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
Us(θ, a,b) = (a + b))2 where b > 0 and the receiver has another von
Neumann-Morgenstern utility function
Ur(θ,α) = 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.

16



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