Definition

Static

For a team of $N$ agents, for each agent, $\mathbf{A} i,i\in\mathcal{N}$ the information function, $\eta^{i}$, maps from some standard Borel space $(\mathit\Xi,\mathcal{B}(\mathit{\Xi}))$ to the observable space $(\mathbb{Y}^{i},\mathcal{Y}^{i})$ (defined in Witsenhausen’s Intrinsic Model): $$\eta^{i}:(\mathit\Xi,\mathcal{B}(\mathit\Xi))\to(\mathbb{Y}^{i},\mathcal{Y}^{i})$$by convention though, we can treat this standard Borel space as being the same as the probability space $(\Omega,\mathcal{F})$ hence $$\eta^{i}:(\Omega,\mathcal{F})\to(\mathbb{Y}^{i},\mathcal{Y}^{i})$$ ## Dynamic In the dynamic case we take past actions into account as well hence $$\eta^{i}:(\Omega,\mathcal{F})\times (\mathbb{U}^{1}_{t},\mathcal{U}^{1}_{t})\times\dots \times (\mathbb{U}^{N}_{t},\mathcal{U}^{N}_{t})\to(\mathbb{Y}^{i},\mathcal{Y}^{i}),\quad t\in\mathcal{T}, i\in\mathcal{N}$$