Jennifer S. Trueblood

A Quantum Theoretical Explanation for Probability Judgment Errors.

A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction and disjunction fallacies, averaging effects, unpacking effects, and order effects on inference. On the one hand, quantum theory is similar to other categorization and memory models of cognition in that it relies on vector spaces defined by features and similarities between vectors to determine probability judgments. On the other hand, quantum probability theory is a generalization of Bayesian probability theory because it is based on a set of (von Neumann) axioms that relax some of the classic (Kolmogorov) axioms. The quantum model is compared and contrasted with other competing explanations for these judgment errors, including the anchoring and adjustment model for probability judgments. In the quantum model, a new fundamental concept in cognition is advanced--the compatibility versus incompatibility of questions and the effect this can have on the sequential order of judgments. We conclude that quantum information-processing principles provide a viable and promising new way to understand human judgment and reasoning.

A Quantum Probability Account of Order Effects in Inference

Order of information plays a crucial role in the process of updating beliefs across time. In fact, the presence of order effects makes a classical or Bayesian approach to inference difficult. As a result, the existing models of inference, such as the belief-adjustment model, merely provide an ad hoc explanation for these effects. We postulate a quantum inference model for order effects based on the axiomatic principles of quantum probability theory. The quantum inference model explains order effects by transforming a state vector with different sequences of operators for different orderings of information. We demonstrate this process by fitting the quantum model to data collected in a medical diagnostic task and a jury decision-making task. To further test the quantum inference model, a new jury decision-making experiment is developed. Using the results of this experiment, we compare the quantum inference model with two versions of the belief-adjustment model, the adding model and the averaging model. We show that both the quantum model and the adding model provide good fits to the data. To distinguish the quantum model from the adding model, we develop a new experiment involving extreme evidence. The results from this new experiment suggest that the adding model faces limitations when accounting for tasks involving extreme evidence, whereas the quantum inference model does not. Ultimately, we argue that the quantum model provides a more coherent account for order effects that was not possible before.

Not Just for Consumers Context Effects Are Fundamental to Decision Making

Context effects—preference changes that depend on the availability of other options—have attracted a great deal of attention among consumer researchers studying high-level decision tasks. In the experiments reported here, we showed that these effects also arise in simple perceptual-decision-making tasks. This finding casts doubt on explanations limited to consumer choice and high-level decisions, and it indicates that context effects may be amenable to a general explanation at the level of the basic decision process. We demonstrated for the first time that three important context effects from the preferential-choice literature—similarity, attraction, and compromise effects—all occurred within a single perceptual-decision task. Not only do our results challenge previous explanations for context effects proposed by consumer researchers, but they also challenge the choice rules assumed in theories of perceptual decision making.

The multiattribute linear ballistic accumulator model of context effects in multialternative choice

Context effects occur when a choice between 2 options is altered by adding a 3rd alternative. Three major context effects--similarity, compromise, and attraction--have wide-ranging implications across applied and theoretical domains, and have driven the development of new dynamic models of multiattribute and multialternative choice. We propose the multiattribute linear ballistic accumulator (MLBA), a new dynamic model that provides a quantitative account of all 3 context effects. Our account applies not only to traditional paradigms involving choices among hedonic stimuli, but also to recent demonstrations of context effects with nonhedonic stimuli. Because of its computational tractability, the MLBA model is more easily applied than previous dynamic models. We show that the model also accounts for a range of other phenomena in multiattribute, multialternative choice, including time pressure effects, and that it makes a new prediction about the relationship between deliberation time and the magnitude of the similarity effect, which we confirm experimentally.

A quantum geometric model of similarity.

No other study has had as great an impact on the development of the similarity literature as that of Tversky (1977), which provided compelling demonstrations against all the fundamental assumptions of the popular, and extensively employed, geometric similarity models. Notably, similarity judgments were shown to violate symmetry and the triangle inequality and also be subject to context effects, so that the same pair of items would be rated differently, depending on the presence of other items. Quantum theory provides a generalized geometric approach to similarity and can address several of Tverskys main findings. Similarity is modeled as quantum probability, so that asymmetries emerge as order effects, and the triangle equality violations and the diagnosticity effect can be related to the context-dependent properties of quantum probability. We so demonstrate the promise of the quantum approach for similarity and discuss the implications for representation theory in general.

Registered Replication Report : Rand, Greene, and Nowak (2012)

In an anonymous 4-person economic game, participants contributed more money to a common project (i.e., cooperated) when required to decide quickly than when forced to delay their decision (Rand, Greene & Nowak, 2012), a pattern consistent with the social heuristics hypothesis proposed by Rand and colleagues. The results of studies using time pressure have been mixed, with some replication attempts observing similar patterns (e.g., Rand et al., 2014) and others observing null effects (e.g., Tinghög et al., 2013; Verkoeijen & Bouwmeester, 2014). This Registered Replication Report (RRR) assessed the size and variability of the effect of time pressure on cooperative decisions by combining 21 separate, preregistered replications of the critical conditions from Study 7 of the original article (Rand et al., 2012). The primary planned analysis used data from all participants who were randomly assigned to conditions and who met the protocol inclusion criteria (an intent-to-treat approach that included the 65.9% of participants in the time-pressure condition and 7.5% in the forced-delay condition who did not adhere to the time constraints), and we observed a difference in contributions of −0.37 percentage points compared with an 8.6 percentage point difference calculated from the original data. Analyzing the data as the original article did, including data only for participants who complied with the time constraints, the RRR observed a 10.37 percentage point difference in contributions compared with a 15.31 percentage point difference in the original study. In combination, the results of the intent-to-treat analysis and the compliant-only analysis are consistent with the presence of selection biases and the absence of a causal effect of time pressure on cooperation.

