Repeated quantum game as a stochastic game: Effects of the shadow of the future and entanglement

TL;DR

This paper investigates quantum repeated games, demonstrating their equivalence to stochastic games, and explores how entanglement and the shadow of the future influence strategic outcomes, revealing fundamental differences from classical game theory.

Contribution

It introduces a novel repeated quantum game protocol, maps quantum games to stochastic games, and analyzes the impact of entanglement and discount factors on strategic stability.

Findings

Quantum repeated games can be modeled as stochastic games.

Entanglement and discount factors significantly influence strategic outcomes.

Quantum prisoner's dilemma shows always-defect can be beaten by tit-for-tat at high discount factors.

Abstract

We present a systematic investigation of the quantum games, constructed using a novel repeated game protocol, when played repeatedly ad infinitum. We focus on establishing that such repeated games -- by virtue of inherent quantum-mechanical randomness -- can be mapped to the paradigm of stochastic games. Subsequently, using the setup of two-player--two-action games, we explore the pure reactive strategies belonging to the set of reactive strategies, whose support in the quantum games is no longer countably finite but rather non-denumerably infinite. We find that how two pure strategies fare against each other is crucially dependent on the discount factor (the probability of occurrence of every subsequent round) and how much entangled the quantum states of the players are. We contrast the results obtained with the corresponding results in the classical setup and find fundamental differences between them: e.g, when the underlying game is the prisoner's dilemma, in the quantum game setup, always-defect strategy can be beaten by the tit-for-tat strategy for high enough discount factor.