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Two-player quantum games: When player strategies are via directional choices

We propose a scheme for a quantum game based on performing an EPR type experiment and in which each player's spatial directional choices are considered as their strategies. A classical mixed-strategy game is recovered by restricting the players' choices to specific spatial trajectories. We show that for players' directional choices for which the Bell-CHSH inequality is violated, the players' payoffs in the quantum game have no mapping within the classical mixed-strategy game. The scheme provides a more direct link between classical and quantum games.


I. INTRODUCTION
Broadly speaking, a quantum game [1][2][3][4] can be considered as a game [5][6][7] in which a player's payoff relations involve a set of quantum probabilities [8] that are obtained from each player's strategic actions or strategies. For instance, in the quantum version of a 2 × 2 game proposed in the Eisert Wilkens Lewenstein (EWL) scheme [2,3], each player's strategies are local unitary transformations performed on a maximally entangled state. The state evolves unitarily and the set of quantum probabilities is obtained by projecting the final quantum state of the game to a basis in 2 ⊗ 2 Hilbert space, in terms of which the payoff relations for each player are then expressed. Quantum games are surveyed in Refs. [9,10] and recent works in this area are in Refs. [12][13][14][15][16][17][18][19][20]. An extensive list of articles in this area are in Ref. [11].
A strategy profile is a Nash equilibrium (NE) [5][6][7]-with one strategy associated with each player-such that there remains no motivation for any player for unilaterally deviating from that profile. In the EWL scheme, a NE is a set of local unitary transformations that satisfies the Nash conditions. A quantization scheme can be viewed as a mechanism that establishes a convincing link between each player's strategies-quantum or classical-and a set of quantum probabilities, obtained from the players' strategies, and in terms of which each player's payoffs are then expressed. As players have access to much larger strategy sets in EWL scheme-relative to the strategy sets available to them in the classical game-Enk and Pike [21] argued that a quantum game in that scheme can be considered as an extended classical game. They argued that the quantized version of a game, in EWL scheme, solves a new classical gamewith players' strategy sets extended-without solving the dilemma within the original game.
This led to suggestions for using EPR type experiment [8,[22][23][24][25][26] in constructing quantum games [29,31] and in which each player's strategy set remain classical while resulting in a set of quantum probabilities-thus circumventing Enk and Pike's argument.
It appears to us that historically there have been two distinct approaches in the literature in the area of quantum games. The first approach considers specially-designed classical games, for instance, the game proposed by Vaidman in Ref. [4] that involves a winning condition, in which a quantum advantage can be demonstrated directly. The second approach, however, develops quantization procedures for a whole class of classical games, as reported in Refs. [2,3]. The second approach is distinct from the first in that a game is not designed in order to demonstrate an advantage in its quantum mechanical implementation-usually tied to crafting a winning condition-but the objective, instead, is to determine how an implementation that allows access to the resources of quantum superposition and entanglement, results in a different outcome of the game. The present paper is along the lines of the second approach.
Non-cooperative games using a tripartite EPR experiment with GHZ states are discussed in Refs. [28,30], and in references therein. A tripartite EPR setting using GHZ states is considered in Ref. [33] that presents a quantum version of a three player non-cooperative game. Each player's strategic choices are three directionsâ,b, andĉ along which the dichotomic observables n · σ are measured, where n =â,b,ĉ and σ is a vector whose components are the standard Pauli matrices σ x , σ y , and σ z .
In this paper, we present a scheme for playing a two-player quantum game in which each player's (classical) strategy sets-consist of orientating his/her unit vector along any direction in three dimensions-and dichotomic measurement outcomes of ±1 along those directions. This scheme therefore uses each player's classical strategies to obtain a set of quantum probabilities in terms of which each player's payoff relations are then expressed.
As the players' strategies are directional choices, Nash equilibria of the game emerge as directional pairs. For the players' directional choices for which the Bell-CHSH inequality is violated, the payoffs in the quantum game cannot be mapped to a classical mixed-strategy game. As the players in our scheme have access to classical strategy sets, it provides a more direct link between classical and quantum games.
The mixed-strategy version of a classical game is to be faithfully imbedded within the corresponding quantum game. When each player's strategies are spatial directions, we find that requiring a classical mixed strategy game to be imbedded in the corresponding quantum game results in placing constraints on each player's available directional choices. That is, we place restrictions on allowed trajectories on the surface of a unit sphere of the heads of the unit vectors representing each player's strategies.

