Simple matrix methods for analyzing diffusion models of choice probability, choice response time, and simple response time

Diffusion processes (e.g., Wiener process, Ornstein-Uhlenbeck process) are powerful approaches to model human information processes in a variety of psychological tasks. Lack of mathematical tractability, however, has prevented broad applications of these models to empirical data. This tutorial explains step by step, using a matrix approach, how to construct these models, how to implement them on a computer, and how to calculate the predictions made by these models. In particular, we present models for binaries choices for unidimensional and multiattribute choice alternatives; for simple reaction time tasks; and for three alternatives choice problems.     

Acceptance and Elimination Procedures in Choice: Noncomplementarity and the Role of Implied Status quo

The present research contrasts two seemingly complementary decision strategies: acceptance and elimination. In acceptance, a choice set is created by including suitable alternatives from an initial set of alternatives, whereas in elimination it is created by removing inappropriate alternatives from that same initial set. The research used realistic career decision-making scenarios and presented to respondents sets of alternatives that varied in their preexperimental strength values. Whereas complementarity of acceptance and elimination is implied by three standard (normative) assumptions of decision theory, we find a systematic discrepancy between the outcomes of these procedures: choice sets were larger in elimination than in acceptance. This acceptance–elimination discrepancy is directly tied to subcomplementarity. The central tenet of the theoretical framework developed here is that acceptance and elimination procedures imply different types of status quo for the alternatives, thereby invoking a different selection criterion for each procedure. A central prediction of the dual-criterion framework is that middling alternatives should be most susceptible to the type of procedure used. The present studies focus on this prediction which is substantiated by the results showing that middling alternatives yield the greatest discrepancy between acceptance and elimination. The implications of this model and findings for various research domains are discussed.     

Multi-attribute,multi-alternative models of choice: Choice,reaction time,and process tracing

The first aim of this research is to compare computational models of multi-alternative, multi-attribute choice when attribute values are explicit. The choice predictions of utility (standard random utility & weighted valuation), heuristic (elimination-by-aspects, lexicographic, & maximum attribute value), and dynamic (multi-alternative decision field theory, MDFT, & a version of the multi-attribute linear ballistic accumulator, MLBA) models are contrasted on both preferential and risky choice data. Using both maximum likelihood and cross-validation fit measures on choice data, the utility and dynamic models are preferred over the heuristic models for risky choice, with a slight overall advantage for the MLBA for preferential choice. The response time predictions of these models (except the MDFT) are then tested. Although the MLBA accurately predicts response time distributions, it only weakly accounts for stimulus-level differences. The other models completely fail to account for stimulus-level differences. Process tracing measures, i.e., eye and mouse tracking, were also collected. None of the qualitative predictions of the models are completely supported by that data. These results suggest that the models may not appropriately represent the interaction of attention and preference formation. To overcome this potential shortcoming, the second aim of this research is to test preference-formation assumptions, independently of attention, by developing the models of attentional sampling (MAS) model family which incorporates the empirical gaze patterns into a sequential sampling framework. An MAS variant that includes attribute values, but only updates the currently viewed alternative and does not contrast values across alternatives, performs well in both experiments. Overall, the results support the dynamic models, but point to the need to incorporate a framework that more accurately reflects the relationship between attention and the preference-formation process.     

Generalized thurstone models for ranking: Equivalence and reversibility

Previous work has determined the conditions under which generalized versions of Thurstone's theory of comparative judgment are formally equivalent (i.e., empirically indistinguishable) for choice experiments. This note solves the analogous problem for ranking experiments: It is shown that if two “Generalized Thurstone Models” are equivalent for choice experiments with n alternatives they are also equivalent for ranking experiments with n alternatives, despite the fact that ranking generates many more preference probabilities. This result in turn allows one to determine which Generalized Thurstone Models are “reversible,” i.e., satisfy the requirement that regardless of whether the subject ranks from best to worst or from worst to best, rankings that express the same preference order will occur with the same probability.     

Replicated alternatives and the role of confusion,chasing, and regret in decisions from experience

The current paper explores choice among alternatives that can be classified into distinct classes. All the members of a particular class were ‘replicated alternatives’: they promised the same payoff distribution. Information to decision makers was limited to feedback concerning the realized (obtained and foregone) payoffs. Experiment 1 demonstrates that increasing the number of replicated alternatives creates confusion (which facilitates random choice) and changes the implications of the tendency to chase recent returns (i.e., select the alternative with the best recent outcomes). This effect, termed ‘confused chasing,’ facilitates risk seeking even when this behavior impairs expected earnings. Experiment 2 reveals that increasing the number of replicated alternatives can reduce (but does not eliminate) the tendency to underweight rare events. Experiment 3 shows that the relative importance of chasing and confusion is sensitive to the likelihood of realizing lower payoffs than the forgone payoffs. The main results are summarized with a simple model assuming that payoff sensitivity decreases with experienced regret. Copyright © 2006 John Wiley & Sons, Ltd.     

