The Influence Limiter: Provably Manipulation-Resistant Recommender Systems (Appendix)

Proof of Lemma 5: Because both D(q||u) = D(1− q||1− u) and GF(q||u) = GF(1− q||1− u), we can assume u ≥ q without loss of generality. Keeping q fixed, we want to show that the result holds for all u. Note that D(q||q) = GF(q||q) = 0. Thus, differentiating with respect to u, it is sufficient to prove that GF ′(q||u) ≥ D′(q||u)/2 for all u ≥ q,u ≤ 1. We change variables by setting y = u−q. We use the notation D′(y) to denote D(q||u)|u=q+y, treating q as fixed and implicit. Likewise, we use the notation GF ′(y). For brevity, we use q to denote (1−q).