Theorem (Weierstrass)

The team problem $\{ J;\mathit\Gamma^{i},i\in\mathcal{N} \}$ admits a team-optimal solution if:

1. The product policy space $\mathbf{\Gamma}$ is a compact subset of a metric space;
2. The cost function $J$ is lower semicontinuous (lsc) on $\mathbf{\Gamma}$ (with respect to the given metric).