A Deferred Proofs

Let m 2 and a,b A such that a is ordered before b in tie-breaking. Suppose PW (P) = { a,b } for some truthful profile P . Suppose PW (P) = { a,b } for some truthful profile P . Agents from the former set will best-respond to rankings whose top preference is a, changing the winner to a, whereas agents from the latter set will best-respond to rankings whose top preference is b, changing the winner back to b. If not, the unique equilibrium winner will be b .