Unilateral Actions the Case of International Environmental Problems

NB_l( 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 NB_i (H, q_i ,0) is decreasing in q_i for q_i>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 q_i^' =suparg{NB_i(H, q_i ,1) = NB_i(H,q_i(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 q_i^' > q_i(L,0) , such that

NB_i(L, q_i^',1) -NB_i(L,q_i (L,0),0) > NB_i(H,q_i^',1)-NB_i(H,q_i(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^{, q}H^, ρ) ≤ ^miⁿ ρ{NBi⁽H^, qi⁽H^,0), ρ), ^} =>

NB_i (H, qH ,1) ≤ NB_i (H, q_i (H,0),0) , which is implied by 6.b.

Define qL = maxarg{NB_i (L, qL ,1) = NB_i (L, q_i (L,0),0)}. We now show that no
qL ≠ q^L is weakly dominated. This amounts to showing that
NB_i (L, qL ,1) > NB_i (L, q_i (L,0),0). Use 6.a. NB_i (L, qL ,1) ≥ NB_i (L, q_i (L,0),0) and the
fact that NB_i(L,q_i,ρ) is decreasing in q_i for q_i>q_i(L,1), for any q¹L = q^L + ε 6.b

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