (=house size), representing the total number of seats to allocate. There is a positive integer ${\displaystyle n}$ representing the number of parties to which seats should be allocated. There is a vector of fractions ${\displaystyle (t_{1},\ldots ,t_{n})}$ with ${\displaystyle \sum _{i=1}^{n}t_{i}=1}$, representing entitlements - ${\displaystyle t_{i}}$ represents the entitlement of party ${\displaystyle i}$, that is, the fraction of seats to which ${\displaystyle i}$ is entitled (out of the total of ${\displaystyle h}$). This is usually the fraction of votes that this party has won in the elections.

The goal is to find an apportionment method is a vector of integers ${\displaystyle a_{1},\ldots ,a_{n}}$ with ${\displaystyle \sum _{i=1}^{n}a_{i}=h}$, called an apportionment of ${\displaystyle h}$, where ${\displaystyle a_{i}}$ is the number of seats allocated to party i.

An apportionment method is a multi-valued function ${\displaystyle M(\mathbf {t} ,h)}$, which takes as input a vector of entitlements and a house-size, and returns as output an apportionment of ${\displaystyle h}$.

We say that an apportionment method ${\displaystyle M'}$ favors small parties more than ${\displaystyle M}$ if, for every t and h, and for every ${\displaystyle \mathbf {a'} \in M'(\mathbf {t} ,h)}$ and ${\displaystyle \mathbf {a} \in M(\mathbf {t} ,h)}$, ${\displaystyle t_{i}<t_{j}}$ implies either ${\displaystyle a_{i}'\geq a_{i}}$ or ${\displaystyle a_{j}'\leq a_{j}}$.

If ${\displaystyle M}$ and ${\displaystyle M'}$ are two divisor methods with divisor functions ${\displaystyle d}$ and ${\displaystyle d'}$, and ${\displaystyle d'(a)/d'(b)>d(a)/d(b)}$ whenever ${\displaystyle a>b}$, then ${\displaystyle M'}$ favors small agents more than ${\displaystyle M}$.[1]^: Thm.5.1 Therefore, Adams' method favors small parties more than Dean's, more than Hill's, more than Webster's, more than Jefferson's.

This fact can be expressed using the majorization ordering on integer vectors. A vector a seats majorizes another vector b, if for all k, the k largest parties receive in a at least as many seats as they receive in b. An apportionment method ${\displaystyle M}$ majorizes another method ${\displaystyle M'}$, if for any house-size and entitlement-vector, ${\displaystyle M(\mathbf {t} ,h)}$ majorizes ${\displaystyle M'(\mathbf {t} ,h)}$. If ${\displaystyle M}$ and ${\displaystyle M'}$ are two divisor methods with divisor functions ${\displaystyle d}$ and ${\displaystyle d'}$, and ${\displaystyle d'(a)/d'(b)>d(a)/d(b)}$ whenever ${\displaystyle a>b}$, then ${\displaystyle M'}$ majorizes ${\displaystyle M}$ . Therefore, Adams' is majorized by Dean's, majorized by Hill's, majorized by Webster's, majorized by Jefferson's.[2]

The shifted-quota method (largest-remainders method) with quota ${\displaystyle q_{i}=t_{i}\cdot (h+s)}$ are also ordered by majorization, where methods with smaller s are majorized by methods with larger s.[2]

To measure the bias of a certain apportionment method M, one can check, for each pair of entitlements ${\displaystyle t_{1},t_{2}}$, 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 ${\displaystyle t_{1},t_{2}}$ are usually rational numbers, it is sufficient to check the house sizes up to the product of their denominators. For each house size, one can check whether ${\displaystyle a_{1}/t_{1}>a_{2}/t_{2}}$ or ${\displaystyle a_{1}/t_{1}<a_{2}/t_{2}}$. If the number of house-sizes for which ${\displaystyle a_{1}/t_{1}>a_{2}/t_{2}}$ equals the number of house-sizes for which ${\displaystyle a_{1}/t_{1}<a_{2}/t_{2}}$, then the method is unbiased. The only unbiased method, by this definition, is Webster's method.[1]^: Prop.5.2

One can also check, for each pair of possible allocations ${\displaystyle a_{1},a_{2}}$, the set of all entitlement-pairs ${\displaystyle t_{1},t_{2}}$ for which the method M yields the allocations ${\displaystyle a_{1},a_{2}}$ (for ${\displaystyle h=a_{1}+a_{2}}$). 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.

In addition to the theoretic analysis, one can check the actual bias in real problems. Using the USA census data, it was found that the ratio of favoring the larger state over the smaller state for Webster's method is nearest to 50% among all common divisor methods. Many other tests indicate that Webster's method is the least biased.[1]