Abstract
The aim of this paper is to investigate Cournot-type competition in the quantum domain with the use of the Li-Du-Massar scheme for continuous-variable quantum games. We derive a formula which, in a simple way, determines a unique Nash equilibrium. The result concerns a large class of Cournot duopoly problems including the competition, where the demand and cost functions are not necessary linear. Further, we show that the Nash equilibrium converges to a Pareto-optimal strategy profile as the quantum correlation increases. In addition to illustrating how the formula works, we provide the readers with two examples.
Highlights
Quantum game theory is an interdisciplinary field that combines quantum theory and game theory
Based on [27], the Cournot duopoly example can be viewed as a strategic form game (N, (Si )i∈N,i∈N ) with the components defined as follows: 1. the set of players is N = {1, 2}, 2. player i’s strategy set is Si = [0, ∞) with typical element qi, 3. player i’s payoff function ui is given by formula ui (q1, q2) = qi P (q1, q2) − cqi, q1, q2 ∈ [0, ∞), (1)
Studies on quantum game theory so far have given us a lot of information about how specific games can be described in the quantum domain
Summary
Quantum game theory is an interdisciplinary field that combines quantum theory and game theory. The Li-Du-Massar scheme appears to be a generally accepted quantum scheme for duopoly examples. It provides a “minimal” quantum structure of a two-player strategicform game with a continuum of strategies. The scheme, originally designed for Cournot duopoly, enables the players to avoid an inefficient Nash equilibrium by means of quantum resources. It preserves the uniqueness of the solution [10, 22]. A natural generalization of the classically played Cournot duopoly is due to [23] (see [24]), where the payoff functions of the players are assumed to depend on the demand and cost functions.
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