Voting Equilibria Under Proportional Representation Online

Proof of Lemma 1 Let vt = i. Take any j ∈ L and consider profile (j, v−t). Since t is neither majority-pivotal nor median pivotal, k(j, v−t) = k(v), and, thus, ph(j, v−t) = ph(v) for every h ∈ L. Moreover, k(v) is a majority party in (j, v−t) if and only if it is so in v. Suppose vt is not strategically sincere for t in v. There exists j ∈ L \ {i} such that u(pj(v); t) > u(pi(v); t). Take any ε ∈ (0, 1). From the discussion in the previous paragraph, we conclude that