A new mathematical representation of Game Theory

In this paper, 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 games have been unified in the 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. This incorporation gives an 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.