Perfect Regular Equilibrium

is both 1) reasonably consistent and 2) sequentially rational⁹ .

3.2 Example

Now, we are ready to exemplify the incapability of a simple perfect Bayesian equilibrium in
a general multi-period game with observed actions. Consider the information transmission
game introduced by Crawford and Sobel (1982). There are two players, a sender and a
receiver. The sender is assigned a type θ that is a random variable from a uniform distribution
on [0,1] and she makes a signal s ∈ [0,1] to the receiver. Then, after observing the signal
s, the receiver chooses his action a ∈ [0,1]. The sender has a von Neumann-Morgenstern
utility function U^s(θ, a,b) = — (θ — (a + b))² where b > 0 and the receiver has another von
Neumann-Morgenstern utility function U^r(θ,α) = — (θ — a)².

In this game, the sender’s strategy s(θ) = θ and the receiver’s strategy a(s~) = max{s —
b, 0} are a simple perfect Bayesian equilibrium together with the receiver’s system of beliefs
μ(max{s — b, 0}; s) = 1 which denotes that given a signal s, the type max{s — b, 0} would be
assigned to the sender with probability one. First, the system of beliefs μ(max{s — b, 0}; s)
= 1 is reasonably consistent with the sender’s strategy s(θ) = θ because it does not violate
the conditions for the reasonable consistency in Definition 2. Under the strategy s(θ) = θ,
each signal θ occurs with probability zero, and thus we cannot employ Bayes’ rule. In this
case, no system of beliefs is considered to violate Bayes’ rule formulated in Definition 2.
Consequently, μ(max{s — b, 0}; s) = 1 is reasonably consistent with s(θ) = θ. Second, the
sender’s strategy s(θ) = θ is the best response to the receiver’s strategy α(s) = max{s — b, 0},

⁹ For a formal definition of the sequential rationality, please refer to Definition 5 in Section f.

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