## Constructing voting paradoxes with logic and symmetry; Arrow's Theorem

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The material is roughly for three one to one-and-half hour Math Circle sessions. Each part is self-contained and could be done independently. The first two parts are suitable for grade 4 and up. Part III could be done with grade 7 and up.

Part I: Voting Paradoxes and Logic. Beginning with a logic puzzle, students are guided through an argument to prove Shapiro's Theorem: If there is no majority of voters who vote unanimously on all propositions then any proposition could be passed.

Part II: Voting Paradoxes and Symmetry deals with the situation when voters need to choose between three or more alternatives. Students learn about popular voting methods, some well-known paradoxes. We follow D.Saari's geometric representation of the voting profile for the case of three alternatives and explore symmetries which contribute to the paradoxes.

Part III: Arrow' Theorem outlines the proof of the Arrow's theorem in a few simple, engaging exercises. This proof is is a combination of proofs by Sridhar Ramesh and by Terrence Tao.