Pareto Efficiency of Mixed Quantum Strategy Equilibria

: The aim of the paper is to investigate Nash equilibria and correlated equilibria of classical and quantum games in the context of their Pareto optimality. We study four games: the prisoner's dilemma, battle of the sexes and two versions of the game of chicken. The correlated equilibria usually improve Nash equilibria of games but require a trusted correlation device. We analyze the quantum extension of these games in the Eisert-Wilkens-Lewenstein formalism with the full SU(2) space of players’ strategy parameters. It has been shown that the Nash equilibria of these games in quantum mixed Pauli strategies are closer to Pareto optimal results than their classical counterparts. The relationship of mixed Pauli strategies equilibria and correlated equilibria is also analyzed.

in games of quantum communication. In the EWL approach with the SU(2) strategy set, obtaining Pareto-efficient solutions is feasible but the problem is that this 3-parameter strategy space yield only trivial Nash equilibria. On the other hand many authors tried to investigate EWL scheme with a 2-parameter strategy space. This however leads to an undesirable dependence of the equilibria on the selected parameterization [12]. To resolve this dilemma we propose using mixed strategies based on 3-parameter SU(2) pure strategies, which allow for non-trivial NE and, at the same time, are not dependent on the choice of 2-dimensional parameter subspace.
In the present work, we study four games in which the problem of suboptimal Nash's equilibrium arises: the prisoner's dilemma, battle of the sexes and two versions of the game of chicken. Thanks to the use of mixed quantum strategies, we obtain both: nontrivial Nash equilibria and that they are closer to Pareto-efficient solutions than classical equilibria. The ultimate goal is to design a quantum device, the input of which is operated by players, parties to the conflict, economic institutions, and the output, through the collapse of the wave function, determines the result of the game, the solution of the dispute or conflict between the parties. The speed with which quantum technologies are currently developing allows us to assume that the efficient quantum strategies may soon be applicable to real practical problems [13].
In the second section the basic concepts of games and their payouts in pure, mixed strategies and general probability distributions are defined. We also define the concepts of the Nash equilibrium, Pareto-efficiency and correlated equilibrium. The third section, presents four classical games, discuss their Nash equilibria and analyzes their Pareto-optimality. We also discuss their correlated equilibria, which thanks to the use of additional mechanisms of correlation of players' behavior, allow for better Pareto optimization of the results of these games. The fourth section is devoted to defining the concept of quantum game in the EWL scheme with the full SU(2) parameter space. Part five of the paper presents our proposals for new Nash equilibria in quantum mixed strategies and their comparison with correlated equilibria. In the last part we discuss the applicability of both correlation mechanisms and the perspective of physical implementation of quantum games. ( 00 , 00 ) ( 01 , 01 ) ( 10 , 10 ) ( 11 , 11 ) ). Let us denote by

Game theory preliminaries
the set of all probability distributions over × . The payoff of a Player , corresponding to a given distribution = { } , =0,1 is Let us now restrict the set of all probability distributions to distributions, that can be factorized, i.e. presented in a form ( 00 01 10 11 They define mixed strategy spaces which are defined by a single number [0,1]. Note that the product of mixed strategy spaces is a subset of the set of all probability distributions ΔS × ΔS ⊂ Δ( × ).
From the viewpoint of mutual efficiency, the concept of Pareto optimality plays an important role. Let be an arbitrary set of strategies. A pair of strategies ( , ) is not Pareto optimal in if there exists another pair, ( ′, ′) that is better for one of the players ΔP ( , ) < ΔP ( ′, ′) , and not worse for the other Player ΔP − ( , ) ≤ ΔP − ( ′, ′), where − is the remaining player for player = , , otherwise the pair ( , ) is called Pareto optimal (or Pareto-efficient) in . A set of all Pareto optimal strategies for a given set of strategies is denoted ( ). For instance a pair of strategies ( , ) ΔS × ΔS is Pareto optimal in ΔS × ΔS if there exist no other set of mixed strategies, that would be better for at least one of players and not worse for the other. Note that the Pareto optimal strategy in a set is not necessarily optimal in a larger set ′ ⊃ .
