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10. Appendix

10.1. Leggett-Garg inequality

The three columns on the left, labeled A, B, C represent three binary valued random variables of a hypothesized 3 − way joint probability model. The eight rows represent the eight possible joint events produced by the eight combinations of values of these three binary random variables. The last three columns on the right are used to compute three probabilities used in the Leggett - Garg inequality. According to the 3-way joint probability model, the probability p(A =B) is computed by summing the probabilities of the events in the rows containing an X in the fourth column; the probability p(B =C) is computed by summing the probabilities of the events in the rows containing an X in the ﬁfth column; the probability p(A =C) is computed by summing the probabilities of the events in the rows containing an X in the sixth column. The Leggett-Garg inequality follows from the fact that columns 4 and 5 contain column 6.

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quantum machine learning

Table 7: Table of probabilities used to prove the Leggett-Garg inequality

A B C p(A =B) p(B =C) p(A =C)

1	1	1
1	1	2		X	X
1	2	1	X	X
1	2	2	X		X
2	1	1	X		X
2	1	2	X	X
2	2	1		X	X
2	2	2

10.2. Quantum Model Parameters

The following parameters were used to almost perfectly ﬁt all the data shown in Table 1.

0.2685

−0.7600

0.0883

ψ =

0.3514

−0.1935

			−0.4260				.
HA =			1.5708		0.5236
			0.5236 1.5708
				−
HB =		3.2987		0		3.2987
	−0.1047			3.2987			0
			0	3.2987		0.1047

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quantum machine learning

Review

Cognitive and Neural Bases of

Multi-Attribute, Multi-Alternative,

Value-based Decisions

Jerome R. Busemeyer,1,* Sebastian Gluth,2 Jörg Rieskamp,2 and Brandon M. Turner3

Researchers have beneﬁted from characterizing evidence-based decision making as a process involving sequential sampling. More recently, sequential sampling models have been applied to value-based decisions – decisions that involve examining preferences for multi-attribute, multi-alternative choices. The application of sequential sampling models to value-based decisions has helped researchers to account for the context effects associated with preferential choice tasks. However, for these models to predict choice preferences, more complex decision mechanisms have had to be introduced. We review here the complex decision mechanisms necessary to account for context effects found with multi-attribute, multi-alternative choices. In addition, we review linkages between these more complex processes and their neural substrates to develop a comprehensive and biologically plausible account of human value-based decision making.

Sequential Sampling and Value-Based[168T$DIF] Decisions

For the past 25 years, models of decision making based on sequential[169T$DIF] sampling and accumulation to threshold (see Glossary)[170T$DIF]principles have become the dominant theory in the cognitive sciences and have also started to play a central role in decision neuroscience [1–3]. The core principle of this class of models is the assumption that the decision maker accumulates evidence for each choice option until a threshold (of sufﬁciently strong evidence) is reached, at which time a decision is made in favor of the ﬁrst to reach the threshold. Thus, sequential sampling models provide a principled account of both choices and response times. There are two major domains of applications of sequential sampling processes – one is for accumulating informative evidence for or against each of several competing hypotheses, and the other is for accumulating affective evaluations for or against each of several courses of action.

Highlights

Accumulation to threshold models describe the dynamics of decision making. They have been highly inﬂuential in perceptual decision making and begin to dominate research on valuebased decisions.

For simple perceptual choices, core mechanisms of these models, such as evidence accumulation and decisions threshold crossing, could be mapped onto a neural circuitry.

Value-based choices, however, often require the comparison of multiple choice options along multiple attributes. Such decisions are prone to context effects that are inconsistent with economic conceptions of rationality]FID$T761[ .

To accommodate the complexity of value-based decision making, a series of novel accumulation to threshold models that assume advanced component processes have been developed.

Most recently, model-based neuroscience studies have started to link these component processes to their neural underpinnings.

From these two domains, the evidence-based approach appeared ﬁrst in cognitive science, with applications to perceptual [4–6], memory [7], and categorization [8,9] tasks. Not much later, neuroscientiﬁc studies discovered that principles of evidence accumulation provided remarkably accurate descriptions for the dynamics of neural activation during the decision process of the animal. Pools of neurons in the frontal eye ﬁelds or lateral parietal areas of the monkey brain appear to increase ﬁring rate up to an unobserved ﬁxed threshold, at which time the behavioral response is produced (but see [10]). The initial neuroscience work focused mainly on simple perceptual–motor decision-making tasks [11,12].

More recently, however, both cognitive scientists and decision neuroscientists have become increasingly interested in applications of sequential sampling models to value-based decision

1Psychological and Brain Sciences,

Indiana University, Bloomington, IN

47405, USA

2Department of Psychology, University

of Basel, 4055 Basel, Switzerland

3Department of Psychology, The Ohio

State University, Columbus, OH

43210, USA

*Correspondence: jbusemey@indiana.edu (J.R. Busemeyer).

Trends in Cognitive Sciences, March 2019, Vol. 23, No. 3 https://doi.org/10.1016/j.tics.2018.12.003 251

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