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quantum machine learning

Received: 10 October 2019 Revised: 21 January 2020 Accepted: 4 February 2020

DOI: 10.1002/wcs.1526

F O C U S A R T I C L E

Comparison of Markov versus quantum dynamical models of human decision making

Jerome R. Busemeyer1

| Peter D. Kvam2

| Timothy J. Pleskac3

1Department of Psychological and Brain

Sciences, Indiana University,

Bloomington, Indiana

2Department of Psychology, University of

Florida, Gainesville, Florida

3Department of Psychology, University of

Kansas, Lawrence, Kansas

Correspondence

Timothy J. Pleskac, Department of Psychology, University of Kansas, Lawrence, KS.

Email: pleskac@ku.edu

Abstract

What kind of dynamic decision process do humans use to make decisions? In

this article, two different types of processes are reviewed and compared: Mar-

kov and quantum. Markov processes are based on the idea that at any given

point in time a decision maker has a definite and specific level of support for

available choice alternatives, and the dynamic decision process is represented

by a single trajectory that traces out a path across time. When a response is

requested, a person's decision or judgment is generated from the current loca-

tion along the trajectory. By contrast, quantum processes are founded on the

idea that a person's state can be represented by a superposition over different

degrees of support for available choice options, and that the dynamics of this

state form a wave moving across levels of support over time. When a response

is requested, a decision or judgment is constructed out of the superposition by

“actualizing” a specific degree or range of degrees of support to create a defi-

nite state. The purpose of this article is to introduce these two contrasting theo-

ries, review empirical studies comparing the two theories, and identify

conditions that determine when each theory is more accurate and useful than

the other.

This article is categorized under:

Economics > Individual Decision-Making

Psychology > Reasoning and Decision Making

Psychology > Theory and Methods

K E Y W O R D S

confidence, decision, interference, quantum cognition, random walk

1 | INTRODUCTION

Imagine watching a murder mystery film with a friend. As you watch, you become aware of your beliefs about guilt or innocence of a suspect. Your beliefs move up and down across time as different kinds of evidence are presented during the movie scenes. At any point in time, your friend may ask which character seems the most guilty, or evaluate the evidence with respect to a particular character's guilt or innocence. When prompted, you can express the likelihood that the suspect is guilty or innocent to your friend.

All authors contributed equally to this work.

WIREs Cogn Sci. 2020;e1526.

wires.wiley.com/cogsci

1 of 19

https://doi.org/10.1002/wcs.1526

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Now imagine trying to decide whether or not to risk passing a car on a two lane highway. As you deliberate, you become aware of your affective evaluations toward taking the risk or not. Your tendency to approach or avoid changes across time as you monitor the driving situation. At each point in time you can decide to take the risk (pass) or not (stay).

These are two different kinds of decision problems. In the first case, the decision variable being monitored is the evidence for or against an hypothesis: an inferential choice problem. In the second case, the decision variable being monitored is your preference for or against an action: a preferential choice problem. Despite their differences, both decisions seem to be based on the same or similar process where evidence or preference is accumulated over time to eventually trigger a choice (Dutilh & Rieskamp, 2016; Pleskac, Yu, Hopwood, & Liu, 2019; Summerfield & Tsetsos, 2012; Usher & McClelland, 2001; Usher & McClelland, 2004; Zeigenfuse, Pleskac, & Liu, 2014). What are the basic dynamics that underlie the changes in these decision variables, evidence or preference, during decision making? And how are the judgments and actions about these decision variables generated from the latent monitoring and accumulation process? This article compares and contrasts two different ways to model the decision processes underlying these choices: a classical Markov process, and a nonclassical quantum process. These two approaches to modeling dynamic decision behavior have been compared on a variety of tasks and measures, including sequential decisions (Busemeyer, Wang, & Lambert-Mogiliansky, 2009a; Wang & Busemeyer, 2016b), decisions with subsequent confidence (Kvam, Pleskac, Yu, & Busemeyer, 2015) or preference ratings (Kvam, 2014; Wang & Busemeyer, 2016a), sequences of judgments and ratings (Busemeyer, Kvam, & Pleskac, 2019), and response time distributions (Busemeyer, Wang, & Townsend, 2006; Fuss & Navarro, 2013). We begin by introducing the basic ideas underlying these two theories using the evidence accumulation problem. Later in the article we consider the preference accumulation problem. In the end, we introduce a more general “open system” quantum—Markov approach that incorporates elements of both frameworks to provide a more complete description of how support for different choice options changes over time.

