Seat bias
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 various ways to compute the bias of apportionment methods. When the agents are federal states, it is particularly important to avoid bias between large states and small states. There are several ways to measure this bias formally. Notation. There is a positive integer formula_1 (=house size), representing the total number of seats to  allocate. There is a positive integer formula_2 representing the number of parties to which seats should be allocated. There is a vector of fractions formula_3 with formula_4, representing "entitlements" - formula_5 represents the entitlement of party formula_6, that is, the fraction of seats to which formula_6 is entitled (out of the total of formula_1). 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 formula_9 with formula_10, called an "apportionment" of formula_1, where formula_12 is the number of seats allocated to party "i . An apportionment method is a multi-valued function formula_13, which takes as input a vector of entitlements and a house-size, and returns as output an apportionment of formula_1. Pairwise comparison of methods. We say that an apportionment method formula_15 favors small parties more than" formula_16 if, for every t and "h", and for every formula_17 and formula_18, formula_19 implies either formula_20 or formula_21. If formula_16 and formula_15 are two divisor methods with divisor functions formula_24 and formula_25, and formula_26 whenever formula_27, then formula_15 favors small agents more than formula_16. 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 formula_16 majorizes another method formula_15, if for any house-size and entitlement-vector, formula_13 majorizes formula_33. If formula_16 and formula_15 are two divisor methods with divisor functions formula_24 and formula_25, and formula_26 whenever formula_27, then formula_15 majorizes formula_16 . Therefore, Adams' is majorized by Dean 's, majorized by Hill's, majorized by Webster's, majorized by Jefferson's. The shifted-quota method (largest-remainders method) with quota formula_42 are also ordered by majorization, where methods with smaller "s" are majorized by methods with larger "s". Counting over all house sizes. To measure the bias of a certain apportionment method M, one can check, for each pair of entitlements formula_43, 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 formula_43 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 formula_45 or formula_46. If the number of house-sizes for which formula_45 equals the number of house-sizes for which formula_46, then the method is unbiased. The only unbiased method, by this definition, is Webster's method. Averaging over all entitlement-pairs. One can also check, for each pair of possible allocations formula_49, the set of all entitlement-pairs formula_43 for which the method "M" yields the allocations formula_49 (for formula_52). 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. The unique proportional divisor method for which this probability is always 50% is Webster. 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. Averaging over all entitlement-vectors. 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". For divisor methods. For each divisor method with divisor formula_53, where there is an electoral threshold formula_54: formula_55 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 formula_56. When  the threshold is negligible, the third term can be ignored. Then, the sum of mean biases is: formula_57, when the approximation is valid for formula_58. Since the mean bias favors large parties when formula_59, 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 formula_42, when entitlement vectors are drawn uniformly at random from the standard simplex, formula_61 In particular, Hamilton's  method is the only unbiased one in this family. Simulations. 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.