Topological Problems in Voting
Back in college, I developed an interest in unexpected impossibility proofs applied to real world systems. The fact that certain abstract mathematical structures inherently have limitations which profoundly impact actual applications is both captivating and sobering. Here’s a cute instance of topological properties applied to voting systems a friend shared with me through this video (we’ll take a slightly different approach).
Background: Arrow’s Theorem
Arrow’s Theorem1 is the most famous impossibility theorem applied to voting systems – lots of articles and papers have spent time introducing and discussing its implications. In essence, Arrow’s Theorem states that a voting process which ranks candidates in an absolute total order cannot simultaneously satisfy:
