y ∈ arg max
m∈[0,1]

Wela(m)(b−(cid:96), v(cid:96)(y)) ∀y ∈ [0, 1].

y ∈ arg max
m∈[0,1]

Wela(m)(b−(cid:96), v(cid:96)(y)) − h(cid:96)(b−(cid:96), v(cid:96)(m)) ∀y ∈ [0, 1].

Deﬁne f (m, y) = Wela(m)(b−(cid:96), v(cid:96)(y)) and g(m) = −h(cid:96)(b−(cid:96), v(cid:96)(m)) ∀m, y ∈ [0, 1]. With these
notations, note that (8), (9) are exactly 1,2 in Lemma 4 (respectively). To apply Lemma 4 we

show that 3 holds. Deﬁne K = arg maxa∈A(cid:12)(cid:12)Eo∼F|a[b(cid:96)(o) − ˜b(cid:96)(o)](cid:12)(cid:12). Since A is a ﬁnite set, K < ∞.
We show that (cid:12)(cid:12) ∂f (m,y)
(cid:12)(cid:12) ≤ K for all m, y ∈ [0, 1] as follows. By deﬁnition of f and of the social
(cid:12)(cid:12)(cid:12)(cid:12) ∂f (m, y)
(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12) ∂ Wela(m)(b−(cid:96), v(cid:96)(y))

(cid:12)(cid:12)(cid:12)(cid:12) ∂Eo∼F|a(m)[v(cid:96)(y, o)]

welfare Wel(·),

(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)

(10)

∂y

∂y

∂y

∂y
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