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The Prisoner's Dilemma game (PDG) is one of the simple test-beds for theprobabilistic nature of the human decision-making process. Behavioralexperiments have been conducted on this game for decades and show a violationof the so-called \textit{sure thing principle}, a key principle in the rationaltheory of decision. Quantum probabilistic models can explain this violation asa second-order interference effect, which cannot be accounted for by classicalprobability theory. Here, we adopt the framework of generalized probabilistictheories and approach this explanation from the viewpoint of quantuminformation theory to identify the source of the interference. In particular,we reformulate one of the existing quantum probabilistic models using densitymatrix formalism and consider different amounts of classical and quantumuncertainties for one player's prediction about another player's action in PDG.This enables us to demonstrate that what makes possible the explanation of theviolation is the presence of \textit{quantum coherence} in the player's initialprediction and its conversion to probabilities during the dynamics. Moreover,we discuss the role of other quantum information-theoretical quantities, suchas quantum entanglement, in the decision-making process. Finally, we propose athree-choice extension of the PDG to compare the predictive powers of quantumprobability theory and a more general probabilistic theory that includes it asa particular case and exhibits third-order interference.

What is \textit{Quantum} in Probabilistic Explanations of the Sure Thing Principle Violation?