Perfect Regular Equilibrium

and a(s) = max{s — b, 0} is the best response to his system of beliefs μ(max{s — b, 0}; s)
= 1. This proves that they satisfy the sequential rationality. Therefore, these strategies and
the system of beliefs are a simple perfect Bayesian equilibrium.

This simple perfect Bayesian equilibrium, however, is incapable of satisfying the weak
consistency and the subgame perfect Nash equilibrium condition. In the scenario of this
equilibrium, the receiver constantly mistakes a true type θ for a wrong type max{θ — b, 0}. As
a result, the sender’s strategy s(θ) = θ and the receiver’s system of beliefs μ(max{s — b, 0}; s)
= 1 do not induce the same probability distribution on the equilibrium path which the players
would actually reach if they were to play according to their strategies s(∙) and α(∙). Since
the weak consistency¹⁰ requires them both to induce the same probability distribution on
the equilibrium path, this simple perfect Bayesian equilibrium does not satisfy the weak
consistency. Moreover, the receiver’s strategy a(s) = max{s — b, 0} is not the best response
to the sender’s strategy s(θ) = θ. So this simple perfect Bayesian equilibrium does not
satisfy the Nash equilibrium condition, and thus it does not satisfy the subgame perfect
Nash equilibrium condition¹¹ .

This incapability of the simple perfect Bayesian equilibrium is caused mainly by the

¹⁰ Definition 7 in Section 5 formally defines this weak consistency in general multi-period games with
observed actions.

¹¹ Crawford and Sobel (1982) tried to solve this problem with a simple perfect Bayesian equilibrium
by adopting a continuous version of Bayes’ rule. Their approach to the problem naturally led them to
only consider the probability density strategies of the sender. That is, they did not consider the overall
strategies of the sender. Their partial consideration of the sender’s strategies might be justified by Lemma
1 in their paper which guaranteed that, in equilibrium, any sender’s strategy can be replaced with her
probability density strategies while preserving the strategies of the receiver. Lemma 1, however, was not
proven correctly, and thus it cannot justify their partial consideration of the sender’s strategies or any other
results. For more information, please refer to Jung (2009).

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