This work, based on the author's MA thesis, concentrates on simultaneous move quantum games of two players. A numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games is constructed. It is then used show that there is a critical degree of entanglement above which a pure strategy Nash Equilibrium does not exist. The behavior of the two payoffs approach the cooperative classical ones. Bayesian quantum games are defined, and it is shown that under certain conditions, there {\it is} a pure strategy Nash equilibrium in such games even when entanglement is maximal. The basic ingredients of a quantum game based on a two-players {\it three} decisions classical games are presented. This requires the definition of trits (instead of bits) and quantum trits (instead of quantum bits). It is shown that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies. An entanglement operator transforming a pure two qutrit state into a maximally entangled one is explicitly constructed.
This work, based on the author's MA thesis, concentrates on simultaneous move quantum games of two players. A numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games is constructed. It is then used show that there is a critical degree of entanglement above which a pure strategy Nash Equilibrium does not exist. The behavior of the two payoffs approach the cooperative classical ones. Bayesian quantum games are defined, and it is shown that under certain conditions, there {\it is} a pure strategy Nash equilibrium in such games even when entanglement is maximal. The basic ingredients of a quantum game based on a two-players {\it three} decisions classical games are presented. This requires the definition of trits (instead of bits) and quantum trits (instead of quantum bits). It is shown that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies. An entanglement operator transforming a pure two qutrit state into a maximally entangled one is explicitly constructed.
This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
In classical (standard) game theory, a useful algorithm for searching Nash equilibrium in games of two players, is to determine the best response functions. For each strategy S1 of player 1 player 2 finds a best response function F2(S1), and vice versa. If the two response functions intersect, the intersection point (S1*, S2*) is a candidate for Nash equilibrium. This method is especially useful when the strategy space of each player is determined by a single variable (discrete or continuous). In the last decade, the concept of quantum games has been developed (hence we distinguish between classical and quantum games). In a quantum game with two players the strategy space of each player is composed of 2 × 2 complex unitary matrices with unit determinant. That is the group SU(2). The corresponding strategy space is characterized by three continuous variables represented by angles: 0 ≤ α ≤ 2π, 0 ≤ β ≤ 2π, 0 ≤ θ ≤ π. That turns the use of response functions impractical. In the present contribution we suggest a method for alleviating this problem by discretizing the variables as: {αi, βj, θk}, i= 1, 2, ..., I; j = 1, 2, ..., J; k = 1, 2, ... K. This enables the representation of every such triple by a single discrete variable, (αi, βj, θk) → xijk. Thereby, the strategy space is characterized by a single discrete variable taking I × J × K values and the method of response functions is feasible. We use it to show the following two results: 1) A two players quantum game with partially entangled initial state has a pure strategy Nash equilibrium. 2) A two player quantum Bayesian game with fully entangled initial state has a pure strategy Nash equilibrium.
The two-players N strategies games quantized according to the Eisert-Lewenstein-Wilkens scheme [1] are considered. It is shown that in the case of maximal entanglement no nontrivial pure Nash equilibrium exists. The proof relies on simple geometric properties of "chiral" group SU(N)×SU(N) and is based on considering the stability subgroup of the initial state of the game. The explicit forms of neither the gate operator nor the payoff matrix are necessary.
In quantum games based on 2-player-N-strategies classical games, each player has a quNit (a normalized vector in an N-dimensional Hilbert space HN) upon which he applies his strategy (a matrix U∈SU(N)). The players draw their payoffs from a state . Here and J (both determined by the game’s referee) are respectively an unentangled 2-quNit (pure) state and a unitary operator such that is partially entangled. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of . Hence, it is practical to design the entangler J= J(β) to be dependent on a single real parameter β that controls the degree of entanglement of , such that its von-Neumann entropy SN(β) is continuous and obtains any value in . Designing J(β) for N=2 is quite standard. Extension to N>2 is not obvious, and here we suggest an algorithm to achieve it. Such construction provides a special quantum gate that should be a useful tool not only in quantum games but, more generally, as a special gate in manipulating quantum information protocols.
Two-player |N|-strategy games quantized according to the Eisert–Lewenstein–Wilkens scheme [Phys. Rev. Lett. 83, 3077 (1999)] are considered. Group-theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three-strategy case.
Game theory is the study of strategic competition used in many branches of science. In general, a game is a match between two or more players and each wants to maximize his or her score. In a game, the player’s score depends on their strategies. The goal is to find the optimized decision, with regard to the payoff, in conformity with the rules of the game. This thesis considers the games with two players who know both their own and their rival’s payoff. However, they do not know the strategy chosen by the opponent.\nIn a classical game, the options of players are referred to as cooperation and defect. Quantum game theory is a generalization of classical game theory. In quantum games a linear combination of two options specify the strategy based on qubit. The quantum domain is characterized by entanglement. The quantum entanglement coefficient (y) varies from 0 to . The game is in the classical regime when y is smaller than a threshold. Quantum entanglement is intrinsically non-local.\nIn an iterated game tournament, 63 different strategies contest with four known strategies. Depending on the result, they are divided to a number of groups. The competition specifies which group of strategies can obtain a higher score.\nOther types of games are discussed in the thesis. They are known as incomplete information games. In these games one player has two payoff matrixes, and the rival does not know which of them is used. We discuss the advantage of using quantum strategy over the classical game, particular in avoiding the dilemma in classical games.