14:00 〜 14:50
[P-25] Democracy and Mathematics
キーワード：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.
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.