Arrow's Impossibility Theorem

Arrow's impossibility theorem is the original, seminal example of a "voting paradox" that limits how "nicely" a method of producing a group decision from individual preferences (like voting) can behave. Specifically, the theorem describes 4 properties that we'd all like our voting system to have and states that no voting system can have all these properties at once.

The beauty of the theorem is how easily it can be tied to common frustrations of democratic politics. I will start by giving a few historical examples of the independence of irrelevant alternatives (the most interesting of the aforementioned 4 properties) in action. I will then give a simplified toy proof of the theorem posed via a game in which one voter does their best to troll another. Finally, I will blast off into outer space and discuss esoteric and impractical topics like set theory and ultrafilter voting.

The Wikipedia article (https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem) on the topic provides a nice introduction while these lecture notes (http://www.ssc.wisc.edu/~dquint/econ698/lecture%202.pdf) give some intuition for a proof. For a deeper dive you can read Kenneth Arrow's original paper (http://sites.duke.edu/niou/files/2014/06/Arrow-Social-Choice-And-Individual-Values.pdf).