Is the following claim written with the right way?

Question

I have a simple question though confusio for me. In game theory we usully write thet a strategy is a mapping from the set of types $T$ to the simplex set of actions (refering to mixed mixed strategies) and we write

$$\sigma:T\to\Delta(A)$$

Is it right to write the following $$\sigma(t)=\mathbb{P}r\times a$$ where $\mathbb{P}r$ is the probability that the profile of strategies $a$ is played?

Answer 1

Would be strange to write it that way.

If you had to define something like that, just do the following:

Start with a type space $(T,\mu)$ with probability measure $\mu$.

Let $\sigma: T \rightarrow \Delta(A)$.

Then $\mathbb{P}(a) =\mu(\sigma^{-1}(a))$.