A game-theoretic approach to dynamic boundary problems for level-set curvature flow equations and applications

This paper is devoted to a game-theoretic approach to the level-set curvature flow equation with nonlinear dynamic boundary conditions. Under the comparison principle for the dynamic boundary problem, we construct a family of deterministic discrete games, whose value functions approximate the unique viscosity solution. We also apply the game approximation to study the convexity preserving properties and the fattening phenomenon for this geometric dynamic boundary problem.

Imposing equilibrium restrictions in the estimation of dynamic discrete games

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 is a fixed‐point iterative procedure, which avoids the computation of these matrices, but—in games—may fail to converge to the consistent NPL estimator. In order to better capture the effect of iterating the NPL algorithm in finite samples, we study the asymptotic properties of this algorithm for data generating processes that are in a neighborhood of the NPL fixed‐point stability threshold. 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 and an empirical application illustrating our theoretical results and the good properties of the spectral algorithm.