Fairness Considerations in Network Flow Problems

In most of the physical networks, such as power, water and transportation systems, there is a system-wide objective function, typically social welfare, and an underlying physics constraint governing the flow in the networks. The standard economics and optimization theories suggest that at optimal operating point, the price in the system should correspond to the optimal dual variables associated with those physical constraint. While this set of prices can achieve the best social welfare, they may feature significant differences even for neighboring agents in the system. This work addresses fairness considerations in network flow problems, where we not only care about the standard social welfare maximization, but also distribution of prices. We first interpret the network flow problem as an economic market problem. We then show that by tuning a design parameter, we can achieve a spectrum of price-fairness, where the gap between prices satisfy certain design objective. We derive the required physical means to implement the fairness adjustment and show that the adjusted optimal solution depends on the original network topology.