This paper develops a dynamic programming method when the one-stage deviation principle in the sense of mechanism design literature doesn’t hold. The commonly used dynamic programming method is valid only if the one-stage deviation principle in the sense of mechanism design literature is satisfied; it doesn't hold in every model, and the one-stage deviation principle in the sense of repeated games does hold but requires the equilibrium strategy of every player off the equilibrium path and is impractical. The dynamic programming method developed in this paper requires transfinite induction, and therefore one needs to specify the stopping times for two dimensions.