dorsal/arxiv
View SchemaA new mathematical representation of Game Theory, I
| Authors | Jinshan Wu |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0404159 |
| URL | https://arxiv.org/abs/quant-ph/0404159 |
Abstract
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.
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"abstract": "In this paper, we introduce a framework of new mathematical representation of\nGame Theory, including static classical game and static quantum game. The idea\nis to find a set of base vectors in every single-player strategy space and to\ndefine their inner product so as to form them as a Hilbert space, and then form\na Hilbert space of system state. Basic ideas, concepts and formulas in Game\nTheory have been reexpressed in such a space of system state. This space\nprovides more possible strategies than traditional classical game and\ntraditional quantum game. So besides those two games, more games have been\ndefined in different strategy spaces. All the games have been unified in the\nnew representation and their relation has been discussed. General Nash\nEquilibrium for all the games has been proposed but without a general proof of\nthe existence. Besides the theoretical description, ideas and technics from\nStatistical Physics, such as Kinetics Equation and Thermal Equilibrium can be\neasily incorporated into Game Theory through such a representation. This\nincorporation gives an endogenous method for refinement of Equilibrium State\nand some hits to simplify the calculation of Equilibrium State. The more\nprivileges of this new representation depends on further application on more\ntheoretical and real games. Here, almost all ideas and conclusions are shown by\nexamples and argument, while, we wish, lately, we can give mathematical proof\nfor most results.",
"arxiv_id": "quant-ph/0404159",
"authors": [
"Jinshan Wu"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"title": "A new mathematical representation of Game Theory, I",
"url": "https://arxiv.org/abs/quant-ph/0404159"
},
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