Probability interference in expected utility theory

Andrei Yu. Khrennikov

Raymond J. Hawkins, Evan Walsh

The innovative application of quantum mechanical concepts of probability to cognitive science has provided new, meaningful ways to more accurately model human decision making. Leveraging the dynamic geometric principles associated with the mathematics of quantum theory, we use the idea of probability interference to explain projection bias and its violations of the classical law of total probability and expected utility theory. In particular, Khrennikov's contextual probability model is successfully applied to the study by Read & van Leeuwen on hunger and projection bias. We conclude that probability interference provides an effective, accurate and meaningful way to model projection bias, thus broadening the reach of quantum cognition in economics.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

Andrei Yu. Khrennikov

This chapter can be considered as an illustration of applications of the mathematical formalism of QM to modeling the process of decision making by humans (see section 1.3 for an introduction and general discussion). In this chapter we present a very general quantum-like model of lottery selection based on the 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. 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 a 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 202 utilities and interference factors (the latter express, in particular, risks related to lottery selection).

Georges Bernard

Jerome R. Busemeyer, Peter Bruza

Can quantum theory provide new answers to puzzling results from decision research that have resisted formal explanations so far? This chapter presents several new contributions that quantum theory provides to decision research. However, in order to locate where quantum theory can make an important new contribution, we first need to provide some necessary background on decision theory and what has already been accomplished. Decision science is a highly developed field that has produced a variety of sophisticated mathematical theories (Gilboa, 2009; Wakker, 2010). Some of these theories are rational theories, which prescribe optimal ways to make decisions. Others are psychological theories that describe the way people actually make decisions. One of the most important rational decision theories is von Neumannn and Morgenstern's expected utility (EU) theory (von Neumann & Morgenstern, 1944). Expected utility theory is designed for decisions under risk ; that is, a choice among actions defined as an objective probability distribution over possible payoffs. An important extension of (EU) theory is Savage's (1954) subjective expected utility (SEU) theory . This extends utility theory to decisions under uncertainty ; that is, a choice among actions defined as functions that map un-certain events into payoffs, and the events may not have objective probabilities. A more general theory designed for decisions under both uncertainty and risk was presented by Anscombe and Aumann (1963). All of these theories are rational, in the sense that they are based on a small set of compelling axioms and the prescribed action logically follows from the axioms.

G. Charles-Cadogan