1. State Space¶

Consider the game state space as $D := \Delta (\mathcal{Z})$ , the set of all probability measures on the finite individual state-space $\mathcal{Z} := \{-N,...,-1,+1,...,+M\}$ , where $0<N,M <+\infty$ . Let $N_{z}:=M+N \equiv |\mathcal{Z}|$ .

1.1. Properties¶

The set $D$ is:

is a unit simplex embedded in $\mathbb{R}^{N_{z}}$ :

$\begin{equation*} \Delta (\mathcal{Z}) := \left\{ \lambda \in \mathbb{R}^{N_{z}}: \lambda_{i} \in [0,1], \forall i = 1,...,N_{z}, \text{ and } \sum_{i=1}^{N_{z}} \lambda_{i} = 1 \right\} \end{equation*}$

represented by a convex polytope (i.e. a unit $N_{z}$ -simplex);
partitioned into $K < +\infty$ equal-area $(N_{z}-1)$ -simplices, $Q_{k}, k \in \{ 1,...,K \} =: \mathbf{K}$ .