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arXiv:quant-ph/0111090 · Physical Review A

Backwards-induction outcome in a quantum game

Azhar Iqbal, A. H. Toor

Published May 13, 2002• 86 citations
arXivHTMLPDF

Abstract

In economics, duopoly is a market dominated by two firms large enough to influence the market price. Stackelberg presented a dynamic form of duopoly that is also called the ``leader-follower'' model. We give a quantum perspective on the Stackelberg duopoly that gives a backwards-induction outcome same as the Nash equilibrium in the static form of duopoly also known as the Cournot's duopoly. We find the two-qubit quantum pure states required for this purpose.

DOI
10.1103/physreva.65.052328

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In economics, duopoly is a market dominated by two firms large enough to influence the market price. Stackelberg presented a dynamic form of duopoly that is also called the ``leader-follower'' model. We give a quantum perspective on the Stackelberg duopoly that gives a backwards-induction outcome same as the Nash equilibrium in the static form of duopoly also known as the Cournot's duopoly. We find the two-qubit quantum pure states required for this purpose.
The development of quantum information theory based on quantum entangled states is advancing rapidly and has promoted the emergence of quantum game theory. Quantum game theory is quantum information processing applied to classical games, which solves problems that classical games cannot solve. This paper explores the quantum Cournot-Bertrand game model of two firms in a duopoly market with bounded rationality and product substitutability. The first and second firms choose production output and sales price as their decision variables, respectively. We first define the quantum Cournot-Bertrand dynamical system and study the existence and locally stable conditions of equilibrium points. Then we explore bifurcation conditions and bifurcation types. Finally, we do the bifurcation analysis and explore the impacts of the model parameters as the potential market size of products, the two firms' linear marginal cost, the players' speed of strategy adjustment, the product substitution index, and the quantum entanglement degree on the Nash equilibrium stability and bifurcation using numerical analysis and simulation, and provide economic explanations for the results.
We study the influence of entanglement and correlated noise using correlated amplitude damping, depolarizing and phase damping channels on the quantum Stackelberg duopoly. Our investigations show that under the influence of an amplitude damping channel a critical point exists for an unentangled initial state at which firms get equal payoffs. The game becomes a follower advantage game when the channel is highly decohered. Two critical points corresponding to two values of the entanglement angle are found in the presence of correlated noise. Within the range of these limits of the entanglement angle, the game is a follower advantage game. In the case of a depolarizing channel, the payoffs of the two firms are strongly influenced by the memory parameter. The presence of quantum memory ensures the existence of the Nash equilibrium for the entire range of decoherence and entanglement parameters for both the channels. A local maximum in the payoffs is observed which vanishes as the channel correlation increases. Moreover, under the influence of the depolarizing channel, the game is always a leader advantage game. Furthermore, it is seen that the phase damping channel does not affect the outcome of the game.
Abstract Recent economic research has focused on applying quantum mechanical models to understand complex markets. Real markets are usually complex with nonlinear inverse demand functions and dynamic participants. Furthermore, when these market games are quantized, players interact via quantum channels that are prone to decoherence. In this work, owing to the complexity of real markets, we introduce noise and memory in a dynamic quantum Stackelberg duopoly game for the first time. We shed light on the significance of two-qubit entangling operators, particularly special perfect entanglers (SPEs), in controlling chaos. The following noteworthy observations are made: First, although entangling operators cannot prevent monopolies in the market, they can shift leader dominance to follower dominance. Second, numerical analysis highlights the significance of the B-gate in a correlated noisy channel as a chaos control. We also highlight the significance of the controlled-NOT (CNOT), and double-CNOT (DCNOT) gates in controlling chaos in noisy, memoryless channel 2. Finally, we show that the CNOT and DCNOT gates are more sensitive than the B-gate. These novel features of two-qubit entangling operators in correlated, noisy and dynamic market games shed new light on the role of SPEs in quantum dynamic game settings.
In this paper, we investigate the influence of entanglement and decoherence on the quantum Stackelberg duopoly (QSD) game. It is shown that the first-mover advantage can be weakened or enhanced due to the existence of entanglement for the QSD game without decoherence. The influence of decoherence induced by the amplitude damping and the phase damping are explicitly studied in the formalism of Kraus operator representations. We show that the amplitude damping drastically changes the Nash equilibrium of the QSD game and the profits of the two players while the phase damping does not affect the Nash equilibrium and the profits of the two players. It is found that under certain conditions there exists a 'critical point' of the damping parameter for the amplitude damping environment. At the 'critical point' the two players have the same moves and payoffs. The QSD game can change from the first-mover advantage game into the follower-mover advantage game when the damping parameter varies from the left-hand-side regime of the 'critical point' to the right-hand-side regime.
We present a quantum model of Bertrand duopoly and study the entanglement behavior on the profit functions of the firms. Using the concept of optimal response of each firm to the price of the opponent, we find only one Nash equilibirum point for the maximally entangled initial state. The presence of quantum entanglement in the initial state gives payoffs higher to the firms than the classical payoffs at the Nash equilibrium. As a result, the dilemma-like situation in the classical game is resolved.