We define cooperative games on general graphs and generalize Lloyd S. Shapley's celebrated allocation formula for those games in terms of stochastic path integral driven by the associated Markov chain on each graph. We then show that the value allocation operator, one for each player defined by the stochastic path integral, coincides with the player's component game which is the solution to the least squares (or Poisson's) equation, in light of the combinatorial Hodge decomposition on general weighted graphs. Several motivational examples and applications are also presented.
翻译:我们定义一般图表上的合作游戏, 并概括Lloyd S. Shapley 的这些游戏的著名分配公式, 由每个图表上相关的 Markov 链驱动的随机路径集成。 然后我们显示, 价值分配操作器, 由随机路径集成的每个玩家各一个, 与玩家的组合游戏相匹配, 即最小方形( 或 Poisson ) 等式的解决方案, 以组合式 Hodge 对一般加权图形的分解为根据。 还介绍了几个激励性实例和应用 。