dorsal/arxiv
View SchemaHamiltotian Formalism of Game Theory
| Authors | Jinshan Wu |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0501088 |
| URL | https://arxiv.org/abs/quant-ph/0501088 |
Abstract
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 give the payoff of players. By this formulism, 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 formulism, when quantum objects instead of classical objects are used as the objects in the game, it's naturally leads to the so-called Quantum Game Theory, 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 reached in this single framework, while traditionally, they are treated separately by Non-cooperative Game Theory and Coalitional Game Theory. Because of 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 so-called Quantum Game Theory. 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.
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"abstract": "A new representation of Game Theory is developed in this paper. State of\nplayers is represented by a density matrix, and payoff function is a set of\nhermitian operators, which when applied onto the density matrix give the payoff\nof players. By this formulism, a new way to find the equilibria of games is\ngiven by generalizing the thermodynamical evolutionary process leading to\nequilibrium in Statistical Mechanics. And in this formulism, when quantum\nobjects instead of classical objects are used as the objects in the game, it\u0027s\nnaturally leads to the so-called Quantum Game Theory, but with a slight\ndifference in the definition of strategy state of players: the probability\ndistribution is replaced with a density matrix. Further more, both games of\ncorrelated and independent players can be reached in this single framework,\nwhile traditionally, they are treated separately by Non-cooperative Game Theory\nand Coalitional Game Theory. Because of the density matrix is used as state of\nplayers, besides classical correlated strategy, quantum entangled states can\nalso be used as strategies, which is an entanglement of strategies between\nplayers, and it is different with the entanglement of objects\u0027 states as in the\nso-called Quantum Game Theory. At last, in the form of density matrix, a class\nof quantum games, where the payoff matrixes are commutative, can be reduced\ninto classical games. In this sense, it will put the classical game as a\nspecial case of our quantum game.",
"arxiv_id": "quant-ph/0501088",
"authors": [
"Jinshan Wu"
],
"categories": [
"quant-ph"
],
"title": "Hamiltotian Formalism of Game Theory",
"url": "https://arxiv.org/abs/quant-ph/0501088"
},
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