9edcc1391c208ba0b503fe9a22574251-Supplemental.pdf

Let m 2 and a,b A such that a is ordered before b in tie-breaking. BR steps will therefore alternate whether they are taken by agents represented in Id(a)(P) or Id(b)(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. Inverse reasoning holds if a and b differ by one initial plurality score and sP(a) = sP(b) 1, implyingr(P0) = b. Without loss of generality let W = {1,2} and suppose u2 > um, since the case where u2 = um is covered in [Brˆanzei et al., 2013]. We believe this proof is challenging due to the dependence in agents' rankings once we condition on profiles that satisfy two-way ties (i.e.