#### Abstract

Throughout the proof, we assume that τ satisfies Property (M); i.e. there is a positive mass of agents who choose from (x, x ε) under τ. Note that Property (M) implies τ (x ε) > 0 since otherwise almost every agent chooses either x or some x ≥ x. Let f and g denote the density for F and G, respectively. Since f and g are bounded by assumption, we let M f… (More)