Quantum game theory
Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:
- Superposed initial states,
- Quantum entanglement of initial states,
- Superposition of strategies to be used on the initial states.
This theory is based on the physics of information much like quantum computing.
Superposed initial states
The information transfer that occurs during a game can be viewed as a physical process. In the simplest case of a classical game between two players with two strategies each, both the players can use a bit (a '0' or a '1') to convey their choice of strategy. A popular example of such a game is the prisoners' dilemma, where each of the convicts can either cooperate or defect: withholding knowledge or revealing that the other committed the crime. In the quantum version of the game, the bit is replaced by the qubit, which is a quantum superposition of two or more base states. In the case of a two-strategy game this can be physically implemented by the use of an entity like the electron which has a superposed spin state, with the base states being +1/2 (plus half) and −1/2 (minus half). Each of the spin states can be used to represent each of the two strategies available to the players. When a measurement is made on the electron, it collapses to one of the base states, thus conveying the strategy used by the player.
Entangled initial states
The set of qubits which are initially provided to each of the players (to be used to convey their choice of strategy) may be entangled. For instance, an entangled pair of qubits implies that an operation performed on one of the qubits, affects the other qubit as well, thus altering the expected pay-offs of the game.
Superposition of strategies to be used on initial states
The job of a player in a game is to choose a strategy. In terms of bits this means that the player has to choose between 'flipping' the bit to its opposite state or leaving its current state untouched. When extended to the quantum domain this implies that the player can rotate the qubit to a new state, thus changing the probability amplitudes of each of the base states. Such operations on the qubits are required to be unitary transformations on the initial state of the qubit. This is different from the classical procedure which chooses the strategies with some statistical probabilities.
Introducing quantum information into multiplayer games allows a new type of "equilibrium strategy" which is not found in traditional games. The entanglement of players' choices can have the effect of a contract by preventing players from profiting from other player's betrayal. 
Quantum MiniMax Theorems
The concepts of a quantum player, a zero-sum quantum game and the associated expected payoff were defined by A. Boukas within the framework of the spectral theorem for self-adjoint operators on finite dimensional quantum probability spaces. Quantum versions of Von Neumann's Minimax Theorem were proved.
- Quantum tic tac toe: not a quantum game in the sense above, but a pedagogical tool based on metaphors for quantum mechanics
- Quantum pseudo-telepathy
- Quantum refereed game
- Jan Sładkowski
- Jens Eisert
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- J. Eisert, M. Wilkens, M. Lewenstein, Phys. Rev. Lett. 83 (1999) 3077–3080, Quantum Games and Quantum Strategies
- N. F. Johnson, Playing a Quantum Game with a Corrupted Source
- L. Marinatto, T. Weber, Phys. Lett. A 272 (2000) 291-303, A Quantum Approach To Static Games of Complete Information
- T. Cheon, I. Tsutsui, Phys. Lett. A 348 (2006) 147-152, Classical and Quantum Contents of Solvable Game Theory on Hilbert Space
- Piotrowski, E. W.; Sładkowski, Jan (2003), "An invitation to quantum game theory", International Journal of Theoretical Physics, Springer, 42 (5): 1089–1099, doi:10.1023/A:1025443111388 (free version here)
- Andreas Boukas, Quantum Formulation of classical two-person zero-sum games, Open Systems and Information Dynamics, 7, 19-32, 2000, MR2001k:81085.