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Quantum games and games with factorizable and non-factorizable probabilities

Azhar Iqbal, James M. Chappell, Derek Abbott

Published Sep 17, 2014• 1 citation
OpenAlex

Abstract

A quantum game-theoretic setting enables the analysis of strategic interaction among agents with access to quantum resources. One of the original motivations for re-expressing quantum algorithms, and quantum communication protocols, in terms of competing quantum agents was to provide new insights into their workings. Looking closely, one finds that quantum strategic interactions utilize quantum probabilities and such probabilities are realizable in quantum mechanical experiments that allow implementation of unitary transformations, quantum measurements etc. This paper presents a perspective on a quantum game-theoretic framework involving quantum probabilities, which are not subject to the strong factorizability constraints that classical probabilities are found to possess. We suggest that a quantum version of a classical game can be obtained more directly if we replace the factorizability relations for probabilities in a classical game with two weaker constraints that are also obeyed by the quantum probabilities. The solutions of symmetric bimatrix games with such `non-factorizable' probabilities are then analyzed and compared to the solutions that appear in the standard version of a quantum game.

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A quantum game-theoretic setting enables the analysis of strategic interaction among agents with access to quantum resources. One of the original motivations for re-expressing quantum algorithms, and quantum communication protocols, in terms of competing quantum agents was to provide new insights into their workings. Looking closely, one finds that quantum strategic interactions utilize quantum probabilities and such probabilities are realizable in quantum mechanical experiments that allow implementation of unitary transformations, quantum measurements etc. This paper presents a perspective on a quantum game-theoretic framework involving quantum probabilities, which are not subject to the strong factorizability constraints that classical probabilities are found to possess. We suggest that a quantum version of a classical game can be obtained more directly if we replace the factorizability relations for probabilities in a classical game with two weaker constraints that are also obeyed by the quantum probabilities. The solutions of symmetric bimatrix games with such `non-factorizable' probabilities are then analyzed and compared to the solutions that appear in the standard version of a quantum game.
A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the outcome of such games. We demonstrate how the standard version of a quantum game can be analysed as a non-factorizable game and determine the limitations of our approach.
A probabilistic framework is developed that gives a unifying perspective on both the classical and quantum versions of two-player games. We suggest exploiting peculiar joint probabilities involved in Einstein-Podolsky-Rosen (EPR) experiments to construct a quantum game when the corresponding classical game is obtained from factorizable joint probabilities. We analyze how nonfactorizability changes Nash equilibria in three well-known games of prisoner's dilemma, stag hunt, and chicken. In this framework we find that for the game of prisoner's dilemma even nonfactorizable EPR joint probabilities cannot be helpful to escape from the classical outcome of the game. For a particular version of the chicken game, however, we find that the two nonfactorizable sets of joint probabilities, which maximally violate the Clauser-Holt-Shimony-Horne sum of correlations, indeed result in new Nash equilibria.
Theory of quantum games is a new area of investigation that has gone through rapid development during the last few years. Initial motivation for playing games, in the quantum world, comes from the possibility of re-formulating quantum communication protocols, and algorithms, in terms of games between quantum and classical players. The possibility led to the view that quantum games have a potential to provide helpful insight into working of quantum algorithms, and even in finding new ones. This thesis analyzes and compares some interesting games when played classically and quantum mechanically. A large part of the thesis concerns investigations into a refinement notion of the Nash equilibrium concept. The refinement, called an evolutionarily stable strategy (ESS), was originally introduced in 1970s by mathematical biologists to model an evolving population using techniques borrowed from game theory. Analysis is developed around a situation when quantization changes ESSs without affecting corresponding Nash equilibria. Effects of quantization on solution-concepts other than Nash equilibrium are presented and discussed. For this purpose the notions of value of coalition, backwards-induction outcome, and subgame-perfect outcome are selected. Repeated games are known to have different information structure than one-shot games. Investigation is presented into a possible way where quantization changes the outcome of a repeated game. Lastly, two new suggestions are put forward to play quantum versions of classical matrix games. The first one uses the association of De Broglie's waves, with travelling material objects, as a resource for playing a quantum game. The second suggestion concerns an EPR type setting exploiting directly the correlations in Bell's inequalities to play a bi-matrix game.
We use the standard three-party Einstein-Podolsky-Rosen (EPR) setting in order to play general three-player non-cooperative symmetric games. We analyze how the peculiar non-factorizable joint probabilities that may emerge in the EPR setting can change outcome of the game. Our setup requires that the quantum game attains classical interpretation for factor- izable joint probabilities. We analyze the generalized three-player game of Prisoner's Dilemma (PD) and show that the players can indeed escape from the classical outcome of the game because of non-factorizable joint probabilities. This result for three-player PD contrasts strikingly with our earlier result for two-player PD for which even non-factorizable joint prob- abilities are not found to be helpful to escape from the classical outcome of the game.
Research in quantum games has flourished during recent years. However, it seems that opinion remains divided about their true quantum character and content. For example, one argument says that quantum games are nothing but 'disguised' classical games and that to quantize a game is equivalent to replacing the original game by a different classical game. The present thesis contributes towards the ongoing debate about quantum nature of quantum games by developing two approaches addressing the related issues. Both approaches take Einstein-Podolsky-Rosen (EPR)-type experiments as the underlying physical set-ups to play two-player quantum games. In the first approach, the players' strategies are unit vectors in their respective planes, with the knowledge of coordinate axes being shared between them. Players perform measurements in an EPR-type setting and their payoffs are defined as functions of the correlations, i.e. without reference to classical or quantum mechanics. Classical bimatrix games are reproduced if the input states are classical and perfectly anti-correlated, as for a classical correlation game. However, for a quantum correlation game, with an entangled singlet state as input, qualitatively different solutions are obtained. The second approach uses the result that when the predictions of a Local Hidden Variable (LHV) model are made to violate the Bell inequalities the result is that some probability measures assume negative values. With the requirement that classical games result when the predictions of a LHV model do not violate the Bell inequalities, our analysis looks at the impact which the emergence of negative probabilities has on the solutions of two-player games which are physically implemented using the EPR-type experiments.
Quantum game theory (QGT) extends traditional game theory by using quantum mechanics to create new models for strategic decision-making. Traditional game theory provides effective tools for analyzing rational human interaction; yet it suffers from three main restrictions because it cannot properly handle uncertain situations, contradictory behavior, and complex relationships among agents. QGT addresses these challenges through its implementation of quantum phenomena, which enable strategic options that go beyond traditional mixed strategies. The extended system introduces fresh equilibrium solutions that allow easier resolution of standard problems. The chapter presents an organized structure to display QGT, which shows its value for strategic decision-making systems. The chapter shows fundamental concepts that link classical theories with quantum theories through its demonstration of how quantum strategies have developed from their original state to their present form, which expands traditional game models. The research investigates optimization methods, cybersecurity measures, economic systems, and artificial intelligence applications. The study identifies implementation obstacles and future research needs. The advanced QGT decision-making systems provide outstanding solutions that allow intelligent systems to interact with their environments and solve complex environmental problems.