Quantum-like model of subjective expected utility

Abstract

We present a very general quantum-like model of lottery selection based on representation of beliefs of an agent by pure quantum states. Subjective probabilities are mathematically realized in the framework of quantum probability (QP). Utility functions are borrowed from the classical decision theory. But in the model they are represented not only by their values. Heuristically one can say that each value ui=u(xi) is surrounded by a cloud of information related to the event (A,xi). An agent processes this information by using the rules of quantum information and QP. This process is very complex; it combines counterfactual reasoning for comparison between preferences for different outcomes of lotteries which are in general complementary. These comparisons induce interference type effects (constructive or destructive). The decision process is mathematically represented by the comparison operator and the outcome of this process is determined by the sign of the value of corresponding quadratic form on the belief state. This operational process can be decomposed into a few subprocesses. Each of them can be formally treated as a comparison of subjective expected utilities and interference factors (the latter express, in particular, risks related to lottery selection). The main aim of this paper is to analyze the mathematical structure of these processes in the most general situation: representation of lotteries by noncommuting operators.