Supplement to “ Robustness and Separation in Multidimensional Screening ”

This supplementary appendix contains additional materials for the paper “Robustness and Separation in Multidimensional Screening.” Section B contains proofs of auxiliary results not included in the main paper. Section C details how the generalized virtual values coincide with traditional (ironed) virtual values in the single-good monopoly problem. Theorems, equations, and sections in the main paper are referenced using the original numbering. B Additional proofs Proof of Lemma 4.3. (Adapted from Madarász and Prat (2012)) It is easy to see that statement (b) of the lemma follows from (a), by integrating over each Sk in the partition; so it suffices to prove (a). As in the proof of Lemma 4.2, we can take ∆ = maxx,θ u(x, θ) − minx,θ u(x, θ), and then in any mechanism, any two types’ payments can differ by at most ∆. Also, put τ = min{ǫ/6∆, 1}. By Lipschitz continuity, there exists δ such that, whenever θ, θ are two types with d(θ, θ) < δ, then |u(x, θ) − u(x, θ)| < τǫ/6 for all x. We show this δ has the desired property. Let (x, t) be any given mechanism. Let t = minθ t(θ). Let S ⊆ ∆(X) × R be the set of values (x(θ), τ t + (1 − τ)t(θ)) for θ ∈ Θ, and let S be its closure, which is compact 1 (by the above observation on payments). Then define (x̃, t̃) by simply assigning to each type θ ∈ Θ the outcome in S that maximizes its payoff, Eu(x, θ) − t. This exists by compactness. This (x̃, t̃) is a mechanism: IC is satisfied by definition, and IR is satisfied since the payments have only been reduced relative to those in (x, t), so each type θ has the option of getting allocation x(θ) for a payment of less than t(θ), which gives nonnegative payoff. Now, let d(θ, θ) < δ. We know that the outcome chosen by θ in the new mechanism can be approximated arbitrarily closely by an element of S corresponding to some type θ; in particular, there exists θ such that |Eu(x̃(θ), θ)− Eu(x(θ), θ)| < τǫ 6 and ∣t̃(θ′)− [τt+ (1− τ)t(θ)] ∣∣ < τǫ 6 . (B.1) Now, we know from IC for the original mechanism Eu(x(θ), θ)− t(θ) ≥ Eu(x(θ), θ)− t(θ), (B.2) and by the definition of the new mechanism (x̃, t̃), Eu(x̃(θ), θ)− t̃(θ) ≥ Eu(x(θ), θ)− [τt+ (1− τ)t(θ)]. Using (twice) the fact that d(θ, θ) < δ, the latter inequality turns into Eu(x̃(θ), θ)− t̃(θ) ≥ Eu(x(θ), θ)− [τt+ (1− τ)t(θ)]− τǫ 3 . Now combining with (B.1) we get Eu(x(θ), θ)− [τt+ (1− τ)t(θ)] > Eu(x(θ), θ)− [τt+ (1− τ)t(θ)]− 2τǫ 3 . (B.3) Adding (B.2) and (B.3), and canceling common terms, we get τt(θ) > τt(θ)− 2τǫ 3 or t(θ) > t(θ)− 2ǫ 3 .