Appropriate quantization of asymmetric games with continuous strategies

Abstract

We establish a new quantization scheme to study the asymmetric Bertrand duopoly with differentiated products. This scheme is more efficient than the previous symmetric one because it can exactly make the optimal cooperative payoffs at quantum Nash equilibrium. It is also a necessary condition for general asymmetric games with continuous strategies to reach such payoffs.

Introduction

Since the initial work by Meyer [1] on the coin tossing game and shortly afterwards, the work by Eisert et al. [2] on the famous Prisoners' dilemma, many interesting aspects on quantum games with discrete strategies have been studied theoretically [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], and the first experimental realization of quantum games was also successfully accomplished on our NMR experimental device [16].

Besides the games with discrete strategies, there are kinds of games with continuous strategies, which are often studied in many fields, such as economics and sociology. It is well known in classical game theory [17] that, in such kinds of games, just like in the game of the above Prisoners' dilemma, the sum of payoffs of the two players at Nash equilibrium is usually not optimal, because the aim of every player is just pursuing his/her own maximum payoff instead of the maximum mutual payoffs. This dilemma-like situation is not what the players want to confront, while it cannot be completely resolved within the framework of classical game theory. Li et al. [18] investigated the particular case of the Cournot's duopoly as an example, and first established a quantization scheme to ensure that the two players can virtually cooperate and get the optimal payoffs at the maximal quantum entanglement between them, though both of them act “selfishly” just as they do in the classical case. Recently Lo and Kiang applied Li et al.'s quantization scheme to the case of the Stackelberg duopoly [19] and the Bertrand duopoly with differentiated products [20], and got similar results. Of course, all such works assume that players can use quantum protocols, which are not to really solve any classical economic problems quantum mechanically.

Both works of Refs. [18], [20] dealt with symmetric games, i.e., the two players have the same conditions, and thus they definitely adopt the same strategies and get the same payoffs at Nash equilibrium. While in practice, games are usually asymmetric since the two players may inevitably have some different conditions. Consequently the two players must have different strategies and different payoffs at Nash equilibrium. It is natural to doubt whether the above quantization scheme—we name it the symmetric quantization scheme, efficient for symmetric games though, can still make the asymmetric games get the optimal cooperative payoffs.

In this Letter, we demonstrate clearly that the symmetric scheme is not efficient for an asymmetric game with continuous strategies, and establish a new quantization scheme—we name it the asymmetric quantization scheme, which is necessary for the game to exactly reach the optimal cooperative payoffs. In Section 2 we retrospect the symmetric Bertrand game with differentiated products, including both classical and quantized version. Then in Section 3 we introduce the asymmetric Bertrand model, clarifying by concise calculation why the symmetric scheme fails to get the optimal cooperative payoffs, and construct the asymmetric quantization scheme to fulfill such payoffs. In Section 4 we extend our scheme to general asymmetric games with continuous strategies. Section 5 is the conclusion.