Advantage of a quantum player over a classical one in 2 x 2 quantum games

We study a general 2×2 symmetric, entangled, quantum game. When one player has access only to classical strategies while the other can use the full range of quantum strategies, there are ‘miracle’ moves available to the quantum player that can direct the result of the game towards the quantum player’s preferred result regardless of the classical player’s strategy. The advantage pertaining to the quantum player is dependent on the degree of entanglement. Below a critical level, dependent on the payoffs in the game, the miracle move is of no advantage.