This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. What has clouded many of the issues is the lack of a mathematical formalism for the subject in which these various issues can be clearly and precisely expressed, and which provides a context in which to present their resolution. Such a formalism is presented here, along with proposed resolutions to some of the issues discussed in the literature. One in particular, the question of whether there can exist equilibria in a quantized version of a game that do not correspond to classical correlated equilibria of that game and also deliver better payoffs than the classical correlated equilibria is answered in the affirmative for the Prisoner's Dilemma and Simplified Poker.
This work is an application of game theory to quantum information. In a state estimate, we are given observations distributed according to an unknown distribution
Pθ (associated with award
Q), which Nature chooses at random from the set
{Pθ:θ∈Θ} according to a known prior distribution
μ on
Θ, we produce an estimate
M for the unknown distribution
Pθ, and in the end, we will suffer a relative entropy cost
R(P;M), measuring the quality of this estimate, therefore the whole utility is taken as
P⋅Q−R(P;M). In an introduction to strategic game, a sufficient condition for minimax theorem is obtained; An estimate is explored in the frame of game theory, and in the view of convex conjugate, we reach one new approach to quantum relative entropy, correspondingly quantum mutual entropy, and quantum channel capacity, which are more general, in the sense, without Radon-Nikodym (RN) derivatives. Also the monotonicity of quantum relative entropy and the additivity of quantum channel capacity are investigated.
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.