II. QUANTIZED PRISONERS' DILEMMA GAME
Consider the symmetric bimatrix game in which S 1 and S 2 are Alice's moves and S ′ 1 and S ′ 2 are Bob's pure strategies and the entries in the brackets are the players' payoffs. For instance, when Alice plays S 1 whereas Bob plays S ′ 2 , Alice's payoff is β and Bob's payoff is γ. Let the players have access to mixed strategies and p is Alice's probability of playing S 1 , and thus (1 − p) is the probability of she playing S 2 . Likewise, q is Bob's probability of playing S ′ 1 , and thus (1 − q) is the probability of he playing S ′ 2 . For the game matrix (1) each players' payoffs in the mixed-strategy game are then obtained as where subscripts A and B are for Alice and Bob, respectively.
For the strategy pair (p * , q * ) to be a NE-corresponding to the two players-neither player is left with any motivation to unilaterally deviate from it, and this is defined by Nash inequalities Π A (p * , q * ) − Π A (p, q * ) ≥ 0, Π B (p * , q * ) − Π B (p * , q) ≥ 0.

A. EWL scheme
In the quantized version of the game (1) developed in Ref. [2]-henceforth referred to as the EWL scheme-each player's strategies consist of local unitary transformations performed on a maximally entangled state. The state evolves and after passing through an unentangling gate, it is measured in a suitable basis. The game (1) is played with two qubits whose quantum state is described in a 2 ⊗ 2 dimensional Hilbert space.
For this game, a measurement basis for the quantum state of two qubits is chosen as An entangled initial quantum state |ψ i is obtained by using a two-qubit entangling gateĴ i.e. |ψ i =Ĵ |S 1 S ′ 1 whereĴ = exp {iγS 2 ⊗ S ′ 2 /2} and γ ∈ [0, π/2] is a measure of the game's entanglement. A separable or a product game has γ = 0 whereas a maximally entangled game has γ = π/2. The players perform their local unitary transformationsÛ A andÛ B on an initial maximally entangled state |ψ i . The transformationsÛ A andÛ B were from the set With the quantum probability rule, the players' payoffs are then obtained as As discussed above, Eqs. (6) show the link that this quantization scheme establishes between each player's strategies-consisting of unitary transformations-and the set of four quantum The NE for the quantum game consists of a pair (Û * A ,Û * B )-corresponding to the two players-of local unitary transformations that satisfy the inequalities That is, it is a pair (Û * A ,Û * B ) from which any unilateral deviation no longer improves player payoff. For (4) a unique quantum NE (Q,Q) was realized whereQ = Benjamin and Hayden [34] noted that when their two-parameter set is extended to include all local unitary operations, i.e. all of SU(2) [8], the strategyQ does not remain an equilibrium and in the full space of deterministic quantum strategies there exists no equilibrium for the quantum Prisoners' Dilemma. This was also discussed further in Ref. [35].

