A version of the Monty Hall problem is presented where the players are permitted to select quantum strategies. If the initial state involves no entanglement the Nash equilibrium in the quantum game offers the players nothing more than can be obtained with a classical mixed strategy. However, if the initial state involves entanglement of the qutrits of the two players, it is advantageous for one player to have access to a quantum strategy while the other does not. Where both players have access to quantum strategies there is no Nash equilibrium in pure strategies, however, there is a Nash equilibrium in quantum mixed strategies that gives the same average payoff as the classical game.

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The application of the methods of quantum mechanics to game theory provides us with the ability to achieve results not otherwise possible. Both linear superpositions of actions and entanglement between the players' moves can be exploited. We provide an introduction to quantum game theory and review the current status of the subject.

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.

Parrondo's Paradox arises when two losing games are combined to produce a winning one. A history dependent quantum Parrondo game is studied where the rotation operators that represent the toss of a classical biased coin are replaced by general SU (2) operators to transform the game into the quantum domain. In the initial state, a superposition of qubits can be used to couple the games and produce interference leading to quite different payoffs to those in the classical case.