We study game-theoretically secure protocols for the classical ordinal assignment problem (aka matching with one-sided preference), in which each player has a total preference order on items. To achieve the fairness notion of equal treatment of equals, conventionally the randomness necessary to resolve conflicts between players is assumed to be generated by some trusted authority. However, in a distributed setting, the mutually untrusted players are responsible for generating the randomness themselves. In addition to standard desirable properties such as fairness and Pareto-efficiency, we investigate the game-theoretic notion of maximin security, which guarantees that an honest player following a protocol will not be harmed even if corrupted players deviate from the protocol. Our main contribution is an impossibility result that shows no maximin secure protocol can achieve both fairness and ordinal efficiency. Specifically, this implies that the well-known probabilistic serial (PS) mechanism by Bogomolnaia and Moulin cannot be realized by any maximin secure protocol. On the other hand, we give a maximin secure protocol that achieves fairness and stability (aka ex-post Pareto-efficiency). Moreover, inspired by the PS mechanism, we show that a variant known as the OnlinePSVar (varying rates) protocol can achieve fairness, stability and uniform dominance, which means that an honest player is guaranteed to receive an item distribution that is at least as good as a uniformly random item. In some sense, this is the best one can hope for in the case when all players have the same preference order.

翻译：我们研究了经典序数分配问题（即单边偏好匹配）中的博弈论安全协议，其中每个参与者对物品具有完全偏好序。为了实现“同等对待同等者”的公平性概念，传统上假设解决参与者之间冲突所需的随机性由可信权威机构生成。然而在分布式环境中，互不信任的参与者需要自行生成随机性。除了公平性和帕累托效率等标准期望性质外，我们研究了博弈论中的极大极小安全性概念，该概念保证遵循协议的诚实参与者即使在腐败参与者偏离协议的情况下也不会受到损害。我们的主要贡献是一个不可能性结果：证明任何满足极大极小安全性的协议都无法同时实现公平性和序数效率。具体而言，这表明Bogomolnaia和Moulin提出的著名概率序列（PS）机制无法通过任何极大极小安全协议实现。另一方面，我们给出了一个能同时实现公平性和稳定性（即事后帕累托效率）的极大极小安全协议。此外，受PS机制启发，我们证明了一种称为OnlinePSVar（变速率）的变体协议能够同时实现公平性、稳定性和均匀优势，这意味着诚实参与者保证获得的物品分配至少不低于均匀随机分配的结果。在某种意义上，这是所有参与者具有相同偏好序情况下所能期待的最优结果。