III. QUANTUM PROBABILITIES FROM PLAYERS' DIRECTIONAL CHOICES
In EWL scheme, the players' unitary transformationsÛ A andÛ B along with the subsequent quantum measurements result in the quantum probability set: The players' payoff relations (6) are then expressed as expectation values of entries in the game matrix (1) over the quantum probability set (8).
For a three-player symmetric game, a more direct approach in obtaining a set of quantum probabilities is proposed in Ref. [33]. More specifically, this approach cosiders tripartite EPR experiment performed on a GHZ state as a three-player non-cooperative quantum game.
Each player's strategies are the three directionsâ,b, andĉ along which the dichotomic observablesn · σ are measured, with the eigenvalues +1 or −1 wheren =â,b, orĉ and σ is a vector whose components are the standard Pauli matrices σ x , σ y , and σ z . A three-player quantum game is developed whose underlying setup is the tripartite EPR experiment.
In the present paper-instead of each player's strategies consisting of local unitary trans-formationsÛ A andÛ B -we consider player A and B strategies as their directional choiceŝ a andb. In an EPR setting, the measurement outcomes alongâ andb are denoted by m = ±1 and n = ±1, respectively. That is, the considered setting requires that a pair of unit vectors (â,b) results in a set of quantum probabilities: where acknowledging that there is no unique way in obtaining the set (9) from each players' strategies (â,b), we propose to obtain this set as follows For instance, Pr Q (S 1 , S ′ 2 ) is the quantum probability that the polarization measurement alongâ gives the outcome m = +1 and polarization measurement alongb gives the outcome n = −1.
The probabilities (10) are obtained as and each players' payoff relations are then A directional pair (â * ,b * ) is a NE when the inequalities are true for any directional choicesâ andb by players A and B, respectively.
Given each player's strategies consisting of directional choices in three dimensions, the classical mixed-strategy game is recoverede from the quantum game if each player's directional choices consist of orientating their respective unit vectorsâ andb along specific trajectories on the surface of a unit sphere. When the players allow their respective unit vectorsâ andb to be orientated along directions beyond these trajectories, it results in obtaining the quantum game.
Although the players' strategy sets consist of classical actions of rotating their respective unit vectors in three dimensions, the considered game is genuinely quantum mechanical because the player's payoff relations have an underlying set of quantum mechanical probabilities. In particular, the players have access to directional choices along which Bell's inequalities can be violated. This indicates genuinely quantum mechanical character of this scheme.
IV. GAME WITH THE QUANTUM STATE |ψ ini = 1 2 (|00 + |01 + |10 + |11 ) For this state we can write i.e. the state is a product state. For this state, we find The payoff to the players (12) can then be expressed as To convert to polar coordinates we letâ = and φ A , φ B ∈ [0, 2π) and have This transformation reduces the independent variablesâ andb to θ A , θ B , φ A , and φ B and players payoffs are then expressed as The transformation (28) reduces the independent variablesâ andb to θ A , θ B , φ A , and φ B and players payoffs are then expressed as Note that EWL used the notation φ A,B to describe one of the two parameters in terms of which their (restricted) local unitary operators are defined. In this paper, we have used notation φ A,B when we change from Cartesian to spherical coordinates in accordance with Eqs. (28) i.e. our context is different. Appendix A and B detail the simplification of the first and the second terms, respectively, in Eq. (29) to obtain Eq. (30). Comparing Eq. (30) with Eq. (2), it is noticed that when we take and thus the quantum payoffs (30) are then reduced to players' classical mixed strategy payoffs (2). This can be interpreted by stating that the quantum game considered here results in the classical mixed strategy game (in which Alice plays the strategy p whereas Bob plays the strategy q) is obtained when the tips of Alice's and Bob's unit vectors (representing their strategic choices) are constrained to trajectories on a unit sphere that are defined by  Consider the Prisoners' Dilemma game, as defined by α = 3, β = 0, γ = 5, δ = 1 in the game matrix (1), a quantized version of which was considered in Ref. [2]. The strategy pair (p, q) is a NE in the classical game and therefore (θ * Alice's mixed strategy p is plotted using Eq. (31) against θ A and φ A for the product state

FIG. 3:
The plot of θ A against φ A for p = 0.6 obtained from the first Equation in (33).
For the maximally entangled state |ψ ini = 1 √ 2 (|00 + i |11 ) considered in Refs. [2,3], following set of quantum probabilities are obtained To express these in polar coordinates, we use Eqs. (28) Players' payoffs are obtained as where We note that these payoffs cannot be reduced to the mixed strategy payoffs of Eq. (2).
Stated alternatively, there do not exist such trajectories for the tips of the players' unit vectors which if followed would result in the mixed-strategy version of the classical game.

VI. GAME WITH THE QUANTUM STATE
This is an entangled state for which we find The payoff to the players can now be expressed as The above transformation reduces the independent variablesâ andb to θ A , θ B , φ A ,and φ B and players payoffs are then expressed as which can be simplified to As an example, Appendix C details the simplification of third term in the payoff relations (50) to obtain the third term of (51). The transformations (28) reduce the independent variablesâ andb to θ A , θ B , φ A ,and φ B and players payoffs can be expressed as We note that as was the case for the state 1 √ 2 (|00 + i |11 ) these payoffs cannot be reduced to the classical mixed strategy payoffs in the game. In other words, there do not exist such trajectories for the tips of each players' unit vectors which if followed can result in the We firstly consider the case when only equalities are involved in the above expressions i.e.

CHSH INEQUALITY
The proposed setup for playing a two-player quantum game uses the setting of an EPR type experiment. Consider such an experiment that is designed to test the Bell-CHSH inequality [8] in which two correlated particles 1 and 2 fly apart in opposite directions from some common source. Subsequently, each of the particles enters its own measuring apparatus which can measure either alongâ orâ ′ for particle 1 andb orb ′ for particle 2. The possible values of these variables may be taken to be +1 and −1 and the source emits a very large number of particle pairs. We let Bell-CHSH inequality can be written as |Λ| ≤ 2 where Now, for the state |ψ ini = 1 √ 2 (|00 + i |11 ), considered above, we have and we obtain that can be expressed as We take, for instance, φ A = φ B = φ ′ A = φ ′ B = π/4 and this reduces (87) to Now, consider the case when θ A = π/4, θ ′ A = 3π/4, θ B = π/2, θ ′ B = π/4 and we obtain Λ = 1 + √ 2 ≥ 2 and Bell's inequality is violated. For the state |ψ ini = 1 To know whether these players' payoffs in the quantum game can be embedded within the classical game,we refer to the players' payoffs (2) in the mixed strategy game. We require Π A (p, q) = Π B (p, q) in accordance with the players' payoff relations (38) in the quantum game. This results in β = γ and the players' payoffs in the mixed-strategy classical game The players' payoffs in the quantum game for the directional choice (π/4, π/4; π/2, π/4), and at which Bell's inequalities are violated, are Π A , B (π/4, π/4; π/2, π/4) = 1 4 Comparing (90) with (91) gives and from which we obtain p + q = 1 and q = , showing that for the directional choice (π/4, π/4; π/2, π/4) on behalf of the two players, and at which the players' payoffs are given by (91), the players' payoffs in the quantum game have no mapping within the classical mixed-strategy game.

