The classic paper of Shapley and Shubik \cite{Shapley1971assignment} showed that the set of imputations in the core of the assignment game is precisely the set of optimal solutions to the dual of the LP-relaxation of this game. Since the worth of this game is given by an optimal solution to the primal LP, this naturally raises the question of studying core imputations through the lens of complementarity. Our exploration yields new insights: we obtain a relationship between the competitiveness of individuals and teams of agents and the amount of profit they accrue. This also sheds light on the phenomenon of degeneracy in assignment games, i.e., when the optimal assignment is not unique. The core is a quintessential solution concept in cooperative game theory. It contains all ways of distributing the total worth of a game among agents in such a way that no sub-coalition has incentive to secede from the grand coalition.

翻译：Shapley 和 Shubik 和 Shubik 和 Shubik 和 Shubik 和 Shapley{Shapley1971traction} 的经典论文显示,分配游戏核心部分的一套估算值恰恰是LP 放松这一游戏双重功能的最佳解决方案组合。由于这一游戏的价值是由原始LP 的最佳解决方案提供的,这自然提出了通过互补透镜来研究核心估算值的问题。我们的探索产生了新的洞察力:我们获得了个人和代理人团队的竞争力和他们积累的利润数量之间的关系。这也揭示了指派游戏中的退化现象,即最佳分配不是独一无二的。核心部分是合作游戏理论中的典型解决方案概念。它包含了在代理人之间分配游戏总价值的所有方法,其方式是没有子联盟的动力。