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.

中文翻译：

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二分法和基数首选项下的公平整数规划


除非控制（可能许多）最优解的选择概率，否则使用整数线性规划无法做出真正公平的决策。为此，我们提出了一个统一的框架，当二元决策变量代表具有二分偏好的代理时，他们只关心他们是否在最终解决方案中被选中。我们开发了几种通用算法来公平地选择最优解决方案，例如，通过最大化 Nash 乘积或最小选择概率，或者使用代理的随机排序作为选择标准 （Random Serial Dictatorship）。我们还详细讨论了当代理具有基本偏好时如何扩展所建议的方法。因此，我们将求解整数线性规划的 “黑盒” 过程嵌入到一个从头到尾都可以解释的框架中。最后，我们在两个特定应用上评估了所提出的方法，即肾脏交换（二分偏好）和最小化单台机器上总迟到的调度问题（基数偏好）。我们发现，虽然最大化纳什乘积或最小选择概率的方法在评估的福利标准上优于其他方法，但随机序列独裁等方法在计算时间内表现得相当好，这与寻找单个最优解的计算时间相似。