Defining a plan for each player

Each player must now have its own plan. As in Section 10.1, it seems best to define a plan as a mapping from states to actions, because it may not be clear what actions will be taken by the other decision maker. In Section 10.1, the other decision maker was nature, and here it is a rational opponent. Let $\pi_1$ and $\pi_2$ denote plans for ${{\rm P}_1}$ and ${{\rm P}_2}$, respectively. Since the number of stages in Formulation 10.4 is fixed, stage-dependent plans of the form $\pi _1: X \times {\cal K}\rightarrow U$ and $\pi _2: X \times {\cal K}\rightarrow V$ are appropriate (recall that stage-dependent plans were defined in Section 10.1.3). Each produces an action $\pi _1(x,k) \in U(x)$ and $\pi _2(x,k) \in V(x)$, respectively.

Now consider different solution concepts for Formulation 10.4. For ${{\rm P}_1}$, a deterministic plan is a function $\pi _1: X \times {\cal K}\rightarrow U$, that produces an action $u = \pi (x) \in U(x)$, for each state $x \in X$ and stage $k \in {\cal K}$. For ${{\rm P}_2}$ it is instead $\pi _2: X \times {\cal K}\rightarrow V$, which produces an action $v = \pi (x) \in V(x)$, for each $x \in X$ and $k \in {\cal K}$. Now consider defining a randomized plan. Let $W(x)$ and $Z(x)$ denote the sets of all probability distributions over $U(x)$ and $V(x)$, respectively. A randomized plan for ${{\rm P}_1}$ yields some $w \in W(x)$ for each $x \in X$ and $k \in {\cal K}$. Likewise, a randomized plan for ${{\rm P}_2}$ yields some $z \in Z(x)$ for each $x \in X$ and $k \in {\cal K}$.