Abstract
The aim of this paper is to discuss in some detail the two different quantum schemes for duopoly problems. We investigate under what conditions one of the schemes is more reasonable than the other one. Using the Cournot's duopoly example, we show that the current quantum schemes require a slight refinement so that they output the classical game in a particular case. Then, we show how the amendment changes the way of studying the quantum games with respect to Nash equilibria. Finally, we define another scheme for the Cournot's duopoly in terms of quantum computation.
Highlights
Quantum game theory, an interdisciplinary field that combines quantum theory and game theory, has been investigated for fifteen years
If we assumed that equal convex hulls of payoff profiles generated by both the classical and quantum games were a necessary condition for the quantum scheme to be a correct one, the protocol (4)–(9) would not be valid
Only one condition is taken into consideration. It says that a quantum game ought to include the classical way of playing the game
Summary
An interdisciplinary field that combines quantum theory and game theory, has been investigated for fifteen years. The general idea (in the case of bimatrix games) was based on identifying the possible results of the game with the basis states |i j ∈ Cn ⊗ Cm. Soon after quantum theory has found an application in duopoly problems [4,5]. Soon after quantum theory has found an application in duopoly problems [4,5] It has been a challenging task as the players’ strategy sets in the duopoly examples are (real) intervals, and there are continuum of possible game results.
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