The following is from Bergemann and Morris (2016)
Suppose that $I$ is a finite set of players, where $i= 1,2,\dots, I$ and $i$ refers to the typical player. Let $\Theta$ be a finite set of states where $\theta\in\Theta$. A game of incomplete information is a $\Gamma = <G, S>$ where $G=<(A_i,u_i), q>$, stands for the payoff environment or basic game, and consists of a finite set of actions $A_i$ for every $i$, where the utility payoff is $u_i:A\times\Theta\to\mathbb{R}$ and $q\in\Delta(\Theta)$ is the full support common prior. Furthermore, the information structure $S=<T,\pi>$ where $T=T_1\times\cdots T_I$ is the product space of private signals and $\pi:\Theta\to\Delta(T)$ the signal probability distribution - also know a-priori.
Also there is a decision rule, that is obedient, that is a mapping
$$\sigma:T\times\Theta\to\Delta(A)$$
By the term obedient we mean that the mechanism $\sigma$ sends a private signal which specifies to every player $i$ to play a strategy $a_i$ where there is no alternative actions $a_i^{'}\in A_i$ that yields strictly higher utility than $a_i$.
The expected payoff for player $i$ is given by the following value function
$$V\left((a_i,a_{-i}),\theta\right)=\underbrace{\sum_{a_{-i},t_{-i},\theta}q(\theta)\pi\left((t_i,t_{-i})|\theta\right)\sigma\left((a_i,a_{-i})|(t_i,t_{-i}),\theta\right)u_i\left((a_i,a_{-i}),\theta\right)}_{\text{Utility payoff}}$$
My questions are the following:
- How could we write the sum term of the value function in terms of expectation, like $\mathbb{E}_{P}$ where $P$ is the probability measure?
- How could we write the sums in terms of integrals in a similar setup?