I have in mind a 2 stage game
Stage 2: $N$ firms simultaneously decide an action. Let $A_i\in \{0,1\}$ denote the action of firm $i$ for $i=1,...,N$. The payoff that firm $i$ gets from choosing $A_i=1$ is $\delta\sum_{j\neq i}A_j+s_i$, where $s_i$ is firm $i$'s private information and $\delta>0$ is a known parameter.
I assume that $\{s_i\}_{i=1}^N$ are i.i.d. random variables.
I use Bayesian Nash equilibrium as solution concept.
Stage 1: The same $N$ firms simultaneously decide whether to discover other firms' types: each firm $i$ has to choose whether to discover firm $j$'s type $s_j$ $\forall j\neq i$. This game is with complete information and I use Nash equilibrium as solution concept.
Discovering types is costly. The benefits that firm $i$ gets from discovering firm $j$' type $s_j$ come from the reduction of the amount of incomplete information at stage 2 and, hence, the increase of the chances of choosing a strategy in stage 2 that is ex-post optimal.
(*)Moreover, I believe that the benefits that firm $i$ gets from discovering firm $j$' type $s_j$ increase with the number of firms $\neq i$ discovering $s_j$.
My intuition for (*) is: higher number of firms $\neq i$ discovering $s_j$ $\rightarrow$ higher number of firms $\neq i$ that will choose their actions in stage 2 based on the "direct observation" of $s_j$ $\rightarrow$ higher number of firms $\neq i$ whose action can be "predicted more precisely" by firm $i$ in stage 2, if firm $i$ knows $s_j$ too $\rightarrow$ discovering $s_j$ becomes more valuable for firm $i$.
Do you think that (*) is correct? If Yes or Not, could you help me with some intuition on why the number of other firms discovering $s_j$ should matter/is irrelevant?
Apologies in advance for the imprecisions or inappropriate vocabulary.
This doesn't pretend to be an answer and I might say some bullshit. But consider this as more like an invitation for a discussion. Also this looks like a research problem, so most probably, there is no direct and truly correct answer.
Consider first situation where none of the agents decide to discover information. Then I would use a framework of global interactions of Brock Durlauf to derive some sort of mean field equilibrium. Essentially agents form rational expectations on probabilities with which others will choose $A=1$, say this expectation is equal to $m$, then in equilibrium $m = \bar{F}(\delta (N-1) m)$ where $N$ is the total number of firms and $\bar{F}(\cdot)$ is complementary cdf of $s$.
Consider then the case where all agents decide to discover information. For simplicity, say we have only 2 agents. Then we have the following payoff matrix of the game with complete information:
\begin{vmatrix} -c,-c & s_1-c, -c \\ -c,s_2-c & \delta + s_1-c, \delta + s_2-c \end{vmatrix}
Here $c$ is cost of acquiring information. Depending on realizations of $s_1$ and $s_2$ we might get different equilibria. And it looks to me that the payoffs that agents get here might be lower than what they get in case 1.
After that I am not sure how to proceed with the scenario where some of the agents acquired information and some of them didn't. What are the payoffs in this case?