VIII. DISCUSSION
This paper presents a quantization scheme for playing two-player games in which each player's strategy consist of orientating a unit vector in three dimensions. In the usual approach, a Nash equilibrium is a pair of unitary operators (Û * A ,Û * B ) defined by the inequalities (7). For the given initial quantum state |ψ ini , the proposed quantum game uses an EPR setting in which player A's and player B's strategies consist of orientating the unit vector a andb, respectively. The polarization (or spin) measurements in an EPR setting result in the outcome m = ±1 alongâ and n = ±1 alongb. The players' payoff relations in the considered scheme involves a set of quantum probabilities that are obtained, according to the Eqs. (11) from each player's strategies, entries of the matrix of the game, and the initial quantum state |ψ ini . The payoff relations in the quantum game are defined in terms of this set as described by Eqs. (12,13). That is, the set of underlying quantum probabilities are generated by each player's strategies-consisting of the players' directional choices-along with the initial quantum state |ψ ini .
With directional choices as player's strategies, the NE in the quantum game consists of a pair of unit vectors (â * ,b * ) in three dimensional space. Also, the classical mixed strategy game is recovered-for certain initial states |ψ ini -when each player's directional choicesâ andb follow the assigned trajectories in space.
The scheme is analyzed for three initial states |ψ ini . We show that playing the game with the quantum state |ψ ini = 1 2 (|00 + |01 + |10 + |11 ) results in the classical mixed strategy game in which Alice's and Bob's directional choices are given by (31,32). These express their strategies p and q in the classical mixed strategy game in terms of the angles Playing the game with the state |ψ ini = 1 2 (|00 + |01 − |10 + |11 ) results in a number of Nash equilibria appearing as the edge cases. For the non-edge cases, we determine that there exist an infinite number of Nash equilibria. At these Nash equilibria we consider φ * A , φ * B ∈ [0, 2π) as independent variables from which the angles θ * A , θ * B ∈ [0, π] can be obtained using Eqs. (80). Corresponding to these angles, the players' payoffs at the Nash equilibria i.e. Π A,B (φ * A ; φ * B ) are obtained by Eq. (82). The φ * A , φ * B plane is found to be divided into rectangular patches with the corresponding variation of the players' payoffs into two distinct values.
We agree with the perspective that if quantum advantage (or an improved gametheoretical outcome) does not emerge in a quantum game, it does not necessarily change a quantum game to a classical game. The games considered in this paper are truly quantum as they involve quantum superposition and entanglement. In particular, the players' payoff relations are defined from underlying quantum mechanical probability distributions and that the corresponding classical games are recoverable by restricting players' directional choices along specific trajectories in three dimensions.
Considering Bell-CHSH inequality for the directional choice (π/4, π/4; π/2, π/4) on behalf of two players, we show that Bell's inequalities are violated. For these directional choices, the players' payoffs in the quantum game are shown to have no mapping within the classical mixed-strategy game. An EPR setting provides the route for the players' access to quantum probability distributions that can violate Bell's inequalities. As the quantum game involves classical strategy sets, Enk and Pike's argument [21] is circumvented.

IX. CONCLUSION
Game theory is widely used in a number of disciplines and this paper presents a scheme for two-player quantum games that establishes a more direct link between a classical game and its quantum version. Players in the quantum game have access to classical strategy sets as is the case in the corresponding classical game, allowing us to circumvent Enk and Pike's argument. As the contribution of this paper to the theory of quantum games is built on the EPR paradox, a possible future research direction can be to interpret the EPR paradox as a strategic quantum game. Also, the proposed scheme motivates studying refinements of the NE concept using an EPR settinga with players' moves consisting of directional choices.

X. DATA AVAILABILITY STATEMENT
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

XI. APPENDIX A
The first term in the payoff relations (29) when the game is played with the state |ψ ini = 1 2 (|00 + |01 + |10 + |11 ) is given as and consider its part Now consider the 2nd term Add Eqs. (99, 100) to obtain which is which is Eq. (99) is therefore reduced to