Hamiltonian Formalism of Game Theory

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 gives the payoff of players. By this formalism, 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 formalism, when quantum objects instead of classical objects are used as the objects in the game, it naturally leads to the Traditional Quantum Game, 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 represented in this single framework, while traditionally, they are treated separately by Non-cooperative Game Theory and Coalitional Game Theory. Because 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 Traditional Quantum Game. 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.