The assignment game models a housing market where buyers and sellers are matched, and transaction prices are set so that the resulting allocation is stable. Shapley and Shubik showed that every stable allocation is necessarily built on a maximum social welfare matching. In practice, however, stable allocations are rarely attainable, as matchings are often sub-optimal, particularly in online settings where eagents arrive sequentially to the market. In this paper, we introduce and compare two complementary measures of instability for allocations with sub-optimal matchings, establish their connections to the optimality ratio of the underlying matching, and use this framework to study the stability performances of randomized algorithms in online assignment games.

翻译：分配博弈模拟了买方与卖方匹配的住房市场，其中交易价格的设定使得最终分配达到稳定。Shapley与Shubik证明，所有稳定分配必然建立在最大化社会福利匹配的基础上。然而在实践中，稳定分配往往难以实现，因为匹配常处于次优状态，尤其在智能体顺序到达市场的在线场景中更为显著。本文针对次优匹配下的分配，提出并比较了两种互补的不稳定性度量方法，建立了它们与底层匹配最优比率之间的关联，并利用该框架研究了在线分配博弈中随机算法的稳定性表现。