Arrow's Impossibility Theorem: Computability in Social Choice Theory
Arrow's Impossibility Theorem establishes bounds on what we can require from voting systems. Given satisfaction of a small collection of "fairness" axioms, it shows votes can only exist as dictatorships in which one voter determines all outcomes. Votes are modelled as maps from a collection of partial orders, the preferences of voters, to a single verdict which is another aggregated partial ordering. This result is classic and has an extension called the Possibility Theorem that shows these dictatorships needn't exist with infinite voter sets. Mihara extends this work by examining the computability of each of these results. He found that the only voting systems that are in any sense computable are necessarily dictatorial, which takes away from the usefulness of the Possibility Theorem. In this paper we primarily survey the results of Mihara, focusing not on applied consequences, as much of the surrounding literature does, but going into greater details on the underlying Mathematics and Computability of the proofs. We give detailed exposition on the methods used and introduce all notation. We first see complete proofs of the classical results and a sufficient introduction to computability that an unfamiliar reader should be able to follow without prior knowledge of the field. We then expand into Mihara's results, and using our established knowledge of computability show the problems with trying to compute non-dictatorial social welfare functions. This involves introducing an extended definition of computability called pairwise computability.