Mathematical study on the election system

分配问题有着200多年的悠久历史，受到了许多应用数学家和运筹学家的积极挑战。在过去的200年里，虽然分配问题本身还没有完全解决，但人们已经提出了许多分配方法，并研究了它们应该满足的各种性质。无偏性是分配方法的一个重要性质，Balinski和Young等人对此进行了深入的研究。在这项研究中，我们研究了参数除数法的无偏性，并与大多数传统的分配方法进行了比较。首先，我们通过显式地给出相应的局部偏差度量来刻画参数除数法，并推导出关于分配方法无偏性的几个有用的事实。然后用日本众议院的数据进行了数值计算，结果表明，在一定范围内适当调整参数值，从无偏的角度来看，参数因子法优于其他分摊方法。我们讨论了参数除数法在未来被社会接受的可能性，并调查了我们将在下议院选举中引入的由单一议员和比例投票制组成的新选举制度的特点和问题。然后从“公平”和“平等”的角度思考了什么样的选举制度在我国日本是合适和可取的。

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".