# Bayesian game and the set of types

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.

In the two type case there is an alternate solution that is frequently used. If $$a \neq c$$ and $$b \neq d$$, that is the types are really different in both attributes, then you can define a function $$f$$ for which $$b = f(a), \hskip 20pt d = f(c).$$ Whatever you need the second attribute for, this way it becomes a function of the first one, and the values of the first attribute completely define the type space.