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). As such, we embed the black-box procedure of solving an integer linear program into a framework that is explainable from start to finish. Moreover, we study the axiomatic properties of the proposed methods by embedding our framework into the rich literature of cooperative bargaining and probabilistic social choice. Lastly, we evaluate the proposed methods on a specific application, namely kidney exchange. 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.

翻译：仅通过整数线性规划无法实现真正公平的决策，除非能控制（可能多个）最优解的选择概率。为此，我们针对二元决策变量代表具有对分偏好（仅关心自身是否被选入最终方案）的决策主体这一场景，提出统一框架。我们开发了若干通用算法以实现最优解的公平选取，例如：最大化纳什积或最小选择概率，或采用决策主体的随机排序作为选择准则（随机序列独裁）。通过这种方式，我们将求解整数线性规划的"黑箱"过程嵌入一个全程可解释的框架中。此外，通过将该框架嵌入合作博弈与概率社会选择领域的丰富文献体系，我们研究了所提方法的公理化特性。最后，我们在具体应用场景——肾脏交换——中对所提方法进行评估。结果表明：虽然最大化纳什积或最小选择概率的方法在评估的福利准则上优于其他方法，但随机序列独裁等方法在计算时间上表现合理，其耗时与求解单个最优解相当。