Playing Prisoner's Dilemma with Quantum Rules

The physical world obeys the rules of quantum, as opposed to classical, physics. Since the playing of any particular game requires physical resources, the question arises as to how Game Theory itself would change if it were extended into the quantum domain. Here we provide a general formalism for {\em quantum} games, and illustrate the explicit application of this new formalism to a quantized version of the well-known prisoner's dilemma game.

We present a perspective on quantum games that focuses on the physical aspects of the quantities that are used to implement a game. If a game is to be played, it has to be played with objects and actions that have some physical existence. We call such games playable. By focusing on the notion of playability for games we can more clearly see the distinction between classical and quantum games and tackle the thorny issue of what it means to quantize a game. The approach we take...

We present a quantization scheme for a three-player Prisoner's Dilemma game. It is shown that entanglement plays a dominant role in the three-player quantum game. Four different types of payoffs are identified on the basis of different combinations of initial state and measurement basis entanglement parameters. A relation among these different payoffs is also established. We also study the communication aspects of the three-player game. By exploiting different combinations of...

This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic be...

A number of recent studies have focused on novel features in game theory when the games are played using quantum mechanical toolbox (entanglement, unitary operators, measurement). Researchers have concentrated in two-player-two strategy, 2x2, dilemma containing classical games, and transferred them into quantum realm showing that in quantum pure strategies dilemmas in such games can be resolved if entanglement is distributed between the players armed with quantum operations. ...

We generalize the quantum Prisoner's Dilemma to the case where the players share a non maximally entangled states. We show that the game exhibits an intriguing structure as a function of the amount of entanglement with two thresholds which separate a classical region, an intermediate region and a fully quantum region. Furthermore this quantum game is experimentally realized on our nuclear magnetic resonance quantum computer.

Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player-two strategy (2x2) dilemma containing classical games into quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of...

We investigate the 3-player quantum Prisoner's Dilemma with a certain strategic space, a particular Nash equilibrium that can remove the original dilemma is found. Based on this equilibrium, we show that the game is enhanced by the entanglement of its initial state.

This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. What has clouded many of the issues is the lack of a mathemati...

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.