# Existence of Bayes Nash equilibria in behavioural strategies

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?

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