Definition. Costed post-training potential [ftip-005M]
Definition. Costed post-training potential [ftip-005M]
For a scalar evaluation utility, the costed post-training potential of \(M_0\) under a finite componentwise budget \(\mathbf B\in \mathbb R_+^7\) is the extended-real supremum
\[ \Phi (M_0,\mathbf B) =\sup _{P\in \mathfrak P(M_0)} \left \{ J_{\rm ev}(P): \mathsf {Trust}(P),\ \mathbb E[C(P)]\leq \mathbf B \right \}. \]Here \(\mathfrak P(M_0)\) has the evaluation domain of Definition [ftip-005J], so each \(J_{\rm ev}(P)\) is real. The expected-cost inequality is componentwise in \([0,+\infty ]^7\); it cannot hold against a finite budget if any coordinate is infinite. Define \(\sup \varnothing =-\infty \), and use \(+\infty \) when feasible performance values are unbounded above. Thus \(\Phi \) takes values in \(\overline {\mathbb R}=\mathbb R\cup \{-\infty ,+\infty \}\).
The potential is finite real exactly when its feasible performance set is nonempty and bounded above. For example, a common finite upper bound on utility and at least one feasible protocol suffice. This does not guarantee that any protocol attains the supremum. Ordinary differences, ratios, or derivatives of potential values require finite real operands and any further regularity hypotheses needed by that operation. The value is conditional on every object named in the display; it is not an intrinsic constant of the model.