Preference Signaling in Matching Markets By Peter Coles , Alexey Kushnir

As in the proof of Proposition 1, we denote a permutation that changes the ranks of w and w′ in a firm preference list as ρ : (..., w, ..., w′, ...)→ (..., w′, ..., w, ...) and construct preference profile θ′ ∈ Θ from θ∗ as follows: • firm f ’s preferences are the same as in θ∗: θf = θ∗, • workers w and w′ are exchanged in the preference lists of firms −f : ∀f ′ ∈ −f, we have θf ′ = ρ(θf ′), • workers w and w′ preference profiles are exchanged θw = θw′ , θw′ = θw, • for any other w0 ∈ W\{w,w′}, θw0 = θw0 . Since firm f ’s preference list is unchanged and since w,w′∈ WS , profile θ′ belongs to Θ̄. Since strategies of firms −f are anonymous, then for any f ′ ∈ −f and for any WS f ′ ⊂ W ∪N we have