An interesting concept of optimizing equilibria beyond the classical game theory was put forward by R. Aumann . By correlated equilibrium we understand a situation in which players make their optimal decisions, guided by an external signal, transmitted to them by a trusted correlating device according to a given probability distribution. Each player maximizes his expected payoff by following this recommendation. Formally, probability distribution { } , =0,1 over the set of action vectors ( , ) , =0,1 of the game is called a correlated equilibrium [15], if for every strategy and respectively, where − is the remaining strategy − ≠ . One of the advantages of correlated equilibria is that they are computationally easier than Nash equilibria. Computing the correlated equilibrium requires only solving the linear problem, while solving the Nash equilibrium requires solving the equations that make each player's payoffs independent of the others.

The efficiency of classical games
The most contrasting example of the lack of Pareto optimality for Nash equilibria is the prisoner's dilemma (PD) game [16]. The game is universal in nature and describes many decision-making dilemma commonly found in different situations of social life. It is defined by = ( , { } , { } ) and the payoffs are defined by the bimatrix in Table 1, where > > > and > + 2 [17]. A typical scenario assumes that two players, Alice and Bob, independently of each other, choose one of two strategies -"cooperation" 0 and 0 or "defection" 1 and 1 .
It is easy to see that regardless of the opponent's choice, the dominant strategy of each player is to "defect" and the pair of mutual defection strategies ( 1 , 1 ) is the Nash equilibrium of the game. On the other hand the Pareto-efficient solutions are all the remaining pairs of pure strategies. Moreover, when allowing the players to randomize their strategies, the Nash equilibrium remains the same and the set of all Pareto optimal strategies is (ΔS × ΔS ) = 0 × ΔS ∪ ΔS × 0 . In case of typical game payoffs: = 5, = 3, = 1, = 0, the Nash equilibrium ( 1 , 1 ) with a payoff of (1,1) is far from the Pareto optimal ( 0 , 0 ) with a payoff of (3,3).
The second game under consideration is battle of the sexes ( ), defined by the payoff bimatrix in Table 2. Alice and Bob plan to spend the evening together, for which they can get paid 2. However, Alice would prefer to go to the theater 0 , whereas Bob would prefer the football game 1 , = , . Going to a preferred place gives players an additional bonus of +1.
This game has two Nash equilibria ( 0 , 0 ) and ( 1 , 1 ) in pure strategies. Both of them form a set of Pareto optimal solutions (ΔS × ΔS ) = ( 0 , 0 ) ∪ ( 1 , 1 ) but the problem, which gives the name to the game, is that they can not be both satisfied with a just solution. One player consistently does better than the other.
has also one NE in mixed strategies, in which players go to their preferred event more often than the other. It is given by a pair of strategies σ = ) is however not Pareto-efficient in ΔS × ΔS because e.g. each of the pure strategy NE is better for both players.
The last of the classical games we consider is the game of chicken (chicken game), with the payoff bimatrix defined in Table 3. This game describes, e.g. the behavior of two drivers approaching, one from the south and one from the west, at the same time to the intersection. They both have two options: to cross the intersection 1 or to stop 0 , = , before it. If both of them choose the option to drive, they will collide and both lose 10. If only one of them passes and the other stops, the passing one wins (1,0). If both of them stop, the result is neutral (0,0).
has two Nash equilibria in pure strategies ( 0 , 1 ) and ( 1 , 0 ), which are Pareto-efficient. However, none of these equilibria, just like in , satisfy both players. The game also has the third equilibrium in mixed strategies: each car passes a crossroads with a probability of 1/11. This equilibrium is fair -both players receive equal payouts, but the trouble is that both payouts are equal to 0, and therefore not optimal in ΔS × ΔS -each player can increase his payout by increasing the frequency of crossing, while the other stops at the junction.
