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. The set of Nash equilibria for a game with generic payoff functions is finite [2]. This implies that the set of totally mixed Nash equilibria for a game with generic payoff functions is also finite. These are the real solutions to a system of polynomial equations and inequalities. The complex solutions to the system of equations are called quasiequilibria. Thus, the set of totally mixed Nash equilibria is a subset of the set of quasiequilibria. In fact, the set of quasiequilibria is also finite in the most generic case, and its cardinality can be computed as a function of the numbers of pure strategies of the players. Thus, this is an upper bound on the number of totally mixed Nash equilibria. Even in a nongeneric case, as long as the set of quasiequilibria is finite, its cardinality will be bounded above by the number in the generic case. For the main theorem of this article, Theorem 1, we hypothesize a set of technical conditions that a system of polynomial equations may satisfy, which are encoded in an associated graph, the polynomial graph, and we prove a formula describing the number of solutions in this case. We then show how to associate such a graph to three special classes of games. The first two are graphical games and extensive-form games. The last is games with emergent node tree structure, a new model for games in which the players can be hierarchically decomposed into groups. Usually such hierarchical decomposition is modelled by cooperative games, and we briefly discuss how our model is related to, yet differs from, the cooperative framework. 1. GENERIC NUMBER OF ROOTS OF A SPARSE POLYNOMIAL SYSTEM The following theorem tells us the number of 0-dimensional complex roots (none of whose components are zero) of a system of polynomial equations which obeys certain sparsity conditions and is otherwise generic. Our formulation of this theorem is motivated by the applications to game theory which follow, although such polynomial systems may arise in other contexts. Theorem 1. Suppose that 0 < d ∈ N and that we are given a partition {1, . . . , d} = ∐N i=1 Ti of {1, . . . , d}. Write di = |Ti |. Suppose further that we are given a directed graph G, the Date: December 15, 2006.