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arXiv:2104.04663 · Frontiers in Artificial Intelligence

Quantum Prisoner’s Dilemma and High Frequency Trading on the Quantum Cloud

Faisal Shah Khan, Ning Bao

Published Nov 3, 2021• 9 citations
arXivHTMLPDF

Abstract

High-frequency trading (HFT) offers an excellent use case and a potential killer application of the commercially available, first generation quasi-quantum computers. To this end, we offer here a simple game-theoretic model of HFT as the famous two player game, Prisoner's Dilemma. We explore the implementation of HFT as an instance of Prisoner's Dilemma on the (quasi) quantum cloud using the Eisert, Wilkens, and Lewenstein quantum mediated communication protocol, and how this implementation can not only increase transaction speed but also improve the lot of the players in HFT. Using cooperative game-theoretic reasoning, we also note that in the near future when the internet is properly quantum, players will be able to achieve Pareto-optimality in HFT as an instance of reinforced machine learning.

DOI
10.3389/frai.2021.769392

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High-frequency trading (HFT) offers an excellent use case and a potential killer application of the commercially available, first generation quasi-quantum computers. To this end, we offer here a simple game-theoretic model of HFT as the famous two player game, Prisoner's Dilemma. We explore the implementation of HFT as an instance of Prisoner's Dilemma on the (quasi) quantum cloud using the Eisert, Wilkens, and Lewenstein quantum mediated communication protocol, and how this implementation can not only increase transaction speed but also improve the lot of the players in HFT. Using cooperative game-theoretic reasoning, we also note that in the near future when the internet is properly quantum, players will be able to achieve Pareto-optimality in HFT as an instance of reinforced machine learning.
Classic game theory is an important field with a long tradition of useful results. Recently, the quantum versions of classical games, such as the Prisoner's Dilemma (PD), have attracted a lot of attention. Similarly, state machines and specifically finite automata have also been under constant and thorough study for plenty of reasons. The quantum analogues of these abstract machines, like the quantum finite automata, have been studied extensively. In this work, we examine some well-known game conditional strategies that have been studied within the framework of the repeated PD game. Then, we try to associate these strategies to proper quantum finite automata that receive them as inputs and recognize them with probability 1, achieving some interesting results. We also study the quantum version of PD under the Eisert-Wilkens-Lewenstein scheme, proposing a novel conditional strategy for the repeated version of this game.
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.
Decision-making by the two negotiating parties is simulated by a prisoner's dilemma game.The game is formulated in a quantum manner, where players strategies are unitary transformations of qubits built over the basis of opposite decision options.Quantum strategies are correlated through the mechanism of quantum entanglement and the result of the game is obtained by the collapse of the resulting transformed state.The range of strategies allowed for quantum players is richer than in case of a classical game and therefore the result of the game can be better optimized.On the other hand, the quantum game is save against eavesdropping and the players can be assured that this type of quantum arbitration is fair.We show that quantum prisoner's dilemma has more favorable Nash equilibria than its classical analog and they are close to the Pareto optimal solutions.Some economical examples of utilizing quantum game Nash equilibria are proposed.
This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game.
What happens when an infinite number of players play a quantum game? In this paper, we will answer this question by looking at the emergence of cooperation, in the presence of noise, in a one-shot quantum prisoner’s dilemma (QuPD). We will use the numerical agent-based model (ABM) and compare it with the analytical Nash equilibrium mapping (NEM) technique. To measure cooperation, we consider five indicators, i.e., game magnetization, entanglement susceptibility, correlation, player’s payoff average, and payoff capacity, respectively. In quantum social dilemmas, entanglement plays a non-trivial role in determining the behavior of the quantum players (or qubits) in the thermodynamic limit, and for QuPD, we consider the existence of bipartite entanglement between neighboring quantum players. For the five indicators in question, we observe first-order phase transitions at two entanglement values, and these phase transition points depend on the payoffs associated with the QuPD game. We numerically analyze and study the properties of both the quantum and the defect phases of the QuPD via the five indicators. The results of this paper demonstrate that both ABM and NEM, in conjunction with the chosen five indicators, provide insightful information on cooperative behavior in an infinite-player one-shot quantum prisoner’s dilemma.