Stop and play models with input-dependent shape functions are equivalent to the nonlinear Preisach model proposed by Mayergoyz. This equivalence engenders the necessary and sufficient conditions for a hysteretic nonlinearity to be represented by stop and play models that have input-dependent shape functions. A representation theorem for the stop model having an input-independent shape function is also given. The latter theorem is; used to improve the representation accuracy of the stop model for a silicon steel sheet.