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Palgrave Macmillan UK eBooks

Quantum Game Theoretical Frameworks in Economics

Edward W. Piotrowski, Jan Sładkowski

Published Jan 1, 2017• 10 citations
Publisher

Abstract

The authors briefly review quantum game theory and its application in economics. This review is addressed at newcomers and some basic ideas of quantum theory are necessary to follow the text—the short introduction in chapter “A Brief Introduction to Quantum Formalism” will be sufficient. Due to the lack of space only the flagship issues will be discussed. Quantum game theory, whatever opinions may be held due to its abstract physical formalism, have already found various applications even outside the orthodox physics domain. We are aware that the implementation of genuine quantum models is not an easy task. Nevertheless, such models are already an interesting although sophisticated theoretical tool. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1057/978-1-137-49276-0_3

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The authors briefly review quantum game theory and its application in economics. This review is addressed at newcomers and some basic ideas of quantum theory are necessary to follow the text—the short introduction in chapter “A Brief Introduction to Quantum Formalism” will be sufficient. Due to the lack of space only the flagship issues will be discussed. Quantum game theory, whatever opinions may be held due to its abstract physical formalism, have already found various applications even outside the orthodox physics domain. We are aware that the implementation of genuine quantum models is not an easy task. Nevertheless, such models are already an interesting although sophisticated theoretical tool. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The authors show how the formal framework of quantum mechanics arises in economics generally and in financial economics in particular as a natural consequence of an information-theoretic approach to these fields. Specifically, extremizing Fisher information, subject to the constraint that the associated probability density reproduces observed economic phenomena, results in Schrödinger-like equations. The authors illustrate the utility of this approach with examples from financial economics. As an emergent consequence of information theory, the use of quantum formalism in economics can be seen as a powerful approach for understanding complex dynamics grounded in the probabilistic view of economics advocated by Knight and Keynes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Purpose The current paper is a brief review of the emerging field of quantum-like modelling in game theory. This paper aims to explore several quantum games, which are superior compared to their classical counterparts, which means either they give rise to superior Nash equilibria or they make the game fairer. For example, quantum Prisoners Dilemma generates Pareto superior outcomes as compared to defection outcome in the famous classical case. Again, a quantum-like version of cards game can make the game fairer, increasing the chance of winning of players who are disadvantaged in the classical case. This paper explores all the virtues of simple quantum games, also highlighting some findings of the authors as regards Prisoners Dilemma game. Design/methodology/approach As this is a general review paper, the authors have not demonstrated any specific mathematical method, rather explored the well-known quantum probability framework, used for designing quantum games. They have a short appendix which explores basic structure of Hilbert space representation of human decision-making. Findings Along with the review of the extant literature, the authors have also highlighted some new findings for quantum Prisoners Dilemma game. Specifically, they have shown in the earlier studies (which are referred to here) that a pure quantum entanglement set up is not needed for designing better games, even a weaker condition, which is classical entanglement is sufficient for producing Pareto improved outcomes. Research limitations/implications Theoretical research, with findings and implications for future game designs, it has been argued that it is not always needed to have true quantum entanglement for superior Nash Equilibria. Originality/value The main purpose here is to raise awareness mainly in the social science community about the possible applications of quantum-like game theory paradigm. The findings related to Prisoners Dilemma game are, however, original.
The authors present briefly the basic notions of the quantum formalism: pure and mixed states; quantum observables; quantum probability; Born’s rule, superposition, and ‘state collapse’; the projection postulate (von Neumann–Lüders postulate); Dirac’s ket and bra-vector notations; elements of quantum information theory and quantum logic; Schrödinger’s and von Neumann’s equations; unitary dynamics; and positive operator valued measures. They also define the basic mathematical notions related to the quantum formalism: Hilbert space; scalar product; norm; Hermitian operator (matrix); projector; and unitary operator and adjoint operator. This chapter may be useful for newcomers to the field, but for those readers who have preliminary knowledge about quantum mechanics (QM) they can proceed directly to advanced chapters. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This chapter reviews the quantum theoretical formalism and sketches its foundational problems. The main function of this chapter is to establish common ground for the interpretive discussions which follow, not to provide a comprehensive introduction to quantum theory. 6 Its first section recapitulates the essential aspects of the Hilbert space formalism of quantum mechanics, the second section covers the textbook account of measurement and measurement collapse, and the third and final section outlines the quantum theoretical treatment of many-component systems and explains the concept of quantum entanglement. Further introductory material is provided in the Appendices A, B, and C, all of which cover important no-go theorems for the interpretation of the theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.