Polynomial Graphs With Applications To Graphical Games, Extensive-Form Games, and Games With Emergent Node Tree Structures

We prove a theorem computing the number of solutions to a system of equations which is generic subject to the sparsity conditions embodied in a graph. We apply this theorem to games obeying graphical models and to extensive-form games. We define emergent-node tree structures as additional structures which normal form games may have. We apply our theorem to games having such structures. We briefly discuss how emergent node tree structures relate to cooperative games.