Figure 2: Different Benefit vs. Accuracy (top) and Accuracy vs. Cost (middle) curves yield optimal solutions that vary widely in cost and accuracy.

The optimum is defined as the maximum of the Benefit/Cost ratio (red markers). We consider cases where Benefit increases monotonically as a function of the Accuracy, and Accuracy increases monotonically as a function of Cost. There is minimum operating cost, which means that Accuracy vanishes if Cost is less than this minimum. We consider four general scenarios for the two functions: (A) convex, (B) concave, (C) sigmoid, and (E), a common scenario in which there is little Benefit below a threshold Accuracy and maximal Benefit is quickly attained above this threshold, while Accuracy is a concave function of Cost. Concave functions encode a law of diminishing returns – e.g., if Accuracy is a concave function of Cost, then doubling the Cost gives less than double the return in Accuracy. In this scenario, different rates of diminishing returns (different blue lines) give optimal solutions with widely different costs (different red markers) [55].