Perfect Bayesian Equilibrium

fleeting EC487_Advanced_Microeconomics

Perfect Bayesian Equilibrium, is mainly mentioned as “PBE” in a lot of context. In PBE, we will then mainly focus on the weak PBE. The weak PBE is defined as:

For the sender, given $\mu (\cdot |a_1)$:

$$\forall \theta \in \Theta ,{\sigma }_1^{\ast }(\cdot |\theta )\in \arg \underset{{\sigma }_1}{\max }\sum _{a^1}\sum _{a^2}{\sigma }_1(a_1|\theta ){\sigma }_2^{\ast }(a_2|a_1)u_1(a_1,a_2,\theta )$$

For the receiver,

$$\forall a_1\in A_1,{\sigma }_2^{\ast }(\cdot |a_1)\in \arg \underset{{\sigma }_2}{\max }\sum _{\theta }\sum _{a_2}\mu (\theta |a_1)\sigma (a_2|a_1)u_2(a_1,a_2,\theta )$$

• $\mu$ is consistent with Bayesian Updating where possible:

$$\mu (\theta |a_1)=\frac{p(\theta ){\sigma }_1^{\ast }(a_1|\theta )}{{\sum }_{{\theta }^{\prime }\in \Theta }p({\theta }^{\prime }){\sigma }_1^{\ast }(a_1|{\theta }^{\prime })}$$

It totally follows the Bayesian Inference.