# A new quantum scheme for normal-form games

## Abstract

We give a strict mathematical description for a refinement of the Marinatto–Weber quantum game scheme. The model allows the players to choose projector operators that determine the state on which they perform their local operators. The game induced by the scheme generalizes finite strategic-form game. In particular, it covers normal representations of extensive games, i.e., strategic games generated by extensive ones. We illustrate our idea with an example of extensive game and prove that rational choices in the classical game and its quantum counterpart may lead to significantly different outcomes.

### Keywords

Normal-form game Centipede game Quantum game Nash equilibrium## 1 Introduction

A 15-year-period research on quantum games results in many ideas of how a quantum game might look like and how it might be played. Certainly, the quantum scheme for \(2 \times 2\) games introduced in [1] (the EWL scheme) has become one of the most common models and it has already found application in more complex games (see, for example, [2]). However, the more complex the classical game is, the more sophisticated techniques are required to find optimal players’ strategies in the EWL-type scheme. While in the scheme for \(2 \times 2\) games the result of the game depends on six real parameters (each players’ strategy is a unitary operator from \(\mathsf {SU}(2)\), and it is defined by three real parameters), the EWL-type scheme for \(3\times 3\) games would already require 16 parameters to take into account [3, 4]. One way to avoid cumbersome calculations when studying a game in the quantum domain was presented in [5] (see also recent papers [6, 7, 8] and [9] based on this scheme). The authors defined a model (the MW scheme) for quantum game where the players’ unitary strategies were restricted to the identity and bit-flip operator. Then, the game became *quantum* if the players’ local operators were performed on some fixed entangled state \(|\varPsi \rangle \) (called the players’ joint strategy). The MW scheme appears to be much simpler than the EWL scheme. The number of pure strategies of each player is the same as in the classical game [10]. Thus, the complexity of finding a rational solution is similar in both a classical game and the corresponding quantum counterpart. Unfortunately, that simple scheme exhibits some undesirable properties that we pointed out in [11]. First, the MW scheme implies *non-classical* game even if the players’ joint strategy is an unentangled state. In particular, if a player’s qubit is in an equal superposition of computational basis states, she cannot affect the game outcome in contrast to her strategic position in the classical game. Moreover, the players have no impact on the form of the initial state. In paper [11], we showed that the above-mentioned drawbacks vanish by allowing the players to choose between the basis state that represents the classical game and the state \(|\varPsi \rangle \). In this paper, we continue that line of research. We give a formal description for players’ strategies to include the choice of the initial state in the MW scheme. It will allow us to move beyond bimatrix games examined in [11] and consider more general normal-form games. Then, we study possible applications of the scheme.

Some knowledge of game theory is required to follow this paper. While theory of bimatrix games is commonly used in quantum game theory, the notion of normal representation of extensive games may not be known for readers that deal with quantum games. Therefore, we encourage the reader who is not familiar with extensive game theory to see one of the textbooks [12, 13].

## 2 Refinement of the Marinatto–Weber scheme

In paper [11], we introduced a new scheme for playing finite bimatrix games in the quantum domain. The idea behind the scheme is that the players can choose whether they play a classical game or its quantum counterpart defined by the MW scheme. In the case of quantum model for \(2\times 2\) bimatrix games, this means that the players choose their local operations: the identity \({\mathbb {1}}\) or the Pauli operator \(\sigma _{x}\) and, additionally, they decide whether the chosen operators are performed on state \(|00k\rangle \) or some fixed state \(|\varPsi \rangle \in \mathbb {C}^2\otimes \mathbb {C}^2\). Now, we give a formal description for the scheme.

