In 1969, John Clauser, Michael Horne, Abner Shimony, and Richard Holt (often referred to collectively as "CHSH") wrote an often-cited paper describing experiments which could be used to prove Bell's theorem.
In one part of this paper, they describe a game where a player could have a better chance of winning by using quantum strategies than would be possible classically.
[1] 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.
The information transfer that occurs during a game can be viewed as a physical process.
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.
When a measurement is made on the electron, it collapses to one of the base states, thus conveying the strategy used by the player.
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.
This is different from the classical procedure which chooses the strategies with some statistical probabilities.
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.
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.
It is shown that for classical mixed strategies setting, there is a unique symmetric Nash equilibrium and the Nash equilibrium is obtained by setting the probability of volunteering for each player to be the unique root in the open interval (0,1) of the degree-n polynomial
Alice gives Bob one chance to "operate" on the box and then allows him to withdraw from the game if he would like, but he can only classically obtain information on one card from this operation, so the game is still unfair.
However, Alice and Bob can play a version of this game adjusted to allow for quantum strategies.
, after shaking the box up, we can describe the state of the face-up part of the cards as:
Now, Bob can take advantage of his ability to operate on the box by constructing a machine as follows: First, he has a unitary matrix defined as
From here, Bob can draw one card, and then choose to either withdraw, or keep playing the game.
His motivation to develop the game was to expose non-physicists to the world of quantum mechanics.
However, the pieces are allowed to obey laws of quantum mechanics such as superposition.
By allowed the introduction of superposition, it becomes possible for pieces to occupy more than one square in an instance.
Check is not included in quantum chess because it is possible for the king, as well as all other pieces, to occupy multiple spots on the grid at once.
When attempting to capture an opponent's piece, a measurement is made to determine the probability of whether or not the space is occupied and if the path is blocked.
The classical Nash Equilibrium has both players taking a mixed strategy with each move having a 50% chance of either flipping or not flipping the penny, and Picard and Q will each win the game 50% of the time using classical strategies.
Then, no matter Picard's move, Q can once again apply a Hadamard gate to the superposition which results in the penny being face up.
Quantum versions of Von Neumann's minimax theorem were proved.
[13][14] Quantum game theory also offers a solution to Newcomb's Paradox.
Because choosing a strategy for the game, then changing it to fool to otherwise omniscient player (corresponding to operating on the game state using a NOT gate) cannot give the ignorant player an additional advantage, as the two Hadamard operations ensure that the only two outcomes are those defined by the chosen strategy.