Arrow's Impossibility Theorem
Arrow's Impossibility Theorem states that it is impossible to create a voting system that satisfies all conditions. This theorem was mathematically proven by Arrow.
As a premise, in order to have a fair majority vote, the following conditions are necessary:
1. There should be no manipulation of the results by any individual (Non-dictatorship).
2. If everyone prefers option A, then the collective preference should also be A (Unanimity).
3. When option A is preferred over option B, the addition of another option C should not change the preference to B over A (Independence from Irrelevant Alternatives).
4. If the options are A and B, then individual opinions should be either A>B or B>A. Additionally, if there are options ABC and A>B or B>C, then it should also be A>C (Completeness).
However, Arrow's theorem proves that it is impossible to create a voting system that satisfies all these conditions.
One famous voting paradox is Condorcet's paradox. As a premise, let's consider options A, B, and C with three voters. Each voter intends to vote as follows:
- Voter A: A>B>C
- Voter B: B>C>A
- Voter C: C>A>B
When using the majority rule, all options (A, B, and C) are tied for first place. If we compare the options in pairs (A vs C), we see that A>C, but for the other pairs, C>A. This creates a Circulation problem when applying the pairwise comparison method.
Arrow's theorem does not deny democracy, but it shows that it is impossible to create a voting system that satisfies all the conditions. As an alternative, the Borda Rule is often suggested. The Borda Rule involves assigning points to each option and voting based on the total points received. You can learn more about the merits and demerits of the Borda Rule here(https://kitaguni-economics.com/bordarule-meritanddemerit/).
According to the theorem, regardless of the preference voting system, if there are three or more independent choices for voters, it is impossible to simultaneously meet specific evaluation criteria when converting individual preferences into the overall ranking. The specific evaluation criteria include Non-universality of the domain, Non-dictatorship, Pareto Efficiency, and Independence from Irrelevant Alternatives.
For more information, you can visit the Wikipedia page on Arrow's Impossibility Theorem(https://ja.wikipedia.org/wiki/アローの不可能性定理).