### 2.1 Quantum model for \(2\times 2\) bimatrix game

- 1.A positive operator \(H\),where \(|\varPsi \rangle \in \mathbb {C}^2\otimes \mathbb {C}^2\) such that \(\Vert |\varPsi \rangle \Vert = 1\),$$\begin{aligned} H = ({\mathbb {1}}\otimes {\mathbb {1}} - |11\rangle \langle 11|)\otimes |00\rangle \langle 00| + |11\rangle \langle 11|\otimes |\varPsi \rangle \langle \varPsi |, \end{aligned}$$(2)
- 2.Players’ pure strategies: \(P^{(1)}_{i}\otimes U^{(3)}_{j}\) for player 1, \(P^{(2)}_{k}\otimes U^{(4)}_{l}\) for player 2, where \(i,j,k,l =0,1\), and the upper indices identify the subspace \(\mathbb {C}^2\) of \((\mathbb {C}^2)^{\otimes 4}\) on which the operatorsare defined. That is, player 1 acts on the first and third qubit and player 2 acts on the second and fourth one. The order of qubits is in line with the upper indices.$$\begin{aligned} P_{0} = |0\rangle \langle 0|,~ P_{1} = |1\rangle \langle 1|, \quad U_{0} = {\mathbb {1}},~ U_{1} = \sigma _{x}, \end{aligned}$$(3)
- 3.Measurement operators \(M_{1}\) and \(M_{2}\) are given by formulawhere \(a_{xy}\) and \(b_{xy}\) are the payoffs from (1).$$\begin{aligned} M_{1(2)} = {\mathbb {1}}\otimes {\mathbb {1}}\otimes \left( \sum _{x,y = 0,1}a_{xy}(b_{xy})|xy\rangle \langle xy|\right) , \end{aligned}$$(4)

*Nash equilibrium*In non-cooperative quantum game theory, Nash equilibrium is the most used solution concept. It is defined as a profile of strategies of all players in which each strategy is a best response to the other strategies. In view of scheme (2)–(4), it is a mixed strategy profile \(\left( (p^*_{ij})_{i,j=0,1},(q^*_{kl})_{i,j=0,1}\right) \) that solves the following optimization problems:

*Bimatrix form*The game given by scheme (2)–(4) can be expressed in terms of bimatrix form. Each entry of the bimatrix is a pair \(\left( \mathrm{tr}(\rho _{\mathrm{f}}M_{1}), \mathrm{tr}(\rho _{\mathrm{f}}M_{2})\right) \) of payoffs that corresponds to a particular profile \(P^{(1)}_{i}\otimes P^{(2)}_{k}\otimes U^{(3)}_{j} \otimes U^{(4)}_{l}\). As a result, we obtainwhere

Note that bimatrix (14) clearly shows the role of components \(P_{i}\) of players’ strategies. Namely, the operations \(U^{(3)}_{j}\otimes U^{(4)}_{l}\) are performed on state \(|\varPsi \rangle \) if and only if both players form profile \(P^{(1)}_{1}\otimes P^{(2)}_{1} \otimes U^{(3)}_{j}\otimes U^{(4)}_{l}\).

### 2.2 Quantum model for general bimatrix games

## 3 Quantum approach to finite normal-form games

In the previous section, we formalized the refinement of the MW scheme that was introduced in [11]. We obtained the scheme that can be applied to any finite bimatrix game. In this section, we construct a framework for general normal-form games. The term of normal-form game has two main meanings. One concerns a strategic game given a priori. It is defined by triple \((N,\{S_{i}\}_{i\in N}, \{u_{i}\}_{i\in N})\), where \(N\) is a set of players and, for \(i\in N\), components \(S_{i}\) and \(u_{i}\) are player i’s strategy set and payoff function, respectively. The second meaning concerns a strategic game \((N,\{S_{i}\}_{i\in N}, \{u_{i}\}_{i\in N})\) that is generated by a game in extensive form. The strategic game obtained in this way is called the normal representation of the extensive game. In what follows, we extend the scheme (2)–(4) to cover both cases.

### 3.1 Strategic-form game

The difference between bimatrix games and finite strategic games is that more than two players (say \(n\) players) are allowed in the latter case. Therefore, operator (2) has to be modified in such a way that it simply outputs a density operator after \(n\) players’ strategies act on it.

*Example 1*

### 3.2 Normal representation of extensive games

Given an extensive-form game, one can construct a representation of that game in the strategic (normal) form. The resulting strategic game and the given extensive game have the same set of players and the same set of strategies for each player. The payoff functions are determined by the payoffs generated by the strategies in the extensive game. The normal representation appears to be a very convenient way to study the extensive game. In particular, while we lose the sequential structure, we obtain the sufficient and easier form of the game to find all the Nash equilibria.

In our earlier paper [16], we introduced a quantum scheme for playing an extensive game by using its normal representation. Based on the MW and EWL schemes, we assigned an action at each information set in an extensive game to a local operation on a particular qubit in the quantum game. As a result, a number of qubits on which each player was allowed to specify local operations were equal to the number of their information sets. In what follows, we extend our idea to the refinement of the MW scheme. This means that in addition to multiple choice of \({\mathbb {1}}\) and \(\sigma _{x}\), the players specify the state on which they perform the local operators.