A Quantum Probability Model of Causal Reasoning

People can often outperform statistical methods and machine learning algorithms in situations that involve making inferences about the relationship between causes and effects. While people are remarkably good at causal reasoning in many situations, there are several instances where they deviate from expected responses. This paper examines three situations where judgments related to causal inference problems produce unexpected results and describes a quantum inference model based on the axiomatic principles of quantum probability theory that can explain these effects. Two of the three phenomena arise from the comparison of predictive judgments (i.e., the conditional probability of an effect given a cause) with diagnostic judgments (i.e., the conditional probability of a cause given an effect). The third phenomenon is a new finding examining order effects in predictive causal judgments. The quantum inference model uses the notion of incompatibility among different causes to account for all three phenomena. Psychologically, the model assumes that individuals adopt different points of view when thinking about different causes. The model provides good fits to the data and offers a coherent account for all three causal reasoning effects thus proving to be a viable new candidate for modeling human judgment.

Registered replication report: Rand, Greene, & Nowak

In an anonymous 4-person economic game, participants contributed more money to a common project (i.e., cooperated) when required to decide quickly than when forced to delay their decision (Rand, Greene & Nowak, 2012), a pattern consistent with the social heuristics hypothesis proposed by Rand and colleagues. The results of studies using time pressure have been mixed, with some replication attempts observing similar patterns (e.g., Rand et al., 2014) and others observing null effects (e.g., Tinghög et al., 2013; Verkoeijen & Bouwmeester, 2014). This Registered Replication Report (RRR) assessed the size and variability of the effect of time pressure on cooperative decisions by combining 21 separate, preregistered replications of the critical conditions from Study 7 of the original article (Rand et al., 2012). The primary planned analysis used data from all participants who were randomly assigned to conditions and who met the protocol inclusion criteria (an intent-to-treat approach that included the 65.9% of participants in the time-pressure condition and 7.5% in the forced-delay condition who did not adhere to the time constraints), and we observed a difference in contributions of −0.37 percentage points compared with an 8.6 percentage point difference calculated from the original data. Analyzing the data as the original article did, including data only for participants who complied with the time constraints, the RRR observed a 10.37 percentage point difference in contributions compared with a 15.31 percentage point difference in the original study. In combination, the results of the intent-to-treat analysis and the compliant-only analysis are consistent with the presence of selection biases and the absence of a causal effect of time pressure on cooperation.

A new framework for modeling decisions about changing information: The Piecewise Linear Ballistic Accumulator model.

In the real world, decision making processes must be able to integrate non-stationary information that changes systematically while the decision is in progress. Although theories of decision making have traditionally been applied to paradigms with stationary information, non-stationary stimuli are now of increasing theoretical interest. We use a random-dot motion paradigm along with cognitive modeling to investigate how the decision process is updated when a stimulus changes. Participants viewed a cloud of moving dots, where the motion switched directions midway through some trials, and were asked to determine the direction of motion. Behavioral results revealed a strong delay effect: after presentation of the initial motion direction there is a substantial time delay before the changed motion information is integrated into the decision process. To further investigate the underlying changes in the decision process, we developed a Piecewise Linear Ballistic Accumulator model (PLBA). The PLBA is efficient to simulate, enabling it to be fit to participant choice and response-time distribution data in a hierarchal modeling framework using a non-parametric approximate Bayesian algorithm. Consistent with behavioral results, PLBA fits confirmed the presence of a long delay between presentation and integration of new stimulus information, but did not support increased response caution in reaction to the change. We also found the decision process was not veridical, as symmetric stimulus change had an asymmetric effect on the rate of evidence accumulation. Thus, the perceptual decision process was slow to react to, and underestimated, new contrary motion information.

Comparison of Quantum and Bayesian Inference Models

The mathematical principles of quantum theory provide a general foundation for assigning probabilities to events. This paper examines the application of these principles to the probabilistic inference problem in which hypotheses are evaluated on the basis of a sequence of evidence (observations). The probabilistic inference problem is usually addressed using Bayesian updating rules. Here we derive a quantum inference rule and compare it to the Bayesian rule. The primary difference between these two inference principles arises when evidence is provided by incompatible measures. Incompatibility refers to the case where one measure interferes or disturbs another measure, and so the order of measurement affects the probability of the observations. It is argued that incompatibility often occurs when evidence is obtained from human judgments.