A 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 advance mathematics, while giving a geometric intuition of the impossibility. This exposes a remarkable connection with distributed computing techniques. We show that Arrow's impossibility is closely related to the index lemma, and expose the geometry behind prior pivotal arguments to Arrow's impossibility. We explain why the case of two voters, n=2, and three alternatives, |X|=3, is where this interesting geometry happens, by giving a simple proof that this case implies Arrow's impossibility for any finite n>= 2,|X|>= 3. Finally, we show how to reason about domain restrictions using combinatorial topology.