Definition (Intrinsic Model)

In a Sequential system let there be $N$ decision makers.

1. Measurable Spaces: We define a collection of measurable spaces $$\{(\Omega,\mathcal{F}),(\mathbb{U}^{i},\mathcal{U}^{i}),(\mathbb{Y}^{i},\mathcal{Y}^{i}),i\in\mathcal{N}\}$$where:

• $(\Omega,\mathcal{F})$ denotes the system’s distinguishable events.
• $(\mathbb{U}^{i},\mathcal{U}^{i})$ denotes the control space for $\text{DM}\ i$.
• $(\mathbb{Y}^{i},\mathcal{Y}^{i})$ denotes the space of observations for $\text{DM} \ i$.

2. Measurement Constraint: Then, there is a measurement constraint which establishes the connection between the observation variables (in $(\mathbb{Y}^{i},\mathcal{Y}^{i})$) and the system’s distinguishable events (in $(\Omega,\mathcal{F})$). Any $\mathbb{Y}^{i}$-valued observation variable is given by $$y^{i}=\eta^{i}(\omega,\mathbf{u}^{[1,i-1]})$$where$$\mathbf{u}^{[1,i-1]}=\{ u^{k},k\le i-1 \}$$where $\eta^{i}$ are measurable functions and $u^{k}$ denotes the action of $\text{DM}\ k$. Hence, the information variable, $y^{i}$, (or information signal) induces a σ-algebra, $\sigma(\mathcal{I}^{i})$ over $\Omega \times \prod_{k=1}^{i-1}\mathbb{U}^{k}$ (where $\mathcal{I}^{i}$ denotes the information available to $\text{DM}\ i$). The collection $$\underline{\eta}:=\{ \eta^{1},\dots,\eta^{N} \}$$is called the Information Structure of the system.
3. Design Constraint: A design constraint which restricts the set of admissible $N-$tuple control policies to the set of all measurable control functions, so that $$u^{i}=\gamma^{i}(y^{i})$$with $$y^{i}=\eta^{i}(\omega,\mathbf{u}^{[1,i-1]})$$with $u^{i},y^{i}$ measurable. Let $\Gamma^{i}$ be the set of all admissible policies for $\text{DM}\ i$ and let $\mathbf{\Gamma}=\prod_{k}\Gamma^{k}$.
4. Probability Measure: A probability measure $P$ defined on $(\Omega,\mathcal{F})$, which describes the measures on the random events of the model.