Definition

The information signal, $y^{i},i\in\mathcal{N}$, is what any agent in a team problem is said to observe. ## Static For each $i\in\mathcal{N}$ we define it in terms of the static information function, $\eta^{i}$ s.t. $$y^{i}=\eta^{i}(\xi)\equiv \tilde{\eta}^{i}(\omega),\quad\omega \in\Omega$$where $\xi\in\mathit\Xi$ is the random state of nature (which is a RV mapping from $\mathit\Xi$ to $\Omega$). Given how we tend to treat $\mathit\Xi\iff\Omega$ in the definition of $\eta^{i}$ we do similarly here with $y^{i}=\tilde{\eta}^{i}(\omega)$.

Dynamic

We know that a decision problem is said to be dynamic if the measurements of at least one of the agents involves past actions hence the information function also takes in past measurements as input: $$\tag{1}y^{i}=\eta^{i}(\xi;\mathbf{u}),\quad i\in\mathcal{N}$$where the dependence on $\mathbf{u}$ (the $N-$tuple of past actions by each agent) is assumed to be Strictly Causal, which means that under a given fixed clock the information received by each agent can depend only on actions taken in the past.

Let $\mathcal{T}:=\{ 1,\dots,T \}$ and $t\in\mathcal{T}$. Let $u_{t}^{i},y_{t}^{i}$ denote the action variable and information signal of agent $\mathbf{A}i$ at time $t\in\mathcal{T}$. Furthermore let $$\mathbf{u}_{t}:\{ u_{t}^{1},\dots,u_{t}^{N} \}\quad \mathbf{y}_{t}=\{ y_{t}^{1},\dots,y_{t}^{N} \}$$and $$\mathbf{u}_{[t_{0},t_{1})}\equiv \mathbf{u}_{[t_{0},t_{1}-1]}:=\{ \mathbf{u}_{t_{0}},\mathbf{u}_{t_{0}+1},\dots,\mathbf{u}_{t_{1}-1} \}\equiv \{ u^{1}_{[t_{0},t_{1})},\dots,u^{N}_{[t_{0},t_{1})} \}$$Then, under the strictly causal assumption, $(1)$ becomes equivalent to $$y_{t}^{i}=n_{t}^{i}(\xi,\mathbf{u}_{[1,t)}),\quad t\in\mathcal{T},i\in\mathcal{N}$$for some information functions $n_{t}^{i},t\in\mathcal{T},i\in\mathcal{N}$. The variable $y_{t}^{i}\in\mathbb{Y}^{i}_{t}$ is the on-line information available to $\mathbf{A}i$ which we can use for the construction of $u_{t}^{i}$ through an appropriate policy $\gamma_{t}^{i}:\mathbb{Y}_{t}^{i}\to \mathbb{U}_{t}^{i}$: $$u_{t}^{i}=\gamma_{t}^{i}(y_{t}^{i})\equiv \gamma_{t}^{i}(\eta_{t}^{i}[\xi;\mathbf{u}_{[1,t)}]),\quad t\in\mathcal{T},i\in\mathcal{N}$$