Optimal apportionment
Optimal apportionment
- Optimal apportionment** is an approach to [[mathematics-of-apportionment|apportionment]] that is based on mathematical optimization.
In a problem of apportionment, there is a resource to allocate, denoted by <math>h</math>. For example, it can be an integer representing the number of seats in a house of representatives. The resource should be allocated between some <math>n</math> agents. For example, these can be federal states or political parties. The agents have different entitlements, denoted by a vector of fractions <math>t_1,\ldots,t_n</math> with a sum of 1. For example, ti can be the fraction of votes won by party i. The goal is to find an allocation - a vector <math>a_1,\ldots,a_n</math> with <math>\sum_{i=1}^n a_i = h</math>.
The ideal share for agent i is his/her quota, defined as <math>q_i := t_i\cdot h</math>. If it is possible to give each agent his/her quota, then the allocation is maximally fair. However, exact fairness is usually unattainable, since the quotas are not integers and the allocations must be integers. There are various approaches to cope with this difficulty (see mathematics-of-apportionment). The optimization-based approach aims to attain, for eacn instance, an allocation that is "as fair as possible" for this instance. An allocation is "fair" if <math>a_i = q_i</math> for all agents i, that is, each agent's allocation is exactly proportional to his/her entitlement. in this case, we say that the "unfairness" of the allocation is 0. If this equality must be violated, one can define a measure of "total unfairness", and try to minimize it.
Minimizing the sum of unfairness levels The most natural measure is the sum of unfairness levels for individual agents, as in the utilitarian rule: * Minimizing <math>\max_{i=1}^n q_i/a_i</math> leads to Adams's method. * Minimizing <math>\max_{i=1}^n a_i/q_i</math> leads to Jefferson's method. * It is also possible to maximize <math>\min_{i=1}^n (a_i - q_i)</math>, or equivalently, minimize <math>\max_{i=1}^n (q_i - a_i)</math>. This method satisfies both quotas. * The minimax method can be generalized to any chosen priority ordering on the fairness criteria.
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