Is the essence of a quantum game captured completely in the original classical game?

van Enk and Pike (2002), argue that the equilibrium solution to a quantum game is not unique but is already present in the classical game itself. In this work, we contest this assertion by showing that a random strategy in a particular quantum (Hawk–Dove) game is unique to the quantum game. In other words the equilibrium solution of the quantum Hawk–Dove game cannot be obtained in the classical Hawk–Dove game. Moreover, we provide an analytical solution to the quantum 2 × 2 strategic form Hawk–Dove game using random mixed strategies. The random strategies which we describe are Pareto optimal with their payoff’s classically unobtainable. We compare the quantum strategies to correlated strategies and find that correlated strategies in quantum Hawk–Dove game or quantum Prisoner’s dilemma yield the Nash equilibrium solution.