Classical and quantum contents of solvable game theory on Hilbert space

Cheon, Taksu; Tsutsui, Izumi

doi:10.1016/j.physleta.2005.08.066

Cited by 64 publications

(75 citation statements)

References 10 publications

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“…An example is provided by the quantum games [41][42][43][44][45][46][47][48][49][50]. With the development of quantum game theory, it has been shown that many quantum games can be reformulated as classical games by allowing for a more complex game structure [51][52][53][54]. But, in the majority of cases, it is more efficient to play quantum game versions, as less information needs to be exchanged.…”

Section: Basic Ideas and Historical Retrospectivementioning

confidence: 99%

Processing Information in Quantum Decision Theory

Yukalov

Sornette

2009

Entropy

113

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A survey is given summarizing the state of the art of describing information processing in Quantum Decision Theory, which has been recently advanced as a novel variant of decision making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intended actions. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. The self-consistent procedure of decision making, in the frame of the quantum decision theory, takes into account both the available objective information as well as subjective contextual effects. This quantum approach avoids any paradox typical of classical decision theory. Conditional maximization of entropy, equivalent to the minimization of an information functional, makes it possible to connect the quantum and classical decision theories, showing that the latter is the limit of the former under vanishing interference terms.

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Section: Basic Ideas and Historical Retrospectivementioning

confidence: 99%

Processing Information in Quantum Decision Theory

Yukalov

Sornette

2009

Entropy

113

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“…In this form, we clearly see that both payoffs are composed of two components. First component, formerly termed as classical family 7) represents essentially classical payoff coming from "altruistic" modification of the game matrix. 18,19) Even with this modification, the payoffs are still constructed from factorizable probabilities p B , and therefore, payoffs will never exceed the limit (25).…”

Section: Pseudo-classical and Quantum Interference Componentsmentioning

confidence: 99%

Bayesian Nash Equilibria and Bell Inequalities

Cheon

Iqbal

2008

J. Phys. Soc. Jpn.

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Games with incomplete information are formulated in a multi-sector probability matrix formalism that can cope with quantum as well as classical strategies. An analysis of classical and quantum strategy in a multi-sector extension of the game of Battle of Sexes clarifies the two distinct roles of nonlocal strategies, and establish the direct link between the true quantum gain of game's payoff and the breaking of Bell inequalities.KEYWORDS: quantum Harsanyi game, nonlocality, entanglement, battle of sexes IntroductionAlthough Bell's inequalities 1, 2) are usually discussed in the context of quantum Bell experiments with spins and observers, they can be established in a far wider variety of settings. Here we bring one such example in a rather unexpected field of the theory of games of incomplete information.3) Game theory now occupies a central place in areas of applied mathematics, economics, sociology, and in mathematical biology. It is well known that Bell inequality can be broken only when the assumption of local realism is abandoned. This result, when considered in the context of game theory of incomplete information, links together the breaking of Bell inequality and the existence of nonlocal correlation between players. To explore this further, a consideration of quantum strategies 4-8) in games of incomplete information becomes both relevant and interesting.With rapid advancement of quantum information technologies, playing games with quantum resources is within the technical reach of advanced laboratories.9, 10) It is quite conceivable that playing games with quantum strategies, using properly coordinated quantum devices, becomes commonplace in the near future. It is therefore timely that we analyze the physical contents of quantum strategies, and examine the relevance of Bell inequality breaking. It is now generally agreed that quantum strategy can shift the classical outcome of the game in favor of all players, but how much of it is due to truly quantum effect, never achievable classically, is still under debate.11) Games with incomplete information synergetic to Bell experiment setup appears to be a good candidate to settle this issue, which is one of the basic unanswered question of quantum game theory.To study quantum strategies in games of incomplete information, we develop a formalism of game theory based on multi-sector probability matrix. We then analyze a game of incomplete information which is an extension of the well known game of Battle of Sexes and find the classical and the quantum Bayesian Nash equilibria. We find two distinct effects of quantum entanglement in games of incomplete information:

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“…In recent times much attention has been paid to the task of extending game theory concepts [3] to quantum mechanics [9][10][11][12][13][14][15][16][17][18][19]. Quantum games may be of interest given that its classical counterpart (CGT) has had such a phenomenal success.…”

Section: Our Goalsmentioning

confidence: 99%