In the page copied from the Handbook of Game Theory with Economic Applications, I have a problem with the condition $$\pi_i(\{\nu\};\omega)>0.$$ I.e. I'm looking for an Event $E$ and states $\nu$ and $\omega$ in the Example on the page 1668 for which $$\pi_i(\{\nu\};\omega)=0,$$ but at the same time $$\pi_i(E,\nu)\neq\pi_i(E,\omega)$$
It looks like $\pi_{Ann}(\beta ; \gamma)=0$ but $\pi_{Ann}(\beta ; \beta)=1/2$ is the part of the example you are looking for. If state $\gamma$ obtains, Ann knows she is in $\gamma$ and not in $\beta$, but if she is in $\beta$ she is not sure which state she is in.