Is there a common topic between politics and mathematics?The most common view is that the two areas are very different.However, Mr. Yukio Hatoyama, who was elected prime minister in the previous House of Representatives election, was Japan’s first politician from a science background.After graduating from the Tokyo University, he became a researcher at Operations Research (OR).Among OR, he specialized in mathematical theory of reliability, and I heard that his doctoral thesis at Stanford University was also on the theme of "system reliability analysis".He is still a current member of the OR Society.In an invited lecture at the 50th year anniversary ceremony of the OR Society held two years ago, he enthusiastically said that rigid bureaucracy must be broken and scientific and rational decision making must be carried out under the title of "science politics".Few in the audience would have expected him to be prime minister two years later, but things have changed a lot.It is hoped that Japan will be led in the right direction using the OR concept.

### Parliamentary seat number problem

This foreword has become longer than I expected.What I would like to write here is a purely mathematical problem that needs to be solved in politics: the quorum problem.Since the founding of the United States, the State has adopted a two-chamber system, and the number of members has been set so that the Senate is represented by two senators from each state and the House of representatives is proportional to the population of the state.Although it is simply said that it is “proportional”, the population is a large number of digits, whereas the number of parliamentarians is about one digit, so it cannot be made strictly proportional.Mathematically, the population of each state p_{ i} ， Number of members x_{ i} ， Total number of members N When, the condition Σ x_{ i} = N Under, ratio p_{ i} / x_{ i} A positive integer so that (= population per member) is approximately equal x_{ i} Is the problem.

### Hamilton method

I think a lot of people can come up with the next method.Sum of population P Then the state i Is

q_{ i} = N (p_{ i} / P)

the right to the number of members.Unfortunately, q_{ i} Is not an integer, but for example q_{ i} If = 3.287, first the integer part of 3 is the state. i Give to N If you can still afford q_{ i} It seems fair to allocate one by one in order from the state with the largest decimal part.In fact, this was not a bad method of calculation, and it was used for some time in the United States.It is called “maximum remainder method” or “Hamilton method”.

### Now it is the matter of mathematics.

However, in 1881, as a result of the renovation of the Capitol, the total capacity N A recalculation as the number of members increased resulted in a decrease in the number of members of Alabama.This is called the “Alabama Paradox”.This is unacceptable to the citizens of Alabama.A close examination revealed that this phenomenon occurs quite often.Therefore, a new calculation method was sought for, and mathematicians also participated in the discussion.As a result, the “divisor method” or “Huntington method” was proposed.Huntington was a professor of mathematics at Harvard University at the time who was consulted for solution.First, the number of members x On the other hand x ≤ d (x) ≤ x + 1 Divisor that meets d (x) To define.d (x) as, x Or x +1 , and take the middle x + 1/2, or geometric mean (x (x + 1)) ^{1/2} Is also influential.The algorithm is as follows.

### Division

- each x
_{ i}Is set to the initial value (usually 1, 0 in some cases). - p
_{ i}/ d (x_{ i}) State to maximize i of x_{ i}Increase by 1, Σ The procedure x_{ i}= N Repeat until

The divisor method increases the total number of members by one, so there is no Alabama paradox.But the divisor d (x) The result depends on how you decide. d (x) = x Then, decide the destination of the next seat based on the number of seats already allocated, d (x) = x If it is +1, it is decided based on the result of increasing the number of seats by one, and it is a standard that makes us think that it is true.The remaining two are in between.Column x ， ， (x (x + 1))^{ 1/2} ， ， x + 1/2, x Smaller states tend to have an advantage as they are located before +1.In the United States, after a long controversy, in 1941, President Roosevelt (x (x + 1))^{ 1/2} Was decided to be adopted, and this method is still used today.It is called the “Hill method” after the name of the proposing statistician.

### Proportional representation system

By the way, for the issue of the number of MPs, change the setting and q_{ i} Political party i Number of votes won, x_{ i} Considering the number of elected members of that party, it becomes a problem to determine the number of elected members of each party in the proportional district.There was lively debate from the perspective of this proportional representation system, but the only conclusion is that it is currently in some European countries. d (x) = x + 1 Has been adopted.From the name of the proposed Belgian lawyer, it is called the “d'Hondt method” or the “maximum divisor method”.The proportional representation section in our country also uses the d'Hondt method.On the other hand, some mathematicians x Some argue that + 1/2 is the least biased and fair.

### Results of the House of Representatives election

What happened in the results of the last House of Representatives election?The proportional division of the House of Representatives is divided into 11 blocks, and each is counted independently.Each political party determines the ranking of a single candidate in a proportional representation constituency and a duplicate candidate in a single-seat constituency by its own rules, and allocates the number of winning candidates from the top of the list.Some strange things happened in this election.In the Kinki block, the DPJ's overlapping candidates almost won in single-seat constituencies, resulting in a shortage of candidates, and the party decided to hand over the winners to other parties.Also, in Minna no tou (everyone’s party), the only duplicate candidate could not get 10% of the voting rate in the constituency, so he had to give up his share for the seat. Anyway, let's take a look at the results of Hokkaido Provincial District for the time being.The number of votes won by each party was 8 out of a total of 3,324,803.

Democratic Party of Japan | LDP | The Komeito | Communist Party | Social Democratic | Daichi |

1,348,318 | 805,895 | 354,886 | 241,345 | 113,562 | 433,122 |

They were.In the proportional division problem, in applying the division method x_{ i} The initial value of is set to 0, but d (x) = x ， ， (x (x + 1))^{ 1/2} In the two d (0) = 0 Become p_{ i} / d (x_{ i} ) Cannot be divided.Therefore, I calculated the number of winners for the remaining three candidates.

Democratic Party of Japan | LDP | The Komeito | Communist Party | Social Democratic | Daichi | total | |

Hamilton method | 3 | 2 | 1 | 1 | 0 | 1 | 8 |

x + 1/2 method | 3 | 2 | 1 | 1 | 0 | 1 | 8 |

d’Hondt method | 4 | 2 | 1 | 0 | 0 | 1 | 8 |

They are.In reality, the Democratic Party won four seats under the d'Hondt method, but it is possible to suggest that the Communist Party was underrepresented.I did not bother to check the rest of the districts since Hokkaido case already prove my point, however, when the overall results are calculated, there seems to be a serious difference for small parties such as the Communist Party and the Social Democratic Party.I don't remember that there was a debate when Japan adopted the d'Hondt method, but the unexpectedly big impact on the results may yield some oppositional viewpoints .As I am planning to take up the issue of the number of seats in the "Advanced topics in System Theory" course at school, I intend to have an assignment of calculating all the seats for each proportional representation district.

This subject matter is also related to the "weight of one vote" of the House of Councilors, by the way.However, this essay will become indefinitely longer once I move onto that topic, so let me stop here.It will be my pleasure if I help you realize that politics and mathematics are linked in unexpected ways.

Toshihide Ibaraki