Mathematical study on the election system

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Apportionment problem has a very long history of more than 200 years and has been challenged actively by many applied mathematicians and operations researchers. In the last 200 years many apportionment methods have been proposed and various types of properties, which they should desirably satisfy, have been proposed and investigated although the apportionment problem itself has not been completely solved yet.Unbiasedness is an important property for the apportionment method, and has been studied intensively by Balinski and Young and others. In this research we investigated the unbiasedness of parametric divisor method, comparing with that of most traditional apportionment methods. First we characterized parametric divisor method by explicitly showing the corresponding local measure of bias and deriving several useful facts concerning the unbiasedness of apportionment methods. Then we showed the numerical results using Japan's House of Representatives data and exhibited parametric divisor method was preferable to other apportionment methods from the viewpoint of the unbiasedness by adjusting the parameter value appropriately in the certain range. We discussed the possibility for the parametric divisor method to be accepted by our society in the future.We investigated the characteristics and problems with respect to our new election system consisting of sigle-member and proportional voting systems, which is to be introduced in the next Lower House election. Then we considered what type of election system was appropriate and desirable to be accepted in our country Japan from the viewpoints of "fairness" and "equality".