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

S. J. van Enk and R. Pike in PRA 66, 024306 (2002) argue that the equilibrium solution to a quantum game isn’t 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, one cannot obtain the equilibrium solution of the quantum Hawk-Dove game in the classical Hawk-Dove game. Moreover, we provide an analytical solution to the quantum $2\times 2$ strategic form Hawk-Dove game using randomly mixed strategies. The random strategy which we describe is Pareto optimal with their payoff classically unobtainable. We compare quantum strategies to correlated strategies and find that correlated strategies in the quantum Hawk-Dove game or quantum Prisoner’s dilemma yield the Nash equilibrium solution.