Classical Game Theory

Alan Bojić

The Quantum game theory is an extension of the classical game theory which describes games in which players share quantum information during the game. The use of the quantum mechanical effects enables players’ better results than they would achieve in the classical game without sharing any information at all. In this paper I would like to describe a new two player cooperative game which is based on the well known CHSH game. A quantum strategy for a new game will be introduced and it will be compared to the classical strategy and CHSH game strategy.

Taksu Cheon

Jinshan Wu

A new representation of Game Theory is developed in this paper. State of players is represented by a density matrix, and payoff function is a set of hermitian operators, which when applied onto the density matrix give the payoff of players. By this formulism, a new way to find the equilibria of games is given by generalizing the thermodynamical evolutionary process leading to equilibrium in Statistical Mechanics. And in this formulism, when quantum objects instead of classical objects are used as the objects in the game, it's naturally leads to the so-called Quantum Game Theory, but with a slight difference in the definition of strategy state of players: the probability distribution is replaced with a density matrix. Further more, both games of correlated and independent players can be reached in this single framework, while traditionally, they are treated separately by Non-cooperative Game Theory and Coalitional Game Theory. Because of the density matrix is used as state of players, besides classical correlated strategy, quantum entangled states can also be used as strategies, which is an entanglement of strategies between players, and it is different with the entanglement of objects' states as in the so-called Quantum Game Theory. At last, in the form of density matrix, a class of quantum games, where the payoff matrixes are commutative, can be reduced into classical games. In this sense, it will put the classical game as a special case of our quantum game.

Ramón Alonso‐Sanz

Jinshan Wu

A new representation of Game Theory is developed in this paper. State of players is represented by a density matrix, and payoff function is a set of hermitian operators, which when applied onto the density matrix give the payoff of players. By this formulism, a new way to find the equilibria of games is given by generalizing the thermodynamical evolutionary process leading to equilibrium in Statistical Mechanics. And in this formulism, when quantum objects instead of classical objects are used as the objects in the game, it's naturally leads to the so-called Quantum Game Theory, but with a slight difference in the definition of strategy state of players: the probability distribution is replaced with a density matrix. Further more, both games of correlated and independent players can be reached in this single framework, while traditionally, they are treated separately by Non-cooperative Game Theory and Coalitional Game Theory. Because of the density matrix is used as state of players, besides classical correlated strategy, quantum entangled states can also be used as strategies, which is an entanglement of strategies between players, and it is different with the entanglement of objects' states as in the so-called Quantum Game Theory. At last, in the form of density matrix, a class of quantum games, where the payoff matrixes are commutative, can be reduced into classical games. In this sense, it will put the classical game as a special case of our quantum game.

Indranil Ghosh