2 | MARKOV AND QUANTUM VIEWS OF EVIDENCE ACCUMULATION

The “classical” dynamical view of evidence accumulation, including the popular decision diffusion models (Ratcliff, Smith, Brown, & McCoon, 2016), asserts that a person's state of belief about a hypothesis at any single moment can be represented as a specific point along some internal scale of evidence, as illustrated in the left panel of Figure 2. This belief state changes moment by moment from one location to another on the evidence scale, carving out a trajectory as illustrated in the top panel of Figure 1. At the point in time when you are asked to report your belief, you simply read out the location on the evidence scale that existed before you were asked. That is, the report is determined by the preexisting location of the belief state. This classical view of belief change is typically formalized as a Markov process (Bhattacharya & Waymire, 1990), which describes a probability distribution over the evidence scale at each moment in

Measurement

Markov model

Quantum model

time

With measurement Without measurement

F I G U R E 1 Markov and quantum random walk models generate diverging predictions for how evidence evolves over time and how measurements like decisions interact with subsequent responses

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time (shown in the left panel of Figure 3). This probability distribution represents a modeler's uncertainty about the location of the state that exists for a decision-maker at any point in time, rather than the decision maker's inherent uncertainty about the location of his or her own. A key assumption of Markov processes is that the probability distribution at the next moment in time only depends on the previous probability distribution and the dynamics of the state.

An alternative “nonclassical” dynamical view of evidence accumulation asserts that your belief about a hypothesis at any single moment is not located at a specific point on the mental evidence scale. Instead, at any moment, it is superposed with some potential (called an amplitude) of realization across the scale, as illustrated by the shades of gray spread across the scale in right panel of Figure 2. As this superposition changes, it forms a wave that flows across levels of support over time. A representation of this wave, where darker regions correspond to greater squared probability amplitudes (corresponding to a greater likelihood of observing the state in that location), is shown in the bottom panel of Figure 1. When a person is asked to report their belief, a location of the evidence must be identified from this superposed state. This collapses the wave to a specific location or range of levels of support, depending on what type of measurement (binary choice, confidence judgment, or some other response) is applied. This “nonclassical” view of belief change has been formalized as a quantum process (Gudder, 1979), which describes the amplitude distribution over the evidence scale across time as shown in the right panel of Figure 3 (but using squared amplitudes). Like the Markov process, it is based on the assumption that the amplitude distribution at the next moment in time only depends on the previous amplitude distribution.

Both Markov and quantum processes are stochastic processes. However, the probabilities used in Markov processes (Figure 3, left panel) and the squared amplitudes used in quantum processes (Figure 3, right panel) are conceptually quite

			Markov random walk										Quantum random walk
0%	10	20	30	40	50	60	70	80	90 100%	0%	10	20	30	40	50	60	70	80	90 100%
‘Certain									‘Certain	‘Certain									‘Certain
left’									right’	left’									right’

F I G U R E 2 Diagram of a state representation of a Markov and a quantum random walk model. In the Markov model, evidence (shaded state) evolves over time by moving from state to state, occupying one definite evidence level at any given time. In the quantum model the decision-maker is in an indefinite evidence state, with each evidence level having a probability amplitude (shadings) at each point in time

Probability

Markov model

0.8
0.7	alpha = 100
0.7	beta = 150
	beta = 150
0.6
				0 s
0.5				.25 s
				.50 s
0.4				.75 s
				1.0 s
0.3
0.2
0.1
0
0	20	40	60	80	100

Confidence

Probability

Quantum model

0.2

0.18drift = 300 diff = 35

	0.16	0 s
		.25 s
	0.14	.50 s
	0.14	.75 s
		.75 s
		1.0 s
	0.12
	0.1
	0.08
	0.06
	0.04
	0.02

100

Confidence

F I G U R E 3 Illustration of Markov (left) and quantum (right) evolution of probability distributions over time. The horizontal axis represents 101 belief states associated with subjective evidence scale values ranging from 0 to 100 in one-unit steps. The vertical axis represents probability corresponding to each evidence level. The separate curves moving from left to right represent increasing processing time

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