lin-0095

7.2 A tangent site for decisions

Let \(\Theta \) be a space of admissible decision models and suppose the local data vary smoothly:

\[ \overline F:\Theta \times \mathcal O\longrightarrow \mathcal E, \qquad F_\theta =\overline F(\theta ,-). \]

The parameter \(\theta \) may encode transition laws, preferences, payoff tables, constraints, forecasts, model parameters, or a mixture of these. For an admitted probe \(v\in T_\theta \Theta \), write

\[ \dot F_v = D_\theta \overline F(\theta ,-)[v]. \]

The induced global decision semantics is

\[ U(\theta ) = \operatorname {Ran}_J \bigl(J^\ast \operatorname {Lan}_JF_\theta \bigr). \]

Definition 7.1 Infinitesimal decision

When the displayed derivative exists, the infinitesimal decision in direction \(v\) is the typed pair

\[ \bigl(U(\theta ),\dot U_v\bigr), \qquad \dot U_v = D_\theta U(\theta )[v]. \]

It records the base decision semantics together with its first-order response to the declared perturbation.

The definition does not identify \(v\) with an action. A direction can change the evidence used by the learner, its local model, a constraint, or the decision context itself. Calling the response a decision derivative is justified only after the probe language says which of these meanings is intended.

Probe	What varies	Tangent question	Additional semantics needed
Evidence probe	local observations or forecasts	how the decision changes with available evidence	sampling and observation model
Preference probe	reward, utility, or ordering	how tradeoffs change the selected behavior	elicitation and scale conventions
Constraint probe	feasible set or continuation test	which constraints become active	constraint qualification
Mechanism probe	transition or structural law	how behavior responds to a changed mechanism	causal or dynamical grounding
Context probe	state, history, information field, or game position	how nearby contexts alter the decision	geometry of the context space

Table 7.1 Infinitesimal decisions are typed by the probes that generate them.