lin-0098

7.5 Decision-relevant quotients

Many tangent changes do not affect the decision. Adding a common baseline to all action values, translating every feasible cost by the same constant, or reparameterizing an internal strategy may change coordinates while preserving behavior.

Let

\[ \rho _\theta :U(\theta )\longrightarrow A_\theta \]

be the declared decision readout into an action, policy, ordering, feasible choice, or behavioral equivalence class. The decision-null directions are

\[ N_\theta = \ker \bigl(T\rho _\theta :T U(\theta )\to T A_\theta \bigr), \]

whenever this kernel is defined. The decision tangent object is then modeled by the quotient

\[ T_{\mathrm{dec}}U(\theta ) = T U(\theta )/N_\theta . \]

Design principle

Differentiate the universal decision semantics, then quotient by variations that the declared decision rule cannot observe. Coordinate sensitivity is not yet decision sensitivity.

The quotient is application-dependent. In reinforcement learning it may remove action-common value shifts. In a game it may remove strategically equivalent mixed-strategy representations. In constrained optimization it may identify parameter changes that leave the active solution set invariant. No universal quotient can be chosen before the behavioral readout is stated.