Abstract
Playing the Cournot duopoly in the quantum domain can lead to the optimal strategy profile in the case of maximally correlated actions of the players. However, that result can be obtained if the fact that the players play the quantum game is common knowledge among the players. Our purpose is to determine reasonable game outcomes when players’ perceptions about what game is actually played are limited. To this end, we consider a collection consisting of the classical and quantum games that specifies how each player views the game and how each player views the other players’ perceptions of the game. We show that a slight change in how the players perceive the game may considerably affect the result of the game and, in the case of maximally correlated strategies, may vary from the inefficient Nash equilibrium outcome in the classical Cournot duopoly to the Pareto optimal outcome. We complete our work by investigating in the same way the Bertrand duopoly model.
Highlights
Quantum game theory [1] unites game theory with quantum mechanics
Our research has shown that a rational result in the quantum duopolies depends on whether the players play the quantum game is common knowledge or not
The Pareto optimal outcome ( a − c)2 /8 is achievable in the quantum Cournot duopoly with maximally correlated strategies if each player knows that he/she plays the quantum game, but he/she has to know that the other player perceives the quantum game, and each player i finds that the other player finds that player i is considering the quantum game and so on
Summary
Quantum game theory [1] unites game theory with quantum mechanics. It is an interdisciplinary research field that assumes games to be played with the use of objects that behave according to the postulates of quantum mechanics. Studying quantum games with limited perception (with unawareness) [16,17] is one of the latest trends. The notion of unawareness provides us with the tools to consider problems in which some of the players perceive quantum games, whereas the other players may think they play the classical game. We shall introduce an element of unawareness to quantum versions of Cournot and Bertrand duopoly already studied by us in papers [18,19]. We shall consider cases in which players play the quantum duopoly game; some of the players may not realize that fact or the players may be aware of playing the quantum game, but at the same time may find that the other player views the classical game. We recall the idea of quantum duopoly introduced in [21]
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