ora-0029

1.16 Infinitesimal intrinsic models

Witsenhausen’s intrinsic model isolates a decision problem by declaring its decision makers, their action spaces, and the information field available to each decision event [ Witsenhausen , 1971 , 1975 ] . A decision law must be measurable with respect to its declared information field. The causal question is consequently structural: can those information-local laws be assembled into a uniquely solvable closed loop? The Universal Decision Model recasts this architecture categorically [ Mahadevan , 2021 ] .

For tangent purposes, a bare sub-\(\sigma \)-algebra is the wrong object to differentiate. Present the information field of agent \(\alpha \) by an observation map

\[ y_\alpha :H\longrightarrow Y_\alpha , \qquad \mathcal I_\alpha =\sigma (y_\alpha ), \qquad H=\Omega \times \prod _{\beta \in A}U_\beta , \]

and factor its decision law as

\[ u_\alpha =\pi _\alpha \bigl(y_\alpha (\omega ,u)\bigr). \]

This presentation exposes maps that can be tangent-lifted while retaining the measurability represented by \(\mathcal I_\alpha \).

Definition 1.28 Fixed-stratum infinitesimal intrinsic model

Let \(\mathfrak {Intr}_{\mathrm{sm}}\) be an internal moduli object of smoothly presented intrinsic models in a setting admitting Weil prolongations. For a Weil algebra \(W\), let \(\mathbb D_W\) denote its infinitesimal probe. A fixed-stratum infinitesimal intrinsic model at \(\mathcal M\) is a map

\[ \widetilde{\mathcal M}:\mathbb D_W \longrightarrow \mathfrak {Intr}_{\mathrm{sm}}, \qquad \widetilde{\mathcal M}(0)=\mathcal M, \]

in which the agent set, action and observation types, information-dependency relation, and causal precedence relation remain fixed. The observation maps, decision mechanisms, exogenous state, and objective may vary through the probe. Equivalently, it is a generalized element of the Weil prolongation \(T^W\mathfrak {Intr}_{\mathrm{sm}}\) lying over \(\mathcal M\).