A Distributed Combinatorial Topology Approach to Arrow's Impossibility Theorem

Baryshnikov presented a remarkable algebraic topology proof of Arrow's impossibility theorem trying to understand the underlying reason behind the numerous proofs of this fundamental result of social choice theory. We present here a novel combinatorial topology approach that does not use advanced mathematics, while giving a geometric intuition of the impossibility. This exposes a remarkable connection with distributed computing techniques. We study in detail the case of two voters, n=2, and three alternatives, |X|=3, and show that Arrow's impossibility is closely related to the index lemma. Also, we study the domain restrictions that avoid the impossibility. Finally, we explain why the case of n=2 and |X|=3 is where this interesting geometry happens, by giving a simple proof that this case implies Arrow's impossibility for any |X| ≥ 3 and any finite n ≥ 2.