*Example 2*

*Four-stage centipede game*) A centipede game is a 2-person extensive game in which the players move one after another for finitely many rounds. In some sense, it can be treated as an extensive counterpart of the Prisoner’s Dilemma. While both players are able to obtain a high payoff, their rationality leads them to one of the worst outcomes. An example of a four-stage centipede game is shown in Fig. 1. Each player has two information sets (in this case, they are represented by the nodes of the game tree) with two available actions at each of them. Each player can stop the game (action S) or continue the game (action C), giving the opportunity to the other player to make her choice. One way to learn how the game may end is by backward induction. If player 2 is to choose at her second information set, she certainly plays action \(S\) since she obtains 5 instead of 4—the result of playing action \(C\). Since players’ rationality is common knowledge, player 1 knows that by playing \(C\) at her second information, she ends up with payoff 3. Thus, player 1 chooses \(S\) that yields 4. Similar analysis shows that the players choose action \(S\) at their first information sets. Consequently, the backward induction predicts outcome \((2,0)\). As we focus on normal-form games, we construct the normal representation associated with the game in Fig. 1. Let us first determine the players’ strategies. We recall that a player’s strategy in an extensive game is a function that assigns an action to each information set of that player. Thus, each player has four strategies in the case of a four-stage centipede game. They can be written in the form \(SS, SC, CS\), and \(CC\), where, for example, \(CS\) means that a player chooses \(C\) at her first information set and \(S\) at the second one. Once the strategies are specified, we determine the payoffs that correspond to all possible strategy profiles. For example, \((SC, CC)\) determines outcome \((2,0)\) since player 1’s strategy \(SC\) specifies action \(S\) at her first information set. On the other hand, profile \((CC,CS)\) corresponds to payoff \((3,5)\) as player 1 always plays \(C\) and player 2 chooses \(S\) at her second information set. The players’ strategies together with the payoffs corresponding to the strategy profiles define the following normal representationBy using bimatrix (42), we can learn that rational players always choose action \(S\) at their first information sets. More formally, there are four pure Nash equilibria: \((SS,SS), (SS, SC), (SC, SS)\), and \((SC, SC)\), each resulting in outcome \((2,0)\).

*Example 3*

*N-stage centipede game*) Let us consider a centipede game where this time the number of stages is any even integer \(n\) for \(n\ge 2\). The extensive form for this game is given in Fig. 2. Similar to the four-stage centipede game, the \(n\)-stage case has also the unique equilibrium outcome \((2,0)\). Rational players choose action \(S\) at their own information sets even though the game enables the players to obtain the payoffs approximate to the number of stages. We have learned from the preceding example that there is a unique, symmetric, and pareto-optimal Nash equilibrium if (42) is extended to (46). It turns out that the result is valid in the general case. That is, there is a Nash equilibrium that implies the payoff \(n + 1/2\) for both players (pair of payoffs \((n + 1/2, n+ 1/2)\) is indeed a pareto-optimal outcome since it is the midpoint of the segment whose endpoints are \((n-1,n+1)\) and \((n+2,n)\)). In order to prove the existence of that equilibrium, let us generalize (43) and (44) to an arbitrary n-stage centipede game. Since there are two players and \(n\) information sets in the game, the positive operator \(H\) and the players’ strategies are given by (40) and (41) for \(k=2\). We assume that players 1 and 2 perform their local operators on qubits with odd and even indices, respectively. Thus, the map \(\xi :\{3,4,\dots , n+2\} \rightarrow \{1,2\}\) is given by formula

*th*qubit, we haveIn the case of \(j_{1} = 1\),

## 4 Conclusions

The aim of our research was to formalize our idea about the MW-type schemes. As a result, we have showed that the players’ strategies do not have to be unitary operators or even superoperators in the quantum game. Apart from unitary operators, they may include projectors that determine the state on which the unitary operations are performed. Thus, the initial state does not have to be a density operator. Certainly, the scheme is in accordance with the laws of quantum mechanics. The resulting state is given by a density operator, and therefore, the payoff measurement is well defined. A positive point of the scheme is the way it can be considered. Given a bimatrix game, the scheme outputs a bimatrix game. Consequently, it implies similar complexity in finding optimal strategies for the players. In addition, our model enables us to consider extensive games via the normal representation. Moreover, the example of the general centipede game has proved that the analysis does not have to be limited to simple games. We suppose that this argument may attract the attention of researchers to the refinement of the MW scheme.

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