Highest averages method

A highest-averages method, also called a divisor method, is a class of methods for allocating seats in a parliament among agents such as political parties or federal states. A divisor method is an iterative method: at each iteration, the number of votes of each party is divided by its divisor, which is a function of the number of seats (initially 0) currently allocated to that party. The next seat is allocated to the party whose resulting ratio is largest. The inputs to a divisor method are the number of seats to allocate, denoted by h, and the vector of parties' entitlements, where the entitlement of party is denoted by (a number between 0 and 1 determining the fraction of seats to which is entitled). Assuming all votes are counted, is simply the number of votes received by , divided by the total number of votes. A divisor method is parametrized by a function , mapping each integer to a real number (usually in the range ) . The number of seats allocated to party is denoted by . Initially, is set to 0 for all parties. Then, at each iteration, the next seat is allocated to a party which maximizes the ratio . The method proceeds for h iterations, until all seats are allocated. An equivalent definition directly gives the outcome of the divisor method as follows. For an election, a quotient is calculated, usually the total number of votes divided by the number of seats to be allocated (the Hare quota). Parties are then allocated seats by determining how many quotients they have won, by dividing their vote totals by the quotient. Where a party wins a fraction of a quotient, this can be rounded down or rounded to the nearest whole number. Rounding down is equivalent to using the D'Hondt method, while rounding to the nearest whole number is equivalent to the Sainte-Laguë method. Rounding up is equivalent to using Adams' method. However, because of the rounding, this will not necessarily result in the desired number of seats being filled. In that case, the quotient may be adjusted up or down until the number of seats after rounding is equal to the desired number.