Example. Finite observations need not identify a frontier [ftip-00LN]
Example. Finite observations need not identify a frontier [ftip-00LN]
Let \(S\subset \mathbb R_{\geq 0}\) be a finite set of measured costs, and choose \(C_0>0\) larger than every element of \(S\). Suppose the exact frontier value observed at each cost in \(S\) is zero. For finite budgets \(C\geq 0\), suppose the declared class of possible frontiers permits both \[ V_0(C)=0, \qquad V_1(C)= \begin {cases} 0,&0\leq C<C_0,\\ 1,&C\geq C_0. \end {cases} \] These nondecreasing frontiers with scores in \([0,1]\) agree on every observed cost and differ at \(C_0\). The observations alone do not distinguish them.
Both possibilities have finite realizations in the framework of Definition [ftip-00JJ]: allow two interventions with costs \(0\) and \(C_0\). Give the first score zero and the second score \(\theta \in \{0,1\}\). For example, on a single deterministic evaluation task with utility equal to the output bit, let the two resulting protocols return \(0\) and \(\theta \). The two possible choices of \(\theta \) give \(V_0\) and \(V_1\), respectively, under the same intervention and cost specification. Every score is finite, and the feasible score maximum is attained at each budget.
This example does not supply a strictly better agreeing frontier for every possible data set or every admissible class. If a proved global score bound is \(1\) and an intervention attains it at a finite cost \(C_*\geq 0\), monotonicity forces \(V_A(C)=1\) for every \(C\geq C_*\); no higher value is admissible. A singleton class of possible frontiers can also identify the frontier without such an alternative. A conditional certificate such as Theorem [ftip-00LM] therefore requires its stated upper-bound evidence; that evidence does not follow merely from the absence of an observed improvement.