Uniqueness of DP-Nash Subgraphs and D-sets in Weighted Graphs of Netflix Games.

Gerke et al. (2019) introduced Netflix Games and proved that every such game has a pure strategy Nash equilibrium. In this paper, we explore the uniqueness of pure strategy Nash equilibria in Netflix Games. Let \\(G=(V,E)\\) be a graph and \\(\\kappa :\\ V\\rightarrow \\mathbb {Z}_{\\ge 0}\\) a function, and call the pair \\((G, \\kappa )\\) a weighted graph. A spanning subgraph H of \\((G, \\kappa )\\) is called a DP-Nash subgraph if H is bipartite with partite sets D, P called the D-set and P-set of H, respectively, such that no vertex of P is isolated and for every \\(x\\in D,\\) \\(d_H(x)=\\min \\{d_G(x),\\kappa (x)\\}.\\) We prove that whether \\((G,\\kappa )\\) has a unique DP-Nash subgraph can be decided in polynomial time. We also show that when \\(\\kappa (v)=k\\in \\mathbb {Z}_{\\ge 0}\\) for every \\(v\\in V\\), the problem of deciding whether \\((G,\\kappa )\\) has a unique D-set is polynomial time solvable for \\(k=0\\) and 1, and co-NP-complete for \\(k\\ge 2.\\)