Sen's proofs of the Arrow and Gibbard theorems

The expanded edition of Sen's classic Collective Choice and Social Welfare contains a new proof of Arrow's impossibility theorem. The proof relies on two lemmas: Spread of Decisiveness and Contraction of Decisive Sets. The first lemma requires that the social preference relation is quasi-transitive, but the second requires full transitivity. I show how Spread of Decisiveness can be used to prove Gibbard's oligarchy theorem. Sen himself sketches how this can be done, but my argument is different and simpler. Arrow's result follows trivially once transitivity is strengthened, omitting the need for Contraction.