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5

The expected posterior is the prior. But for this to hold, you have to take the expectation with respect to the prior. In your example, you describe a Bayesian but ev aluate their update with respect to your belief. Ann and Bob might be perfect Bayesians and their expected posteriors might be exactly their priors. But that does not mean that Ann's ...

4

Suppose you are an analyst studying a Bayesian game. You know the players, the possible states of nature, the common prior, the action spaces, the payoff functions, and you know about some information channels available to the players, the latter given via some information structure. However, you can't rule out that the players have additional information ...

0

First, $l(\tau)(y)$ is a function of both $\tau$ and $y$, and the dependence on $\tau$ is essential; this is how communication happens. Since $T$ and $Y$ are finite, there is no point in introducing integrals. $\mathbb{E}_{l(\tau)}g(F(y))$ is the expectation over the function when the random value $y$ is distributed according to $l(\tau)$. That is, \mathbb{...

6

We have that ${\cal I} = ((X^i)_i, \mu)$​ and ${\cal J} = ((Y^i)_i, \nu)$​ are two information structures. An Interpretation mapping for player $i$​​ is a mapping $\phi^i: X^i \to \Delta(Y^i)$​ so it associates with every $x^i$​ a distribution over $Y^i$​. Let $x^i \in X^i$​. Then $\phi^i(x^i)$​ is a distribution over $Y^i$​ so $\phi(x^i)(y^i)$​ is the ...

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