Fig. 5: A schematic example demonstrating the rationale and workings of fairness-based solutions.

A, B Let us assume that each of three patients has two tumor cells (columns), each displaying five membrane receptors that are highly expressed only on the tumor cells and not on the non-tumor ones (rows). If we target {APP, MET} (panel A, cost unfairness parameter \(\alpha =1\)) in all patients, then this achieves a cohort target set (CTS) size of 2, which is the minimum possible. Employing the original individual-based optimizing objective, each patient could instead be treated by an individual target set (ITS) of size 1 by targeting the distinct receptors called Target 1 (specific to patient 1), Target 2, and Target 3, respectively, but this would result in an optimal CTS of size 3 (panel B, cost unfairness parameter \(\alpha =0\)). The solution in panel A has an unfairness value \(\alpha =1\) because the worst difference among all patients is that a patient receives 1 more treatment than necessary. C Heatmaps showing how the CTS size varies as \(\alpha\) increases (y-axis), starting from its baseline value of 0 where each patient is assigned a minimum-sizes individual treatment set (top row). The lower bound on tumor cells killed (x-axis) is also varied while the upper bound on non-tumor cells killed is kept fixed at 0.1. We are particularly interested in finding the smallest value on the y-axis at which the CTS size reaches its minimum value, which is circled for the baseline \({lb}=0.8,\) because this bounds the tradeoff between the achievable reduction in the number of targets needed to treat the whole cohort and the number of extra targets above the ITS minimum that any patient might need to receive. Source data for panel C are provided as a Source data file.