# Remarks on quantum duopoly schemes

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## 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.

### Keywords

Cournot duopoly Quantum game Nash equilibrium## 1 Introduction

Quantum game theory, an interdisciplinary field that combines quantum theory and game theory, has been investigated for fifteen years. The first attempt to describe a game in the quantum domain applied to finite noncooperative games in the normal form [1, 2, 3]. The general idea (in the case of bimatrix games) was based on identifying the possible results of the game with the basis states \(|ij\rangle \in \mathbb {C}^{n}\otimes \mathbb {C}^{m}\). Soon after quantum theory has also 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 therefore, there are continuum of possible game results. The scheme presented in [4] adapts the Marinatto–Weber quantum \(2\times 2\) game scheme [3] to the Stackelberg duopoly example. The model relates actions in the duopoly to the probabilities of applying the bit-flip operators. On the other hand, the model introduced in [5] is a new framework compared with the quantum \(2\times 2\) game schemes. It was defined to consider the Cournot duopoly problem where each strategy in the classical game corresponds to a specific unitary operator. The model entangles the players’ quantities, and the relation depends on a degree of the entanglement.

The Iqbal-Toor [4] and Li–Du–Massar [5] schemes undoubtedly brought new ideas to the field of quantum games. These remarkable schemes have found an application in many duopoly problems. The former scheme was further investigated, for example, in [6, 7, 8, 9, 10], the latter one in [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. The aim of the paper is to pay attention to some properties of the quantum duopoly schemes that might be thought unsuitable (in the case of the Iqbal-Toor scheme) and specify the Nash equilibrium analysis (in the case of the Li–Du–Massar scheme).

## 2 Cournot’s duopoly model

- 1.
\(N=\{1,2\}\) is a set of players,

- 2.
\(S_{i} = [0,\infty )\) is a player

*i*’s strategy set, - 3.\(u_{i}\) is a player
*i*’s payoff function given by formulaHere \(P(q_{1}, q_{2})\) represents the market price of the product,$$\begin{aligned} u_{i}(q_{1}, q_{2}) = q_{i}P(q_{1}, q_{2}) - cq_{i} ~~\text{ for }~~ q_{1}, q_{2} \in [0,\infty ). \end{aligned}$$(1)and$$\begin{aligned} P(q_{1}, q_{2}) = {\left\{ \begin{array}{ll} a-q_{1} - q_{2} &{}\quad \text{ if } q_{1} + q_{2} \leqslant a \\ 0 &{}\quad \text{ if } q_{1} + q_{2} > a, \end{array}\right. } \end{aligned}$$(2)*c*is a marginal cost with \(a>c > 0.\)

It is appropriate at this point to note that in the literature one can find (2) with requirement \(a>c\geqslant 0\) and a statement that the Cournot duopoly has the unique Nash equilibrium. In fact, \(c>0\) is crucial to the uniqueness of the equilibrium. If \(c=0\) and one of the players, say player 1, chooses \(q_{1}\geqslant a\), then the player 2’s set of best replies is \([0,\infty )\). By completely symmetric arguments, \(q_{1} \in [0,\infty )\) is player 1’s best reply to \(q_{2}\geqslant a\). Thus, there would be continuum many equilibria \((q^*_{1}, q^*_{2})\) such that \(q^*_{1}, q^*_{2} \geqslant a\) with payoff 0 for both players.

## 3 Remarks on existing quantum duopoly schemes

In this section, we discuss the two main quantum approaches to the problem of duopoly. Both schemes show how to define game with uncountable sets of strategies.

### 3.1 The Iqbal-Toor quantum duopoly scheme

*x*and \((1-x)\)\((y ~\text{ and }~ (1-y))\) are the probabilities of choosing by player 1 (player 2) the identity operator \(\mathbbm {1}\) and the Pauli operator \(\sigma _{x}\), respectively. In order to associate player

*i*’s actions \(q_{i} \in [0,\infty )\) for \(i=1,2\) with final state (4), the authors defined the following probability relations:

*x*and

*y*are given by Eq. (5), the final state (4) can be written as

*i*’s payoff is of the form

*i*would benefit by choosing \(q'_{i} > q_{i}\). Thus, one may question, if the players are able to obtain superior payoffs compared to the classical case. However, as it has been mentioned, that model does not take into account the proper price function (2). It cannot then be considered in terms of a generalization of the Cournot model. Let us now replace (7) with (9). This gives

### 3.2 The Li–Du–Massar quantum duopoly scheme

The quantum protocol introduced in [5] is another way to define the problem of duopoly in the quantum domain. Let us recall the formal description of this scheme [5, 25].

