事后有效随机分配的结构和复杂性
Structure and complexity of ex post efficient random assignments
摘要：
在随机分配问题中，对象被随机分配给代理，同时考虑代理对对象的偏好。随机分配指定代理获得对象的概率。我们研究随机分配的事后效率的结构和计算方面。我们首先表明，而事后有效的分配可以很容易地计算，检查是否一个给定的随机分配是事后有效的是NP完全的。因此，通过确定性帕累托最优分配来实现给定的随机分配是NP难的。然后，我们形式化的另一个概念的效率称为鲁棒事后效率，是弱于随机优势效率，但比事后效率强。我们提出了一个强大的事后效率的表征，并表明，它可以在多项式时间内进行测试，如果有一个常数的代理类型。结果表明，著名的随机序列独裁规则是不鲁棒事后有效的。最后，我们表明，而强大的事后效率仅取决于哪些条目的分配矩阵是零/非零，事后效率的分配取决于实际值。
英文摘要：
In the random assignment problem, objects are randomly assigned to agents keeping in view the agents' preferences over objects. A random assignment specifies the probability of an agent getting an object. We examine the structural and computational aspects of ex post efficiency of random assignments. We first show that whereas an ex post efficient assignment can be computed easily, checking whether a given random assignment is ex post efficient is NP-complete. Hence implementing a given random assignment via deterministic Pareto optimal assignments is NP-hard. We then formalize another concept of efficiency called robust ex post efficiency that is weaker than stochastic dominance efficiency but stronger than ex post efficiency. We present a characterization of robust ex post efficiency and show that it can be tested in polynomial time if there are a constant number of agent types. It is shown that the well-known random serial dictatorship rule is not robust ex post efficient. Finally, we show that whereas robust ex post efficiency depends solely on which entries of the assignment matrix are zero/non-zero, ex post efficiency of an assignment depends on the actual values.
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