NBl( H, q i ,0) > NBi( H, q ( H ,0),1)}. Use 3.1. NBi(H,qH f0) ≥ NBi(H ,q( H ,0),1)
and the fact that NBi (H, qi ,0) is decreasing in qi for qi>q(H,0), for any
q 1H = qH - ε, 3.1 holds with strict inequality. Hence, no q 1H ≠ qH can be
weakly dominated. Therefore, given qH = qH - ε, posterior beliefs will be up-
dated to ρ(qH ) = 0, and consequently, given this new set of beliefs, only
qH = qH is an undominated separating equilibrium.
Proof of result 3: Define qi' =suparg{NBi(H, qi ,1) = NBi(H,qi(H,0),0)} . If there
exists a q'i satisfying 6.a with strict inequality, then existence is guaranteed. If
NBi(L,qi',1)>NBi(L,qi(L,1),0), then also NBi(L,qi',1)>NBi(L,qi(H,0),0), since
qi(L,1) is the unique maximizer of NBi(L,qi,θ), which implies that if there ex-
ists a qi' > qi(L,0) , such that
NBi(L, qi',1) -NBi(L,qi (L,0),0) > NBi(H,qi',1)-NBi(H,qi(H,0),0)}, then a separating
equilibrium exists. But this is guaranteed via the single crossing condition.
Proof of result 4:
Weak domination of qH for the high cost type in this case corresponds to the
following inequality:
maxp NBi(H, qH, ρ) ≤ min ρ{NBi(H, qi(H,0), ρ), } =>
NBi (H, qH ,1) ≤ NBi (H, qi (H,0),0) , which is implied by 6.b.
Define qL = maxarg{NBi (L, qL ,1) = NBi (L, qi (L,0),0)}. We now show that no
qL ≠ qL is weakly dominated. This amounts to showing that
NBi (L, qL ,1) > NBi (L, qi (L,0),0). Use 6.a. NBi (L, qL ,1) ≥ NBi (L, qi (L,0),0) and the
fact that NBi(L,qi,ρ) is decreasing in qi for qi>qi(L,1), for any q1L = qL + ε 6.b
44
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