Let's consider again the chicken game but with different, positive payoffs: Table 4. The payoff matrix of the game of chicken II As in the previous game, the winner is the player who chooses the 1 option while the other one plays 0 , = , . The best solution is for both players to choose ( 0 , 0 ) but it is not an equilibrium. As before, this game has three Nash equilibria: two in pure strategies ( 1 , 0 ) and ( 0 , 1 ) and one in a mixed strategy, in which both players choose 0 and 1 with equal probabilities σ = ) respectively. As before, Pareto-efficient equilibria are not fair (in the sense that one player wins and the other loses), and the fair equilibrium is not Pareto-efficient (because both players can score better in ΔS × ΔS by choosing ( 0 , 0 ). It follows from (4) that the correlated equilibrium for this game should obey four inequalities: 00 ≤ 01 , 00 ≤ 10 , 11 ≤ 01 and 11 ≤ 10 . It is easy to show that the correlated equilibrium corresponding to the highest equal payoffs for both players is = ( 00 01 and the corresponding payoffs are (3 ), better then in the symmetric Nash equilibrium. Aumann [18] proposed the following mechanism of correlated equilibrium realization. Let's consider the third side (or some natural event), which with a probability of 1/3 draws one of three cards marked: (0,0), (0,1) and (1,0). After the card is drawn, the third party informs the players about the strategy assigned to them on the card (but not about the strategy assigned to the opponent). Suppose one player is assigned "1", knowing that the other player saw "0" (because there is only one card that assigns him "0"), he should play "1" because he will receive the highest possible payout 5. Let's assume that the player was assigned "0". Then he knows, that the other player has received "0" or "1" commands, with probabilities 1/2. The expected payoff for playing "1" (contrary to the recommendation) is therefore 5 * . Because none of the players has motivation to play differently than was recommended by the third party, the result of the draw is the correlated equilibrium. The probability distribution Δ( × ) can not be factorized as in equation (3) and therefore is not a mixed game strategy ∉ ΔS × ΔS . It is also not Pareto-efficient ∉ (Δ( × )) in the set of all probability distributions. Back to the first example of a chicken game, the traffic light installed at the intersection may act as a correlation mechanism given by the matrix of the drivers with a probability of ½ meets the green light. It is a correlated equilibrium because none of the drivers is interested in running a red light, knowing that the other one is green at that time. If they both comply with the traffic rules, they will receive a payment of ½, i.e. higher than the mixed strategy Nash equilibrium. It has the highest, equal for both players payoff because it is Pareto-efficient in the set of all probability distributions (Δ( × )) but not accessible by any mixed strategy as ∉ ΔS × ΔS .
One can also find an optimal correlated equilibrium for battle of the sexes game. The correlated equilibrium definition (4) yields inequalities: 3 00 ≥ 01 , 00 ≥ 3 10 , 3 11 ≥ 01 and 11 ≥ 3 10 . The equal payoff optimal solution is then (2 . It means that they come out together to the theater or the football game depending on the coin toss. This payout is higher than the Nash equilibrium in mixed strategies and is Pareto-optimal (Δ( × )) in the set of all probability distributions and again, not accessible by any mixed strategy ∉ ΔS × ΔS . In case of the prisoner's dilemma the conditions (4) show that there are no correlated equilibria other that its NE. It is because both cooperation strategies 0 and 0 are strictly dominated and therefore can never be played in a correlated equilibrium.
The disadvantage of correlated equilibria is the need to use an external signal that must be generated by an independent device that can be manipulated. It is not difficult to imagine that if one of the drivers is, for example, an important politician, then when he passes through an intersection where a policeman (acting as a correlating device) is standing, he has priority and the other driver has to wait. Therefore, it is worth looking for correlation mechanisms that would be safe and not susceptible to manipulation. As in the field of cryptography [19], such a solution may be transferring games to the quantum domain

Quantum games
In recent years, we have witnessed the rapid development of research on quantum information processing [20,21] and successful experiments related to the engineering of entangled qubits [22,23]. In the laboratories of Google Quantum AI [24], IBM [25], D-wave and several other companies [26], there is a race to achieve the so-called quantum supremacy. Google AI Quantum managed to construct a quantum processor based on 53 qubits, which in 200 seconds solved a problem that a classical computer would solve in 10 thousand years [24]. In the field of possible applications of quantum engineering, quantum games are also attracting much attention [27,28]. Apart from their own intrinsic interest, quantum games explore the fascinating world of quantum information [29][30][31].
The idea of quantum computers use to extend classical games to the quantum domain was put forward at the end of the 20th century. In his groundbreaking work on the theory of quantum games [32], Meyer proposed a simple coin toss game and showed that a player using quantum superposition will always win against a classical player. A general protocol for quantum games was proposed by Eisert, Wilkens and Lewenstein (EWL) [11]. This model has been widely discussed [33] and, e.g. extended to multiplayer games [34].
