Fair integer programming under dichotomous and cardinal preferences
Tom Demeulemeester, Dries Goossens, Ben Hermans, Roel Leus
TL;DR
This work introduces a unified framework for fair integer programming where binary decisions encode agent dichotomies, focusing on controlling selection probabilities across the set of optimal ILP solutions rather than relying on a single deterministic optimum. It connects to probabilistic social choice and cooperative bargaining, and develops distribution rules (Uniform, Leximin, Custom Criteria, Random Serial Dictatorship) with column-generation-based implementations to avoid full enumeration. The approach is illustrated on kidney exchange (dichotomous) and a single-machine tardiness scheduling problem (cardinal), revealing that Nash-product and minimum-probability rules often excel on welfare criteria, while Random Serial Dictatorship provides favorable computation times. The framework supports extensions to near-optimal solutions and cardinal preferences, offering a transparent, explainable alternative to black-box ILP solvers for high-stakes decisions.
Abstract
One cannot make truly fair decisions using integer linear programs unless one controls the selection probabilities of the (possibly many) optimal solutions. For this purpose, we propose a unified framework when binary decision variables represent agents with dichotomous preferences, who only care about whether they are selected in the final solution. We develop several general-purpose algorithms to fairly select optimal solutions, for example, by maximizing the Nash product or the minimum selection probability, or by using a random ordering of the agents as a selection criterion (Random Serial Dictatorship). We also discuss in detail how to extend the proposed methods when agents have cardinal preferences. As such, we embed the black-box procedure of solving an integer linear program into a framework that is explainable from start to finish. Lastly, we evaluate the proposed methods on two specific applications, namely kidney exchange (dichotomous preferences), and the scheduling problem of minimizing total tardiness on a single machine (cardinal preferences). We find that while the methods maximizing the Nash product or the minimum selection probability outperform the other methods on the evaluated welfare criteria, methods such as Random Serial Dictatorship perform reasonably well in computation times that are similar to those of finding a single optimal solution.