Fairness With Low Resentment in Distributed Sensor Systems to Detect Emitters

Consider a single distributed sensor system to detect the occurrence of rare emitters in multiple regions, each representing a different community. Alarms are sent to a common dispatch center, which dispatches units to each alarmed community. We assume that all communities contribute equally to the cost of the system; however, the probability of detecting an emitter may vary among communities, raising the issue of fairness. We adopt in here the concept of envy-free fairness in which the goal is to equalize the worst-case probability of detection in each community. As shown in our previous work, envy-free fairness can be achieved by adjusting the probabilities of false alarm at each community. In here, we extend our results by addressing a concern that may arise from envy-free fairness: resentment. After precisely defining the concept of resentment, we show that it is possible to design an envy-free fair detection system while keeping the maximum resentment bounded by combining poorly-served communities with a high enough number of well-served communities. We also present algorithms to allocate sensors to communities to design envy-free fair systems with bounded resentment while considering different optimization goals and constraints. Our examples illustrate that our algorithms often produce close-to-optimum allocations.