Theorem 2.4.5

Let, - $\{ J;\Gamma^{i},i\in\mathcal{N} \}$ be a static stochastic team problem where $\mathbb{U}^{i}\equiv \mathbb{R}^{m_{i}},i\in\mathcal{N}$ (i.e. uncountable); - the loss function $L(\xi,\mathbf{u})$ is convex and continuously differentiable in $\mathbf{u}$ a.s.; - $J(\underline{\gamma})$ is bounded from below on $\mathbf{\Gamma}$; - $\underline{\gamma}^{*}$ be a policy $N$-tuple with a finite cost and suppose that for every $\underline{\gamma}\in\mathbf{\Gamma}$ s.t. $J(\underline{\gamma})<\infty$, $$\tag{⭐}\sum_{i\in\mathcal{N}}E\left[\nabla_{u^{i}}L(\xi;\underline{\gamma}^{*}(\mathbf{y}))[\gamma^{i}(y^{i})-\gamma^{i*}(y^{i})]\right]\ge 0$$where $\nabla_{u^{i}}L(\xi;\gamma^{*}(\mathbf{y}))$ stands for the partial derivatives under the policy $\underline{\gamma}^{*}$. Then, $\underline{\gamma}^{*}$ is a team-optimal policy, and it is unique if $L$ is strictly convex in $\mathbf{u}$.

Remark

As noted in the textbook, this theorem arises due to us now considering uncountable (but still finite dimensional) measurement spaces. This causes hypothesis one, $(c.1)$, to no longer imply our policy space $\mathbf{\Gamma}$ is compact.