A new approach to modeling pedestrian's avoidance dynamics based on a Fokker-Planck Nash game framework is presented. In this framework, two interacting pedestrian are considered, whose motion variability is modeled through the corresponding probability density functions (PDFs) governed by Fokker-Planck equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. Existence of Nash equlibria solution is proved and characterized as solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to output of real experiments (conducted with humans) for 4 test cases.