Definition

Given a static stochastic team problem $\{ J;\Gamma^{i},i\in\mathcal{N} \}$, a policy $N$-tuple $\underline{\gamma}\in\mathbf{\Gamma}$ is stationary if: 1. $J(\underline{\gamma})$ is finite. 2. The $N$ partial derivatives in the following equations are well-defined. 3. $\underline{\gamma}$ satisfies these equations: $$\tag{🌝}\nabla_{u_{i}}\left(E_{\xi|y^{i}}[L(\xi;\underline{\gamma}^{-i}(\mathbf{y}^{-i}),u^{i})]\right)|_{u^{i}=\gamma^{i}(y^{i})}=0\text{ a.s. }\quad i\in\mathcal{N}$$where$$\underline{\gamma}^{-i}(\mathbf{y}^{-i})=\{ \gamma^{m}(y^{m}):m\in\mathcal{N} \setminus i \}$$ ## Remark A pbp solution $\underline{\gamma}^{*}\in\mathbf{\Gamma}$ for a static team problem $(J,\mathbf{\Gamma})$ would be given by $$\min_{\beta\in\Gamma^{i}}J(\underline{\gamma}^{-i,*},\beta)=J(\underline{\gamma}^{*})\quad i\in\mathcal{N}$$which can be equivalently written as $$\tag{⭐}\min_{u\in S^{i}}E_{\xi|y^{i}}\left[ L(\xi;\underline{\gamma}^{-i,*}(\mathbf{y}^{-i}),u) \right]=E_{\xi|y^{i}}\left[ L(\xi;\underline{\gamma}^{*}(\mathbf{y})) \right],\quad i\in\mathcal{N}$$ We can then see that 🌝 is equivalent to ⭐ if $L(\xi;\mathbf{u})$ is convex in $u^{i}$ and $S^{i}$ are convex sets (as the derivative equalling zero of a convex function means we’ve achieved the minimum).

We can then relax the above conditions to: - $L(\xi;\mathbf{u})$ is continuously differentiable in each agent’s action variable for every $\xi\in\Xi$, and; - $S^{i},i\in\mathcal{N}$, are open subsets of finite-dimensional vector spaces. to then say that 🌝 is a necessary condition for ⭐.

This then leads us to Theorem 2.4.4 where we apply this idea along with Lemma 2.4.1.