The known Hawk and Dove game is analyzed from quantum mechanics with another two possible behaviors, Bully and Retaliator. The formalism used in the development of the strategies is not Dirac's due to its complex implementation in Computer Algebra System (CAS) but the matrix analysis is proposed. Both are completely alike in the results given, so the matrix method used is not less efficient than Dirac's. The classical game with the four strategies is also described and compared. As results of the quantum game, are presented pay-offs matrixes for possible players, the density matrix and its relation to quantum information and communication. Applications such as finances and biology are also presented and proposed.