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Follow up from this question: Perturbing distributional strategies on a measure zero set

In the context of a Bayesian game, $\Gamma = \left< (\Theta_i,\mu_i)_{i \in N}, (A_i)_{i \in N}, (u_i)_{i \in N}\right>$ where type of agent $i$ is drawn from $\Theta_i$ according to a probability measure $\mu_i$, recall that a distributional strategy $\sigma_i$ is a probability measure on $\Theta_i \times A_i$ such that the marginal on $\Theta_i$ is $\mu_i$.

For any distributional strategy $\sigma_i \in \Delta(\Theta_i \times A_i)$, we can define a behavioural strategy as regular conditional distributions $\beta_i(B,\theta) = \sigma(B \vert \theta)$ where $B$ is a measurable subset of $A$. Note each distributional strategy corresponds to an equivalence class of behavioural strategies.

The question is as follows:

Suppose $(\sigma_i)_i$ is a BNE of $\Gamma$. Does there exist a set of behavioural strategies $(\beta_i)_i$ where $\beta_i(B,\theta_i) = \sigma_i(B \vert \theta_i)$ such that they constitute a BNE?

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  • $\begingroup$ Under the only reasonable way to define expected payoff for behavior strategies, yes. $\endgroup$ Nov 29, 2020 at 22:51

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