We explore the uniqueness of pure strategy Nash equilibria in the Netflix Games of Gerke et al. (arXiv:1905.01693, 2019). Let $G=(V,E)$ be a graph and $\kappa:\ V\to \mathbb{Z}_{\ge 0}$ a function, and call the pair $(G, \kappa)$ a capacitated graph... A spanning subgraph $H$ of $(G, \kappa)$ is called a $DP$-Nash subgraph if $H$ is bipartite with partite sets $X,Y$ called the $D$-set and $P$-set of $H$, respectively, such that no vertex of $P$ is isolated and for every $x\in X,$ $d_H(x)=\min\{d_G(x),\kappa(x)\}.$ We prove that whether $(G,\kappa)$ has a unique $DP$-Nash subgraph can be decided in polynomial time. We also show that when $\kappa(v)=k$ 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 $k\ge2.$ (read more)