Suppose, two players I and II, given a state space of three states$\{a,b,c\}$ with a common prior, $p(a) = p(b) =p(c) =1/3$, are endowed with two partitions of state space, $\mathscr{P}_\text{I} = \{\{a,b\},\{c\}\}$ and $\mathscr{P}_\text{II} = \{\{a\},\{b,c\}\}$. State space, prior probability distribution on state space, the information structure, $\mathscr{P}_{\text{I}}$ and $\mathscr{P}_{\text{II}}$ are common knowledge to I and II.
I and II's objectives are to deduce the posterior probability of an event $E $ as accurate as possible. They announce his own posterior probability in an alternating fashion.
Let's assume the state $b$ is obtained and $E = \{b\}$. At the first stage, I will announce $p(E) = 1/2$, because given his partition, he can't distinguish $a$ and $b$ when $a$ or $b$ actually happens. At the second stage II will announce $p(E) = 1$, because he can deduce from I's announcement that the actual state must be $a$ or $b$, and he knows that from his own partition the actual state is $b$ or $c$. at the third stage, I will announce $p(E) = 1$ too.
If II's objective is to make I's posterior probability as inaccurate as possible and such objetive is common knowledge. I's estimation of $P(E)$ will be $1/2$, because I can always turn a deaf ear to II to guarantee the result is not worse than that, and II can always choose to announce a random number from $[0,1]$ to make sure I's estimation of $E$ is not more accurate than that.
Here's my question. What if II's objective type is uncertain to I. To be precise, it's a common knowledge that there is a 50/50 chance II is a honest type or a dishonest type in the eyes of I. What are best strategies for I and II?
The different types of player II have to be encoded in the state space, the answer will depend on the specific encoding.