Fairness Considerations in Network Flow Problems Ermin Wei ∗ , Chaithanya Bandi † Abstract In most of the physical networks, such as power, wa- ter and transportation systems, there is a system-wide ob- jective 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 cor- respond 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 ad- dresses fairness considerations in network flow problems, where we not only care about the standard social welfare maximization , but also distribution of prices. We first inter- pret the network flow problem as an economic market prob- lem. We then show that by tuning a design parameter, we can achieve a spectrum of price-fairness, where the gap be- tween prices satisfy certain design objective. We derive the required physical means to implement the fairness adjust- ment and show that the adjusted optimal solution depends on the original network topology. 1. Introduction Network flow problem naturally emerges from many important applications, such as communication networks, transportation networks and energy supply networks [7], [11], [4], [6], [1]. This problem of optimally routing and distribution the flow to minimize the transportation cost has been well studied. In this work, we look at the economical interpretation of this problem, where prices associated with the flows arise from the optimal solution as a dual variable associated with the flow constraints. In a competitive mar- ket implementation of the network flow problem, the prices ∗ Department of Electrical Engineering and Computer Science, North- western University † Managerial Economics and Decision Sciences, Northwestern Univer- sity at each node correspond to how much each agent should pay/receive for one unit of flow. Due to the cost structure of the arcs, at the optimal solution, a very wide range of optimal prices may arise, resulting in drastically differences from suppliers and consumers as well as among the suppli- ers and consumers. The important aspect of fairness in the system is often neglected in the existing studies. We first present some background on the studies of fairness and then highlight some of our contributions in this work. 1.1. Literature Review The question of fair allocation of resources has been ex- tensively studied through the years in many areas, notably in social sciences, welfare economics, and engineering. A va- riety of ways to measure what is fair have been proposed, and no single principle has come out to be universally ac- cepted. However, there are general theories of justice and equity on which most fairness schemes have been based. In this section, we briefly review the most important theories and define special notions of fairness called proportional and max-min fairness, the two criteria that emerge from these foundational theories, which are also widely used in prac- tice. For more details, see [10] and [5]. Among the most prominent theories of justice is Aris- totle’s equity principle, according to which resources should be allocated in proportion to some preexisting claims, or rights to the resources that each player has. Another theory, widely considered in economics in the 19th century, is clas- sical utilitarianism, which dictates an allocation of resources that maximizes the sum of utilities. A third approach is due to [9], where the key idea is to give priority to the players that are the least well off, so as to guarantee the highest min- imum utility level that every player derives. Finally, Nash introduced the Nash standard of comparison, which is the percentage change in a player’s utility when he receives a small additional amount of the resources. A transfer of re- sources between two players is then justified if the gainer’s 2015 IEEE 54th Annual Conference on Decision and Control (CDC) December 15-18, 2015. Osaka, Japan 978-1-4799-7886-1/15/$31.00 ©2015 IEEE 6909