SCIENCE CASTLE2017

Presentation information

関西大会

[WS4] ポスター発表 奇数

Sat. Dec 23, 2017 2:00 PM - 2:50 PM ポスターエリア (B1F第二体育館)

2:00 PM - 2:50 PM

[P-25] Democracy and Mathematics

住吉 レオナルド弘至, 爲政 真由 (AICJ中学・高等学校 Windows Of Mathematics (WOM))

Keywords:Democracy(民主主義)、Preference(選好)、Social welfare function(社会厚生関数)

Democracy in the society and mathematics seems not to have any relationship. However, because of social choice theory and some other theorems, there was a movement to capture democracy mathematically.

Kenneth Joseph Arrow described society as a set including more than 3 people, and democracy as a way of thinking that individual preference of more than 3 choices are fairly reflected to the preference of society, which is the social welfare function. Fairly reflected means to fill up the following 4 conditions.



1)Unrestricted domain: All the combinations of preference possible have to be

considered as a domain.

2)Pareto principle: If the preference of all the members of society is the same,

it will be considered as the social welfare function.

3)Binary independence: The choice in social preference is only affected by the choice

of each individual preference.

4)Non-dictatorship: There must not exist a dictator ― a person whose

individual preference is always reflected to the social preference.



Arrow demonstrated the impossibility of constructing the social welfare function satisfying all 4 conditions. This is called Arrow’s paradox (or Arrow's impossibility theorem).

We have gone through the proof by thinking of 3 people, A, B and C, with 3 choices, x, y and z. Here, we would try to find an example of a violation regarding the 4 conditions. Indeed, there exists someone’s individual preference wholly effects the social welfare function, and this violates the non-dictatorship.