In this approach, players' strategies are operators in a certain vector space known as a Bloch sphere [35]. This space is a set of qubits -normalized vectors with complex coefficients spanned on a two-element basis {|0⟩, |1⟩} which, up to the phase, can be represented in the form | ⟩ = cos 2 |0⟩ + e i sin 2 |1⟩, where ∈ [0, ] and ∈ [− , ]. Qubits | ⟩ represent superposition of the basis states |0⟩ and |1⟩ are pure quantum states. A qubit in a state (5) does not have any value "between" |0⟩ and |1⟩. It means that before the measurement is carried out, it is not defined and only the measurement yields a value of |0⟩ or |1⟩ with probabilities cos 2 2 and sin 2 2 respectively. This process is called the collapse of the wave function. For example, all qubits representing states with = /2, i.e. at the equator of the Bloch sphere represent a quantum state which, after measurement, collapses to the state |0⟩ or |1⟩ with probabilities equal to (|00⟩ + |11⟩) is the maximally entangled (Bell) state [36]. From now on we assume that = 2 , i.e. the initial state is fully entangled. We also assume that the initial state | 0 ⟩ is known to both players.
In quantum game theory, players' strategies are unitary transformations ̂ i ̂ operating on the initial state | 0 ⟩. Transformations ̂∈ SU (2)  is constant. However, Benjamin & Hayden [37] observed that the set of 2 dimensional quantum strategies is not closed under composition and Frąckiewicz [38] showed, that only the full (2) strategy parameter space provides a strong isomorphism of the classical and quantum game.
In the special case where the players' strategies are defined only by the angle θ, with fixed = = 0, they can be expressed by ̂( , 0,0) = cos 2̂+ sin 2 . In this case, ̂( 0,0,0) =̂ is the unit matrix corresponding to the classical 0 ( 0 ) strategy and respectively. In this way the classical game becomes a special case of the quantum game. Quantum games can be physically implemented by a quantum computer operating according to the above algorithm. Such an algorithm was carried out experimentally [39,40] in EPR-type experiments based on measurements of the Stern Gerlach effect. The players initially share an entangled pure quantum state | 0 ⟩. Each of them apply his strategy by performing arbitrary local unitary operations on his own qubit, but no direct communication between players is allowed. The result of the game is revealed, by measuring the final state (7) which, as a result of the collapse of the wave function, will give one of the four possible states with the appropriate probability. Due to the fact that players use quantum strategies, entanglement offers opportunities for players to interact with each other, which has no analogue in classical games.
The probability distribution leading to the payoff of the quantum game (8) is, in general non-factorizable and therefore can play a role of the external device correlating player actions proposed by Aumann. There is no need to use cryptographic protocols to replace the trusted mediator [41]. In this case, quantum mechanics offers the possibility of randomizing players' strategies better than classical methods.

Nash equilibria of quantum games
Let's go back to optimization of game equilibria. In the classical prisoner's dilemma (Table 1), the only Nash equilibrium is the mutual defection ( 1 , 1 ). In the EWL  quantization scheme with 2D parameter space, there is a new Nash equilibrium, the "magic" strategy denoted by ̂≡̂( 0, 2 , 0) = ( 0 0 − ) corresponding to the Pareto-efficient payoff (3,3). However, if we consider the above strategy in the full SU (2) space, then the "Nash equilibrium" obtained in this way ceases to be the equilibrium. Indeed, for any strategy ̂( , , ), ∈ SU(2) there is a strategy ̂=̂( + , − /2, ) which "cancels" the action ̂ of the Player A and changes the game result to (0, 5) in favor of the Player B. The result is the same if the answer of the Player B is ′ =̂( + , - . It is easy to show, that in (3) case of EWL the Nash equilibrium can exist only if the original game bimatrix has the result with maximal payoffs for both players. However, a Nash equilibrium can be built by mixing pure quantum strategies, what leads to mixed quantum strategies [42,43]. Let us consider a set of quantum strategies: The names of these strategies refer to their similarity to the Pauli matrices ̂= −̂, ̂= −̂ and ̂=̂, and therefore can be named Pauli strategies. They form a basis of infinitesimal generators of SU (2).
