we consider the mixed-strategy zero-sum game such that each player’s objective function is quadratic in its own variables. by considering each player’s value function and duality, the bi-quadratic games are reformulated as linear programs over the cone of copositive (cop) and completely positive (cp) matrices. we apply moment and sos relaxations for the conic constraints of cp and cop matrices, respectively, and obtain a hierarchy of semidefinite relaxations. under certain conditions, the finite convergence for this hierarchy is guaranteed, and the tightness can be checked via flat truncation. we present numerical experiments to show the effectiveness of our approach.