Quantum game theory
Encyclopedia
Quantum game theory is an extension of classical game theory
to the quantum domain. It differs from classical game theory in three primary ways:
This theory is based on the physics of information much like quantum cryptography
.
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.
s allows a new type of equilibrium strategy which is not found in traditional games. The entanglement of players's choices can have the effect of a contract
by preventing players from profiting from betrayal
.
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
to the quantum domain. It differs from classical game theory in three primary ways:
- SuperposedQuantum superpositionQuantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...
initial states, - Quantum entanglementQuantum entanglementQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
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 cryptography
Quantum cryptography
Quantum key distribution uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages...
.
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
Qubit
In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....
, which is a quantum superposition
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...
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
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
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 of assigning different probabilities to the act of selecting each of the strategies.Multiplayer games
Introducing quantum information into multiplayer gameMultiplayer game
A multiplayer video game is one which more than one person can play in the same game environment at the same time. Unlike most other games, computer and video games are often single-player activities that put the player against preprogrammed challenges and/or AI-controlled opponents, which often...
s allows a new type of equilibrium strategy which is not found in traditional games. The entanglement of players's choices can have the effect of a contract
Contract
A contract is an agreement entered into by two parties or more with the intention of creating a legal obligation, which may have elements in writing. Contracts can be made orally. The remedy for breach of contract can be "damages" or compensation of money. In equity, the remedy can be specific...
by preventing players from profiting from betrayal
Betrayal
Betrayal is the breaking or violation of a presumptive contract, trust, or confidence that produces moral and psychological conflict within a relationship amongst individuals, between organizations or between individuals and organizations...
.
See also
- Quantum tic tac toeQuantum tic tac toeQuantum tic-tac-toe is a "quantum generalization" of tic-tac-toe in which the players' moves are "superpositions" of plays in the classical game...
- Quantum pseudo-telepathyQuantum pseudo-telepathyQuantum pseudo-telepathy is a phenomenon in quantum game theory resulting in anomalously high success rates in coordination games between separated players. These high success rates would require communication between the players in a purely classical world; however, the game is set up such that...
- Jan Sładkowski
- Jens EisertJens EisertJens Eisert, born October 9, 1970, is a physicist notable as one of the co-pioneers of quantum game theory with Maciej Lewensteinand PhD advisor Martin Wilkens.He is also known for his research in quantum information science and related fields....