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Responding to OP's comment For the babbling equilibrium, it's important to note that how S randomizes over his messages matters. In particular, the messages should be randomized in the same way across all states. For example, if S sends messages $\{o, g, m\}$ with probabilities $(p,q,r)$ (where $p,q,r\in(0,1)$ and $p+q+r=1$) when the state is $o$, he needs ...

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