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 ⭐.