In a finite Bayesian game, in most textbooks, a type of player $i$ is defined as $\theta_i\in\Theta_i$. Little is said about the "nature" of the set of types.
For example, could we have a two-element set $\Theta_i=\{(a,b),(c,d)\}$? If I am not mistaken this means that $\Theta_i$ is a subset of $\mathbb{R^2}$, right?
For the common prior then, we could have, for example, $\mathbb{P}((a,b))=p$ and $\mathbb{P}((c,d))=1-p$. Both probabilities describe the joint probability distribution that Nature assigns the types to the players.
Is this allowed in the definition of a Bayesian game?
I'd appreciate any help. Thank you.