Seat bias
Seat bias
- Seat bias** is a property describing [[mathematics-of-apportionment|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,
Notation There is a positive integer <math>h</math> (=house size), representing the total number of seats to allocate. There is a positive integer <math>n</math> representing the number of parties to which seats should be allocated. There is a vector of fractions <math>(t_1,\ldots,t_n)</math> with <math>\sum_{i=1}^n t_i = 1</math>, representing *entitlements*, that is, the fraction of seats to which some party <math>i</math> is entitled (out of a total of <math>h</math>). 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 <math>a_1,\ldots,a_n</math> with <math>\sum_{i=1}^n a_i = h</math>, called an apportionment of <math>h</math>, where <math>a_i</math> is the number of seats allocated to party i.
An apportionment method is a multi-valued function <math>M(\mathbf{t},h)</math>, which takes as input a vector of entitlements and a house-size, and returns as output an apportionment of <math>h</math>.
Majorization order We say that an apportionment method <math>M'</math> favors small parties more than <math>M</math> if, for every t and h, and for every <math>\mathbf{a'}\in M'(\mathbf{t},h)</math> and <math>\mathbf{a}\in M(\mathbf{t},h)</math>, <math>t_i < t_j</math> implies either <math>a_i'\geq a_i</math> or <math>a_j'\leq a_j</math>.
If <math>M</math> and <math>M'</math> are two divisor methods with divisor functions <math>d</math> and <math>d'</math>, and <math>d'(a)/d'(b) > d(a)/d(b)</math> whenever <math>a > b</math>, then <math>M'</math> favors small agents more than <math>M</math>.
This fact can be expressed using the majorization ordering on vectors. A vector a 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 <math>M</math> majorizes another method <math>M'</math>, if for any house-size and entitlement-vector, <math>M(\mathbf{t},h)</math> majorizes <math>M'(\mathbf{t},h)</math>. If <math>M</math> and <math>M'</math> are two divisor methods with divisor functions <math>d</math> and <math>d'</math>, and <math>d'(a)/d'(b) > d(a)/d(b)</math> whenever <math>a > b</math>, then <math>M'</math> majorizes <math>M</math>. Therefore, Adams' method is majorized by Dean's, which is majorized by Hill's, which is majorized by Webster's, which is majorized by Jefferson's.
The shifted-quota methods (largest-remainders) with quota <math>q_i = t_i\cdot (h+s)</math> are also ordered by majorization, where methods with smaller s are majorized by methods with larger s.
<math>\text{MeanBias}(r, k, t) = (r-1/2)\cdot\left(\sum_{i=k}^n(1/i) -1\right)\cdot(1-n t)</math>
In particular, Webster's method is the only unbiased one in this family. The formula is applicable when the house size is sufficiently large, particularly, when <math>h\geq 2n</math>. When the threshold is negligible, the third term can be ignored. Then, the sum of mean biases is:
<math>\sum_{k=1}^n \text{MeanBias}(r, k, 0) \approx (r-1/2)\cdot (n/e-1)</math>, when the approximation is valid for <math>n\geq 5</math>.
Since the mean bias favors large parties when <math>r>1/2</math>, there is an incentive for small parties to form party alliances (=coalitions). Such alliances can tip the bias in their favor. The seat-bias formula can be extended to settings with such alliances.
For shifted-quota methods For each shifted-quota method (largest-remainders method) with quota <math>q_i = t_i\cdot (h+s)</math>, when entitlement vectors are drawn uniformly at random from the standard simplex,
<math>\text{MeanBias}(s, k, t) = \frac{s}{n}\cdot\left(\sum_{i=k}^n(1/i) -1\right)\cdot(1-n t)</math>
In particular, Hamilton's method is the only unbiased one in this family.