Modeling behavior of decision makers with the aid of algebra of qubit creation–annihilation operators

Polina Khrennikova

doi:10.1016/j.jmp.2016.10.003

Abstract

We present a general model of the process of decision making based on the representation of the basic behavioral variables with the aid of an algebra of qubit creation–annihilation operators, adopted from the quantum information theory. In contrast to the genuine quantum physical systems, which are divided into either bosons or fermions and modeled with the aid of operators, satisfying canonical commutation or anti-commutation relations, decision makers preferences for possible actions are constructed with the aid of operators satisfying the so-called qubit commutation relations. Systems described by operators, satisfying such commutation relations, combine the features of bosons and fermions. Thus, one of the basic consequences of the presented model is that decision makers mimic the combined bosonic–fermionic behavior. By using the algebra of qubit creation–annihilation operators, we proceed with the construction of the concrete operators, describing the process of decision making. In particular, the generators of the quantum Markov dynamics, which is used for modeling human decision making process, are expressed as polynomials of the qubit creation–annihilation operators. The devised coefficients have a natural cognitive and social meaning.

Highlights

During the last two decades, the formalism of quantum mechanics was actively pursued, to model the process of decision making in cognitive psychology, sociology, economics, finance and politics, see, e.g., (Busemeyer et al., 2006, 2012; Pothos and Busemeyer, 2009, Accardi et al, 2008, 2009, Asano et al, 2011ab, 2012, Basieva et al, 2011, Aerts et al, 2012, Bagarello, 2012, 2015; Khrennikova et al, 2014, 2016, Khrennikova, 2014a, b, 2015, 2016, Bagarello and Haven, 2016).[1]
One of the problems of this approach is the absence of an analog of the procedure of canonical quantization, which is used in physics to transfer classical physical quantities defined as functions on the phase space, f = f (q, p), into the corresponding operators acting in complex Hilbert space of states of quantum systems (Schrödinger quantization procedure: f = f (q, p))
We do not have a kind of classical mechanics on the phase space for mental variables

Summary

Introduction

During the last two decades, the formalism of quantum mechanics was actively pursued, to model the process of decision making in cognitive psychology, sociology, economics, finance and politics, see, e.g., (Busemeyer et al., 2006, 2012; Pothos and Busemeyer, 2009, Accardi et al, 2008, 2009, Asano et al, 2011ab, 2012, Basieva et al, 2011, Aerts et al , 2012, Bagarello, 2012, 2015; Khrennikova et al, 2014, 2016, Khrennikova, 2014a, b, 2015, 2016, Bagarello and Haven, 2016).[1]. We do not have a kind of classical mechanics on the phase space for mental variables. We were not able to identify the mental analogs of the position and momentum variables (q, p) and to construct a type of a “mental phase space.”. One cannot exclude the possibility that such observables would not exist at all. Their existence in physics is closely related to the real manifold geometry of physical space used in classical physics. There are no reasons to expect that the “mental space” has the same geometry. We are neither able to construct the “quantum phase space” for cognition with the “coordinates” (q, p)

Objectives

Conclusion