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：分配博弈模拟了一个买卖双方进行匹配的住房市场，其中交易价格的设定使得最终分配达到稳定。Shapley 和 Shubik 证明，任何稳定分配必然建立在最大化社会福利的匹配之上。然而在实践中，稳定分配很少能够实现，因为匹配往往是次优的，尤其是在智能体顺序到达市场的在线场景中。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，建立了它们与底层匹配最优比率之间的联系，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。