User:Lukef442/Quantum game theory

Quantum game theory is an extension of classical game theory to the quantum domain, meaning that participants now have access to quantum resources. It differs from classical game theory in three primary ways:
 * 1) Superposed initial states,
 * 2) Quantum entanglement of initial states,
 * 3) Superposition of strategies to be used on the initial states.

This theory is based on the physics of information much like quantum computing and has roots in quantum information.

History
In 1999, a professor in the math department at the University of California at San Diego named David A. Meyer first published Quantum Strategies which details a quantum version of the classical game theory game, matching pennies. In the quantum version, players are allowed access to quantum signals through the phenomenon of quantum entanglement.

Quantum Prisoner's Dilemma
The Classical Prisoner's Dilemma is a game played between two players with a choice to cooperate with or betray their opponent. Classically, the dominant strategy is to always choose betrayal. When both players choose this strategy every turn, they each ensure a suboptimal profit, but cannot lose, and the game is said to have reached a Nash equilibrium. Profit would be maximized for both players if each chose to cooperate every turn, but this is not the rational choice, thus a suboptimal solution is the dominant outcome. In the Quantum Prisoner’s Dilemma, both parties choosing to betray each other is still an equilibrium, however, there can also exist multiple Nash equilibriums that vary based on the entanglement of the initial states. In the case where the states are only slightly entangled, there exists a certain unitary operation for Alice so that if Bob chooses betrayal every turn, Alice will actually gain more profit than Bob and vice versa. Thus, a profitable equilibrium can be reached in 2 additional ways. The case where the initial state is most entangled shows the most change from the classical game. In this version of the game, Alice and Bob each have an operator Q that allows for a payout equal to mutual cooperation with no risk of betrayal. This is a Nash equilibrium that also happens to be Pareto optimal.

Additionally, The quantum version of the Prisoner's Dilemma differs greatly from the classical version when the game is of unknown or infinite length. Classically, the infinite Prisoner's Dilemma has no defined fixed strategy but in the quantum version it is possible to develop an equilibrium strategy.

Quantum Chess
Quantum Chess was first developed by a graduate student at the University of Southern California named Chris Cantwell. His motivation to develop the game was to expose non-physicists to the world of quantum mechanics.

The game utilizes the same pieces as classical chess (8 pawns, 2 knights, 2 bishops, 2 rooks, 1 queen, 1 king). However, the pieces are allowed to obey laws of quantum mechanics such as superposition.