Unilateral Actions the Case of International Environmental Problems

Sequential equilibrium

We restrict attention to pure strategy equilibria. Consequently, there remains
two different kinds of equilibria; separating equilibria, where each type send
different signals, and pooling equilibria where the two types send the same sig-
nal. Since the analysis focuses on the prospects of unilateral actions, focus is
exclusively on sequential separating equilibrium, or more precisely, interest is
on finding condition for a separating equilibrium to exist. In a separating equi-
librium, the receiver can perfectly infer the type of the sender. Formally, a col-
lection of reduction levels and beliefs {qH ,qL,ρ(q)} forms a sequential equi-
librium if the following conditions are satisfied:

i) Optimality for the country with costs θ:

Qθ ∈ arg max NB_i (θ, Qi, p(Qi ))

ii) Beliefs are Bayes-consistent:

a) If q_H ≠ q_L then ρ(qH ) = 0 and ρ(qL) = 1

b) I^f Qh = Ql ^{then ρ}(qH) = ^ρ( qL) = P^o

c) If q_θ ≠ {q_h , q_l } then any ρ(q_θ ) is admissible

After observing qi, the receiver must form a belief about which types could
have sent q_i. These posterior beliefs are denoted ρ(θ∣q_i) with
ρ(L ∣q_i)+ρ(H ∣q_i)=1. The first requirement of strategies that form a sequential
equilibrium is sequential rationality, which amounts to saying that for each qi, qj
must maximise expected payoffs, given the beliefs. Regarding the optimality of
beliefs, if q_H ≠ q_L and the receiver observes, e.g., qH, then it must be that costs
are high, and the only consistent belief is ρ(qH ) = 0 and given this belief, it is
optimal for the high cost type to play q_H . In this particular signalling game the
requirement of consistency of beliefs does not place any restrictions on beliefs

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