Theorem

We have that the policies have the following relationships $$\Gamma_{S}\subset\Gamma_{M}\subset\Gamma_{A}$$and so we also have that $\forall \{ N,\beta,\infty \}$ $$\inf_{\gamma\in\Gamma_{A}}J_{a}(X,\gamma)\subset\inf_{\gamma\in\Gamma_{M}}J_{a}(X,\gamma)\subset\inf_{\gamma\in\Gamma_{S}}J_{a}(X,\gamma)$$also for $a\in\{ N,\beta,\infty \}$ the first is an equality and for $a\in\{ \beta,\infty \}$ the second is an equality.