Let us consider a quantum game Γ , where the set of unitary strategies is = { 0 , ,̂,̂}. The final state of the game | ⟩ = † (̂ ⨂ ̂)̂ |00⟩, where , ∈ {0, , , }, can be expanded in terms of a single vector of an observational basis. Therefore payoffs corresponding to this game (Table 5.) are single bimatrix pairs of the original classical game. Note that for any strategy of Player A, there is such a strategy of Player B, that the result of the quantum game is any pair of payoffs of the original game. Having this matrix, one can now construct mixed Pauli strategies defined by quad- ). Note that this quantum equilibrium gives both players a much higher payoff than the Nash equilibrium and the best correlated equilibrium, both yielding a payoff of (1,1). Similarly, we can find a Nash equilibrium for battle of the sexes game from Table 2. Likewise the quantum PD, this game has no equilibrium in pure quantum strategies. One can check that, the highest payouts of the game occur in two subgames defined by pairs of quantum strategies { 0 ,̂} and {̂,̂}. Therefore, one can be built two pairs of equilib- ). It is however worse than maximal correlated equilibrium (3 ). The comparison of the obtained results is presented in Table 6. Interestingly in the family of all mixed Pauli strategic equilibria, there is e.g. = (

Conclusions
In this paper, we were looking for game solutions that would be closer to the Paretoefficient results than classical game solutions. We took into account: the prisoner's dilemma game, battle of the sexes and two versions of the chicken game. For most of these games (apart from PD), we have shown that there are correlated equilibria that improve Nash equilibria of these games. However, obtaining results in this way requires the introduction of an external device that correlates the actions of players. Such a device, sending signals to players, would be vulnerable to manipulation and difficult to use. Therefore, we proposed to use the quantum domain extension of games. In the paper we adopted the most common formalism of Eisert-Wilkens-Lewenstein quantum games, extended to the full space of SU(2) strategies. We have shown that the games under consideration have, in the mixed strategies, Nash equilibria much closer to Pareto-efficient solutions than the equilibria of classical games. These equilibria are comparable to correlated equilibria.
In the case of the prisoner's dilemma, the Nash equilibrium of the quantum game corresponds to mixing with equal probability of cooperation and defection. Although this result is not Pareto-efficient, like mutual cooperation, the players' payoffs obtained in this way are better than the best correlated equilibrium, which is equal to the Nash equilibrium of the classical game. In the case of battle of the sexes, the best correlated equilibrium coincides with the quantum NE, it is fully fair for both partners and Pareto-efficient. For the chicken game, the best Pareto optimal, correlated equilibrium also coincides with the Nash equilibrium of quantum game. This solution is unattainable in classical mixed strategies. In the second version of the chicken game, the best equal solution obtained in mixed Pauli strategies is better than classical NE but worse than the one achievable in correlated equilibria. However, there is also an asymmetric solution with payoffs, the sum of which is greater than the sum for the correlated equilibrium. Neither of these solutions is Pareto efficient.
The question, whether quantum versions of games can contribute to solving practical economic situations, naturally arises. Examples from other fields indicate that quantum strategies can solve the problems of market games [44], duopoly problems [45,46], auctions and competitions [47] or gambling [48] better than classical strategies. As is clear from this study, a solution of games with the help of quantum strategy can give better results than conventional solutions.
A general question can be asked: are there any connections between classical games and quantum phenomena? As a mathematical theory, classical games turn out to be a special case of quantum games. Do real classical games played by people every day have anything to do with physical quantum processes? The answer to this question may be surprising. A quantum phenomenon "suspected" of combining both realities is the collapse of the wave function. According to a recent hypothesis, the quantum fluctuations cause macroscopic phenomena that we consider random, such as, for example, tossing a coin or a die [49]. Moreover, every practical use of probability has its source in quantum phenomena. If this point of view were taken, any use of mixed strategy in a classical game would in fact be a quantum phenomenon.
In quantum games, an important element of the game mechanism is a quantum coherence. This phenomenon, by nature, has no analogue to the classical game. Problems with the decoherence of the wave function make it difficult to maintain two entangled qubits even at the level of strictly controlled experiments, taking place under extreme conditions of isolation from the environment. Building a quantum computer based on a register of many entangled qubits, subjected to unitary quantum gate operations and capable of solving practical problems or simulating quantum games with quantum algorithms is a real challenge. However, in recent years we have seen more and more successful attempts to build such a computer and use it to implement quantum games.