LP-duality theory has played a central role in the study of cores of games, right from the early days of this notion to the present time. The classic paper of Shapley and Shubik \cite{Shapley1971assignment} introduced the "right" way of exploiting the power of this theory, namely picking problems whose LP-relaxations support polyhedra having integral vertices. So far, the latter fact was established by showing that the constraint matrix of the underlying linear system is {\em totally unimodular}. We attempt to take this methodology to its logical next step -- {\em using total dual integrality} -- thereby addressing new classes of games which have their origins in two major theories within combinatorial optimization, namely perfect graphs and polymatroids. In the former, we address the stable set and clique games and in the latter, we address the matroid independent set game. For each of these games, we prove that the set of core imputations is precisely the set of optimal solutions to the dual LPs. Another novelty is the way the worth of the game is allocated among sub-coalitions. Previous works follow the {\em bottom-up process} of allocating to individual agents; the allocation to a sub-coalition is simply the sum of the allocations to its agents. The {\em natural process for our games is top-down}. The optimal dual allocates to "objects" in the grand coalition; a sub-coalition inherits the allocation of each object with which it has non-empty intersection.
翻译:LP- 质量理论在游戏核心研究中发挥了核心作用, 从这个概念的早期到现在。 经典的 Shapley 和 Shubik 和 Shubik\ cite {Shapley1971traction} 的论文引入了利用这一理论力量的“ 右” 方法, 即选择LP- Relax 支持多环形组合曲的难题。 到目前为止, 后一事实是通过显示基础线性系统的制约矩阵是完全单一的} 。 我们试图将这一方法带到其逻辑的下一个步骤 -- -- 使用完全的双重整体性游戏联盟, 从而解决新一轮游戏的起源于组合优化中的两种主要理论, 即完美的图表和多类机器人。 在前者, 我们处理的是稳定的组合和球形游戏, 在后一种独立的游戏中, 我们通过这些游戏, 我们证明核心的内置精度是双双双轨的优化解决方案的下方。 另一种新颖的办法是将游戏的顶级分配方式分配到下层的下方。 。 最优的双向的下层分配过程是“ ” 。 它的尾的尾的尾的尾的尾的尾分配过程是排序。 它的尾部的尾部的尾部的尾部的尾部, 。