Imposing equilibrium restrictions provides substantial gains in the estimation of dynamic discrete games. Estimation algorithms imposing these restrictions have different merits and limitations. Algorithms that guarantee local convergence typically require the approximation of high-dimensional Jacobians. Alternatively, the Nested Pseudo Likelihood (NPL) algorithm avoids the computation of these matrices, but – in games – may fail to converge to the consistent NPL estimator. We study the asymptotic properties of the NPL algorithm treating the iterative procedure as performed in finite samples. We find that there are always samples for which the algorithm fails to converge, and this introduces a selection bias. We also propose a spectral algorithm to compute the NPL estimator. This algorithm satisfies local convergence and avoids the approximation of Jacobian matrices. We present simulation evidence illustrating our theoretical results and the good properties of the spectral algorithm.