The classic paper of Shapley and Shubik \cite{Shapley1971assignment} characterized the core of the assignment game using ideas from matching theory and LP-duality theory and their highly non-trivial interplay. Whereas the core of this game is always non-empty, that of the general graph matching game can be empty. This paper salvages the situation by giving an imputation in the $2/3$-approximate core for the latter. This bound is best possible, since it is the integrality gap of the natural underlying LP. Our profit allocation method goes further: the multiplier on the profit of an agent is often better than ${2 \over 3}$ and lies in the interval $[{2 \over 3}, 1]$, depending on how severely constrained the agent is. Next, we provide new insights showing how discerning core imputations of an assignment games are by studying them via the lens of complementary slackness. We present a relationship between the competitiveness of individuals and teams of agents and the amount of profit they accrue in imputations that lie in the core, where by {\em competitiveness} we mean whether an individual or a team is matched in every/some/no maximum matching. This also sheds light on the phenomenon of degeneracy in assignment games, i.e., when the maximum weight matching 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. Our imputation, in the $2/3$-approximate core, implies that a sub-coalition will gain at most a $3/2$ factor by seceding, and less in typical cases.
翻译:经典的 Shapley 和 Shubik 和 Shubik 和 Shubik 的 Shapley 和 Shapley 1971traction} 的 经典 论文, 利用来自匹配理论和 LP- 质量理论及其高度非三重互动的理念, 确定了任务游戏的核心。 虽然游戏的核心总是非空的, 普通图形匹配游戏的核心可能是空的。 本文通过在2/3美元与后者的近似核心中给出一个估算来挽救情况。 这是最好的选择, 因为它是自然基础LP 的整体性差距。 我们的利润分配方法更进一步: 代理人利润的乘数通常比${ 2\ 超过 3} 3 美元, 并且在于 $ $2{ 2\ over 3} 和 1] 之间的间隔, 取决于该游戏的游戏的核心约束程度。 接下来, 我们提供新的洞察力显示, 如何通过补差的透镜来研究一个任务的核心因素。 我们展示的是, 个人和整个代理人的竞争力和团队的竞争力之间的竞争力和利润是多少, 。 。 以正态的数值中的 。 。 在每一个的数值中, 的数值中, 的数值中, 的数值中, 将意味着, 每一个的数值中, 以每一体的数值中, 的数值中, 的数值中, 将意味着, 的数值中, 的数值中, 的数值中, 。