A connection between quantum decision theory and quantum games: The Hamiltonian of Strategic Interaction

• Entanglement between space of actions and beliefs.
• Interaction modeled via Hamiltonian based on the theory of quantum angular momentum.
• The model has a unique parameter understood as a non-revealed type of the decision-maker.
• We obtain a probabilistic analysis of the behavior of the decision-maker.

Experimental economics and studies in psychology show incompatibilities between human behavior and the perfect rationality assumption which do not fit in classical decision theory, but a more general representation in terms of Hilbert spaces can account for them. This paper integrates previous theoretical works in quantum game theory, Yukalov and Sornette’s quantum decision theory and Pothos and Busemeyer’s quantum cognition model by postulating the Hamiltonian of Strategic Interaction which introduces entanglement in the strategic state of the decision-maker. The Hamiltonian is inherited from the algebraic structure of angular momentum in quantum mechanics and the only required parameter, θ̃∈[0,π], represents the strength of the interaction. We consider it as a non-revealed type of the decision-maker. Considering θ̃ to be a continuous random variable, phenomena like learning when participating in repeated games and the influence of the amount of disposable information could be considered as an evolution in the mode and shape of the distribution function fθ̃(t,I). This modeling is motivated by the Eisert–Wilkens–Lewenstein quantization scheme for Prisoner’s Dilemma game and then it is applied in the Ultimatum game, which is not a simultaneous but a sequential game. Even when this non-revealed type θ̃ cannot be directly observed, we can compute observable outcomes: the probabilities of offering different amounts of coins and the probability of the different offers being accepted or not by the other player.