博弈论的一种新的数学表示，
A new mathematical representation of Game Theory, I
摘要：
本文介绍了一种新的博弈论数学表示框架，包括静态经典博弈和静态量子博弈。其思想是在每一个单人策略空间中寻找一组基向量，并定义它们的内积，使它们形成一个Hilbert空间，从而形成系统状态的Hilbert空间。博弈论的基本思想、概念和公式在这样一个系统状态空间中得到了重新表达。这个空间提供了比传统经典博弈和传统量子博弈更多的可能策略。所以除了这两个博弈之外，更多的博弈被定义在不同的策略空间中。所有的对策都统一在新的表示中，并讨论了它们之间的关系。一般纳什均衡的所有游戏已经提出，但没有一个普遍的存在性证明。除了理论描述外，统计物理学的思想和技术，如动力学方程和热平衡，可以很容易地通过这种表示方法融入博弈论。这种结合给出了一种细化平衡态的内生方法，并对简化平衡态的计算作了一些尝试。这种新的表示法的更多特权取决于更多理论和实际游戏的进一步应用。在这里，几乎所有的想法和结论都是通过例子和论证来展示的，而我们希望，最近，我们可以为大多数结果提供数学证明。
英文摘要：
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 every single-player 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, concepts 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 traditional quantum game. So besides those two games, more games have been defined in different strategy spaces. All the games have been unified in the new representation and their relation has been discussed. General Nash Equilibrium for all the games has been proposed but without a general proof of the existence. Besides the theoretical description, ideas and technics from Statistical Physics, such as Kinetics Equation and Thermal Equilibrium 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. The more privileges of this new representation depends on further application on more theoretical and real games. Here, almost all ideas and conclusions are shown by examples and argument, while, we wish, lately, we can give mathematical proof for most results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
