Quantum strategic game theory

Abstract

We propose a simple yet rich model to extend strategic games to the quantum setting, in which we define quantum Nash and correlated equilibria and study the relations between classical and quantum equilibria. Unlike all previous work that focused on qualitative questions on specific games of very small sizes, we quantitatively address the following fundamental question for general games of growing sizes:How much "advantage" can playing quantum strategies provide, if any?Two measures of the advantage are studied.1. A natural measure is the increase of payoff. We consider natural mappings between classical and quantum states, and study how well those mappings preserve the equilibrium properties. Among other results, we exhibit a correlated equilibrium p whose quantum superposition counterpart s p(s)|s is far from being a quantum correlated equilibrium; actually a player can increase her payoff from almost 0 to almost 1 in a [0, 1]-normalized game. We achieve this by a tensor product construction on carefully designed base cases.2. Another measure is the hardness of generating correlated equilibria, for which we propose to study correlation complexity, a new complexity measure for correlation generation. We show that there are n-bit correlated equilibria which can be generated by only one EPR pair followed by local operation (without communication), but need at least log 2 (n) classical shared random bits plus communication. The randomized lower bound can be improved to n, the best possible, assuming (even a much weaker version of) a recent conjecture in linear algebra. We believe that the correlation complexity, as a complexity-theoretical counterpart of the celebrated Bell's inequality, has independent interest in both physics and computational complexity theory and deserves more explorations.

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

“…Their work was later debated by others, such as Benjamin and Hayden in [20] and Zhang in [21], where it was pointed out that players in the game setting of [19] were restricted and therefore the resulting Nash equilibria were not correct. The work in [22] gave an elegant introduction to quantum game theory, along with a review of the relevant literature for the first years of this newborn field.…”

Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation

“…Now consider a Bell scenario where |X| = |Y | = 1, i.e., Alice and Bob each have only one choice of measurement. In this setting we have that any correlation p = (p(ab)) is quantum and D(p) is known as the quantum correlation complexity of p [31]. In [32] it is shown that in this special case, D(p) is equal to the PSD-rank of the corresponding correlation matrix a,b p(ab)|a b|, where the vectors are in the computational basis.…”

Payoffs and Coherence of a Quantum Two-Player Game in a Thermal Environment

“…A quantum game described as a closed system is always an idealization. In this work, we attempt to take one step beyond this approximation and consider the prisoner's dilemma in the presence of a thermal environment modeled via rigorous Davies approach Equation (23). Such a description is most general when applied to Markovian open systems weakly coupled to the thermal bath.…”

“…J (this operation is introduced for technical reasons only; often, its use is criticized, as "quantum strategy" can be understood as any manipulation of the system in question, cf. [23]) describes the process of the creation of entanglement in the system and D the possible destructive noise effects that will be neglected here. The use of entanglement is one of the possible ways to utilize the power of quantum mechanics in quantum games.…”

“…Note that we follow here the "scenario" put forward in [5]. This approach can be criticized as unduly restricting the setting and strategies [24,25], but is widely used and convenient for our aims. One of the possibilities is that the states of players are being entangled by using the transformation:…”