Density Matrix Formalism of Game Theory

A new representation of Game Theory is developed in this work. 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.

The two Motivations

First, in order to apply Statistical Mechanics onto Game Theory

Second, from the requirement of games on quantum objects

The object of Game Theory is a game, a multi-player decision making situation, usually with conflicts between players. For example, in a Penny Flipping Game (PFG), two players play with a coin, say initially with head state. The strategies can be used by players are Non-flip and Flip, which, in the language of Physics, are operators acting on the coin. The payoff is defined such as player 1 wins one dollar for head state after both players applied their strategies, and lose one dollar for tail state. For such static strategy games, Nash Theorem has given a closed conclusion that at least one mixture-strategy Nash Equilibria (NE) exists for any games. Here the NE is defined that under such state no more players will like to change its own strategy state, and mixture strategy is defined as a probability distribution function (PDF) over the strategy space of every player.

Our question is how about we replace the two-side coin here with a ½-quantum spin? What's the effect of this on Game Theory? It's still a game-theory question. Players can still choose strategies to act on the spin, although they have much more choices now. Compared with Non-flip and Flip , in Quantum Mechanics, any unitary 2×2 matrices can be used as operators, and I, X, Y, Z are the four typical matrices of them. Now the Game Theory must answer how to define the strategy state for this game, how to define NE, and the existence of NE. At last, we have to ask whether such game can be studied within the framework of Traditional Game Theory (TGT), or should we develop a new framework but still with the same spirit of Game Theory? In this work, we will construct a new framework, which can be used both TGT and the game on quantum objects, named Quantum Game Theory (QGT).

The general framework: Operator Representation of Game Theory

In this work, we introduce a framework of new mathematical representation of Game Theory, including static classical game and static quantum game. The idea is to find a set of base vectors in strategy space and to define their inner product so as to form them as a Hilbert space, and then form a Hilbert space of system state. Basic ideas and formulas in Game Theory have been reexpressed in such a space of system state. This space provides more possible strategies than traditional classical game and quantum game. All the Game Theories have been unified in a new representation and their relation has been discussed. It seems that if the quantized classical game has some independent meaning other than traditional classical, a payoff matrix with non-zero off-diagonal elements is required. On the other hand, when such new representation is applied onto quantum game, the payoff matrix gives non-zero off-diagonal elements. Also in the new representation of quantum games, a set of base vectors are naturally given from the quantum strategy (operator) space. This gives a kind of support for our approach in classical game. Ideas and technics from Statistical Physics can be easily incorporated into Game Theory through such a representation. Such incorporation gives some endogenous method for refinement of Equilibrium State and some hits to simplify the calculation of Equilibrium State. Kinetics Equation and thermal equilibrium has been introduced as an efficient way to calculate the Equilibrium State. Although we have gotten some successful experience on some trivially cases, the progress of such a dynamical equation for the general case is still waiting for more exploration.

Reference

M. J. Osborne and A. Rubinstein, A course in Game Theory, MIT Press, 1994.
J.W. Friedman, Game Theory with application in Economics, New York: Oxford University Press, 1990.
M.E.J. Newman and G.T. Barkema, Monte Carlo Method in Statistical Physics, New York: Oxford University Press, 1999.
J.F. Nash, "Equilibrium points in N-Person Games", PNAS, 48-49(1950); "The Bargaining Problem", Econometrica, Vol. 18, Issue 2, 155-162(1950).
J. Wu, arXiv:quant-ph/0501088; arXiv:quant-ph/0503094; arXiv:quant-ph/0404159.
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press��2000), page 20.
D.A. Meyer, Phys. Rev. Lett. 82 (1999), 1052.
J. Eisert, M. Wilkens and M. Lewenstein, Phys. Rev. Lett. 83(1999), 3077.
J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, R. Han, Phys. Rev. Lett. 88 (2002), 137902.
S. J. van Enk and R. Pike, Phys. Rev. A 66(2002), 024306.
C.F. Lee and N.F. Johnson, Phys. Rev. A 67(2003), 022311.
L. Marinatto and T. Weber, Phys. Lett. A 272 (2000), 291.
D.H. Wolpert, arXiv:cond-mat/0402508.
E.W. Piotrowski and J. Sladkowski, An invitation to Quantum Game Theory, Int. J. Theor. Phys. 42(2003), 1089.
A.P. Flitney and D. Abbot, An introduction to quantum game theory, Fluctuation and Noise Letters Vol.2, No.4(2002), R175-R187.