In this paper we study one of the most well known examples of evolutionary games, the Hawk and Dove problem, in the dynamic framework of Markov Decision Evolutionary Games. We associate with each player an extra individual state depending on the age and on the strength of the individual. This state may change as a function of the actions taken by those it encounters. The goal of a player is to maximize the expected sum of its immediate tness during his life time. We identify a unique stationary equilibrium in the game and compute its value. We further extend the Hawk and Dove game by introducing Group Markov Decision Evolutionary Game Theory (GMDEG), in which a player does not necessarily represent a single interacting individual but a whole class of such individuals. The tness to be maximized is the one of the group: this approach shows novel results on the structure of the equilibria and moreover the non uniqueness of the equilibria.