Definition. Iso-quality cost ratio [ftip-00JP]
Definition. Iso-quality cost ratio [ftip-00JP]
For a finite real target \(q\in \mathbb R\) and the frontier of Definition [ftip-00JJ], define the inverse cost \[ C_A(q)=\inf \{C\in \mathbb R_{\geq 0}:V_A(C)\geq q\} \in [0,+\infty ], \qquad \inf \varnothing =+\infty . \] The iso-quality ratio is defined only on the domain \[ \rho _{A/B}(q)=\frac {C_A(q)}{C_B(q)}, \qquad 0\leq C_A(q)<+\infty , \quad 0<C_B(q)<+\infty . \] The architectures must use the same target, evaluation interface, cost units, and comparison arm. The ratio compares efficiency under these choices.
An inverse cost is a threshold infimum, not an executable minimum. The infimum over budgets may be unattained; even if a budget satisfies \(V_A(C)\geq q\), its performance supremum may be unattained at \(q\). An actual target-achieving intervention requires a separate witness.
Positive individual costs do not guarantee a positive inverse cost. For interventions \(\eta _n\), \(n\geq 1\), with score \(1\) and cost \(1/n\), the target \(q=1\) has \(C_A(1)=0\), although every intervention costs more than zero. If both architectures have this family, the putative ratio is \(0/0\) and is excluded by the displayed domain.