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

Abstract

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 $k\ge 2.$