Basic Framework for Games with Quantum-like Players

We develop a framework for the analysis of strategic interactions under the constructive preference perspective à la Kahneman and Tversky formalized in the Type Indeterminacy model. The players are modeled as systems subject to measurements and characterized by quantum-like uncertain preferences. The decision nodes are modeled as, possibly non-commuting, operators that measure preferences modulo strategic reasoning. We define a Hilbert space of types spanned by the players' eigentypes representing their potential preferences in different situations. We focus on pure strategy TI games of maximal information where all uncertainty stems from the intrinsic indeterminacy of preferences. We show that preferences evolve in a non-deterministic manner with actions along the play: they are endogenous to the interaction. We propose the notion of cashing-on-the-go to compute a player's utility, and the Type Indeterminate Nash Equilibrium as a solution concept relying on best-replies at the level of the eigentypes. We illustrate an example exhibiting the phenomenon of the manipulation of rivals' preferences.