Let $ A_i $ be the set of strategies of player i and $ \Theta_i $ the set of possible types of player i. The payoff function in a static Bayesian Game is defined as
$ u_i: A_i \times \Theta_i \rightarrow \mathbb{R} $
I do understand that the cross product of the types and actions determine the payoff in a (static) Bayesian Game, but I wonder why it is actions cross player's types? So far, I thought that the player's types are decisive for the action as they fix what action is a best response for each type of any player. Hence, I assumed that the payoff function is determined as
$ u_i: \Theta_i \times A_i \rightarrow \mathbb{R} $
where for every type of player i an action is choosen and mapped in the set of real numbers.
Can anybody tell me what the rationale is behind defining the payoff function as actions cross player's types and not the other way around?
A convention. A type is chosen exogenously by "Nature" while an action is chosen by the player after observing the type. So often one would write the payoff function as $u(a_i;\theta_i)$ using semi-colon to emphasize that players do not control what type they are when they are choosing their actions.
In addition this convention is similar to the notation in multivariate calculus where a variable is typically written first followed by a parameter, say $f(x,t)=tx^2$, where $t$ is a parameters with$t\in\{1,2,3\}$.