On a parametric divisor method for the apportionment problem

Apportionment problem has been focused upon for more than 200 years by many applied mathematicians and operations researchers. Balinski and Young have done quite extensive works for this problem. This paper proposes a parametric divisor method for solving an apportionment problem. The present method is shown to ‘cover’ most traditional apportionment methods. First, it introduces the idea of stable regions for the allocation of seats to each political constituency, then shows the explicit relation between apportionment methods and corresponding stable regions. Then the paper looks at these apportionment methods from the viewpoints of constrained optimization problem, and it shows the corresponding optimization problem for the present parametric divisor method. Finally using Japan’s House of Representative data, the paper shows the numerical results for the application of various apportionment methods. It concludes by suggesting appropriate parameter values for our parametric apportionment method.