Informed traders do know the cross section of the privately known signal between each other

Question

I am having the following setup of privately known signals and I am trying to understand an assumption. Here, I quote the setup.

Consider two agents idexed by $i=\{1,2\}$ and each one observes some privately noisy signal, that is \begin{align}s_i=\tilde{v}+\tilde{u_{i}}\end{align} where $(\tilde{v},\tilde{u_{i}})$ are identically distributed, with $\tilde{v}\sim N(v,\sigma_{v}^2)$ stand for the payoff of the risky asset and $\tilde{u_{i}}\sim N(0,\sigma_{u_{i}}^2)$ denotes the error term of each of the privately known signals. Also, it holds that $\sigma_{(u_1,u_2)}\neq 0$ and $\sigma_{(v,u_1)}=0=\sigma_{(v,u_2)}$.

The assumption that I am trying to understand following the aforementioned setup is the following:

``Every informed agent does know the cross sectional distribution of the privately knonw noisy signal that the other agent observes." Also in the market, there are uninformed traders etc. Does this mean, that in fact they know exactly the signal that each other observes or do I catch it wrong?

Answer 1

Yes, that assumption just means that every informed agent knows the true distributions where signals come from (i.e. roughly speaking, they know the parameters that define them).

The terminology "cross-sectional distribution" comes from the usual interpretation that instead of having literally 2 agents, in reality, you have a continuum of agents that come in 2 types. If that is the case, due to the law of large numbers, the empirical distribution of signals at any point in time will be equal to the theoretical distribution of signals. Therefore, knowing the cross-sectional distribution is the same as knowing the theoretical distribution. Since these distributions are normal, the latter corresponds exactly with knowing the mean and variance of signals of the other (type of) agent.