Definition Monoidal structure on minmax problems [lcc-001N]

There is a monoidal structure on minmax problems, given by (L \otimes L') = (X \otimes X', Y \otimes Y', (x,x',y,y') \mapsto L(x,y) + L(x',y')). The unit here is (*,*,0).

A state is a point x_0 so that L(x_0,y) \leq 0 for all y. More interesting is asking for a state of L \otimes L^*. This is a pair x \in X, y \in Y so that the inequality L(x,y') \leq L(x',y) holds for all y',x'

Note that \sup _{y'} L(x,y') \geq \inf _x L(x',y) for all x,y, this is the minmax inequality (or "weak duality").

Thus a choice of x,y giving a state gives equality in that inequation---it is a solution of the minmax game. In other words, L \otimes L^* has a state if and only if strong duality holds for L, and the state is given by an optimal and dual optimal pair in that case.

(By duality, and since (L \otimes L^*)^* \cong L \otimes L^*, such an object has a state if and only if it has a costate)