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.

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.

The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal-form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure, nor in mixed strategies is given.

We present the unique solution to the Quantum Battle of the Sexes game. We show the best result which can be reached when the game is played according to Marinatto and Weber's scheme. The result which we put forward does not surrender the criticism of previous works on the same topic.