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A Combinatorial Topology Approach to Arrow's Impossibility Theorem

Rajsbaum, Sergio and Raventós-Pujol, Armajac (2022): A Combinatorial Topology Approach to Arrow's Impossibility Theorem.

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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 a combinatorial topology proof that does not use advance mathematics, and gives a very intuitive geometric reason for Arrow’s impossibility. The geometric proof for the basis case of two voters, n=2, and three alternatives, |X|=3, is based on the index lemma, that counts the absolute number of times that a closed curve in the plane travels around a point. This yields a characterization of the domain restrictions that allow non-dictatorial aggregation functions. It also exposes the geometry behind prior pivotal arguments to Arrow’s impossibility. We explain why the basis case of two voters, 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.

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