*i*’s strategies depend on \(x_{i}\in [0,\infty )\), and they are given by formula

*A*and

*B*satisfy the relation \([A, [A,B]] = \beta B\), \(\beta \)-constant. Given (28) the payoff functions are

The question is now: Whether the quantum scheme with payoff functions (29) and (30) has different sets of Nash equilibria. Similar to the classical case, if \(c=0\), each profile \((x^*_{1},x^*_{2})\) such that \(x_{1}, x_{2} \geqslant \mathrm {e}^{-\gamma }a\) is a Nash equilibrium in the case (30). The players obtain the payoff 0, and any unilateral deviation from the equilibrium strategy does not change the player’s payoff. If \(c>0\), then there is a unique Nash equilibrium \((x^*_{1}, x^*_{2})\) that coincides with the one determined in [5]. However, the proof of existence and uniqueness of the equilibrium need a more sophisticated reasoning compared with [5]. In what follows, we give a rigorous proof of the following fact:

**Proposition 1**

*c*is positive, the quantum Cournot duopoly defined by the Li–Du–Massar scheme with the payoff function (30) has the unique Nash equilibrium \((x^*_{1}, x^*_{2})\) such that

*Proof*

## 4 Another example of the quantum Cournot duopoly scheme

The Li–Du–Massar scheme with the refined payoff function (30) is defined in accordance with the quantum protocols for finite games. The scheme generalizes the classically played Cournot duopoly, and it keeps the set of feasible payoff profiles unchanged. We saw in Sect. 3.1 that the scheme based on the Marinatto–Weber approach for bimatrix games does not satisfy the latter condition. The question now is whether these two requirements on a quantum scheme imply the unique quantum model for the Cournot duopoly problem. The two well-known and quite different quantum schemes for bimatrix games introduced in [2] and [3] suggest that the answer ought to be negative. In fact, this is the case. We can define another scheme that is consistent with the requirements above. In what follows, we give an example of a scheme that is similar in concept to the Li–Du–Massar scheme. The idea is based on ‘entangling’ the players’ quantities \(x_{1}\) and \(x_{2}\) in order to obtain \(q_{1} = ax_{1} + bx_{2}\) and \(q_{2} = ax_{2} + bx_{1}\) for some real numbers *a* and *b*.

*X*, respectively, defined on \(\mathbb {C}^2\). The resulting state is then given by

*i*’s strategies with \(x_{i} \in [0,\infty )\) for \(i=1,2\). The values \(x_{1}\) and \(x_{2}\) determine two positive operators \(\left\{ M_{1}(x_{1},x_{2}),M_{2}(x_{1},x_{2})\right\} \) given by formula

**Proposition 2**

*N*of Nash equilibria in the game defined by scheme (40)–(45) is as follows:

*Proof*

## 5 Conclusions

The theory of quantum games has no rigorous mathematical structure. There are no formal axioms and definitions that would give clear directions of how a quantum game ought to look like. In fact, only one condition is taken into consideration. It says that a quantum game ought to include the classical way of playing the game. As a result, this allows one to define a quantum game scheme in many different ways. The schemes we have studied in Sect. 3 are definitely ingenious. They make a significant contribution to quantum game theory. Our work has shown which of the two schemes (the Iqbal-Toor scheme or the Li–Du–Masasr scheme) might be considered more reasonable. The payoffs in the game determined by the Iqbal-Toor scheme can go beyond the classical set of feasible payoffs compared with the Li–Du–Masasr scheme. Therefore, one may question whether the former scheme outputs the game played in a quantum manner or just another classical game. It might not mean that the latter scheme with (30) gives the definitive form of the quantum Cournot duopoly. At the end of the paper, we defined another scheme in terms of quantum theory.

We can conclude that the question of quantum duopoly is still an open problem even in the simplest case of the Cournot duopoly. It is definitely worth investigating as other new schemes may bring us closer to specify a strict definition of a quantum game.

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