Democracy is Mathematically Impossible

These are mostly stuff I found interesting from this video: https://youtu.be/qf7ws2DF-zk?si=zSchFLfLce09egCg

• Most countries use a First-Past-The-Post type of democracy (first to 50%), which is flawed.
• Also, candidates with similar goals and ideologies end up stealing votes from each other and the worse candidate would win.
• Another flaw is that the government could be formed by candidates that won because of population differences in states. A candidate from a state of large population with more votes could be overshadowed by candidates who won from smaller states.
• One more issue is that there could be no clear winners if no one reaches 50%.
• To get a clear winner, the candidate with lowest vote is eliminated and the election should be tried again. This would be mathematically similar to having a preferential style voting, where people rank candidates as first, second, third preferences.
• Minneapolis Mayoral Election in 2013 did this, and surprisingly, all of the 30+ candidates behaved very friendly with each other during the debates, just to get on the preference of the voters of the other candidates.
• I am not going to remember the names, but some French mathematicians tried to rework the whole structure, and came up with different ideas. In one, all the candidates are again ranked, but each of them are compared against each other candidate to see who the people preferred between two candidates.
• This was also proposed 450 years earlier by a monk who was looking at how Church leaders were chosen. The book was lost and rediscovered in 2001.
• The issue with this method is that there could be transitive errors. The paradox is named after the scientist. Say three people chose A>B>C and B>C>A and C>A>B. There are no clear preferences here.
• The French mathematician was later executed during Reign of Terror and died before this could be solved.

The Impossibility Theorem

Kenneth Arrow stated five rules for a rational election. He then proved no election could satisfy all these conditions. He won the Nobel for this.

1. Non-dictatorship: No single voter should have the power to determine the outcome regardless of others' preferences.
2. Unrestricted Domain: Voters should be able to rank all possible alternatives in any order.
3. Pareto Efficiency: If every voter prefers one option over another, the group preference should reflect the same.
4. Independence of Irrelevant Alternatives: The group's preference between any two options should not be affected by the introduction or removal of a third option.
5. Transitivity (or Non-Cyclic Collective Preferences): If option A is preferred over B, and B over C, then A should be preferred over C.

Arrow's theorem shows that no voting system can perfectly satisfy all these conditions simultaneously when there are three or more alternatives.