Loss aversion and inhibition in dynamical models of multialternative choice

The roles of loss aversion and inhibition among alternatives are examined in models of the similarity, compromise, and attraction effects that arise in choices among 3 alternatives differing on 2 attributes. R. M. Roe, J. R. Busemeyer, and J. T. Townsend (2001) have proposed a linear model in which effects previously attributed to loss aversion (A. Tversky & D. Kahneman, 1991) arise from attention switching between attributes and similarity-dependent inhibitory interactions among alternatives. However, there are several reasons to maintain loss aversion in a theory of choice. In view of this, an alternative theory is proposed, integrating loss aversion and attention switching into a nonlinear model (M. Usher & J. L. McClelland, 2001) that relies on inhibition independent of similarity among alternatives. The model accounts for the 3 effects and makes testable predictions contrasting with those of the Roe et al. (2001) model.     

The time course of perceptual choice: the leaky, competing accumulator model

The time course of perceptual choice is discussed in a model of gradual, leaky, stochastic, and competitive information accumulation in nonlinear decision units. Special cases of the model match a classical diffusion process, but leakage and competition work together to address several challenges to existing diffusion, random walk, and accumulator models. The model accounts for data from choice tasks using both time-controlled (e.g., response signal) and standard reaction time paradigms and its adequacy compares favorably with other approaches. A new paradigm that controls the time of arrival of information supporting different choice alternatives provides further support. The model captures choice behavior regardless of the number of alternatives, accounting for the log-linear relation between reaction time and number of alternatives (Hick's law) and explains a complex pattern of visual and contextual priming in visual word identification.     

Knowledge, relevant alternatives and missed clues

The classic version of the relevant alternatives theory (RAT) identifies knowledge with the elimination of relevant alternatives (Dretske 1981, Stine 1976, Lewis 1996, inter alia). I argue that the RAT is trapped by the problem of the missed clue, in which the subject sees but does not appreciate decisive information.     

PROGRAM SELECTION WITH ABUNDANCE OF CHOICE

A model of program choice focusing on the decision-making process is developed and applied to cable television viewing. The process concepts of orienting search (becoming aware of alternatives to watch) and revaluation (reconsidering a choice) are found to be positively correlated with channel familiarity (awarensss of available cable channels) and channel repertoire (use of cable channels). Viewers appear to have their own routine means of choosing what to watch that vary according to personality characteristics and demographics. Young, male adults with a high need for stimulation engage in more active choice processes.     

On the tautology of the matching law

The generalized matching law, that organisms divide their time between alternatives in proportion to the value of the reinforcement consequent on the choice, is derivable from the assumption that an organism choosing between alternatives is under no constraints except those the contingencies of reinforcement impose. Hence, the law is not subject to empirical test. Its value lies in its simple codification of assumptions underlying choice experiments.     

Some results on random utility models

This paper extends earlier work on the properties of Thurstone choice models to the general random utility model (RUM), in which the components of the random utility vector are not necessarily independent and identically distributed, even apart from location shifts. The condition for two RUMs to be equivalent is established, and relationships between the choice axiom, decomposition, and other conditions are explored. In the special case of the Thurstone model, the conditions considered are all equivalent to the double exponential distribution.     

Human reasoning: Deduction rules or mental models, or both?

A good deal of energy is currently being expended into discovering the fundamental machinery underlying deductive reasoning. Is it based upon mental models (arrays) or deduction rules (propositions)? The appeal of finding a “grand unified theory” of reasoning is obvious, but the likelihood of achieving this must also be considered.

This paper discusses the use of experimental psychology in attempts to discover the processes associated with the fundamental reasoning mechanism. One particular problem is that individuals can use different strategies to solve reasoning problems. The consequences of this are assessed in relation to: (1) the assumptions underlying the experiments, (2) the choice of tasks and task presentations intended to enable the fundamental reasoning processes to be viewed directly, and (3) the power status of the theories and the nature of the evidence required to show that either theory is superior. Under close scrutiny the debate appears to be unresolvable by using empirical techniques. However, although the main conclusions are negative, it is suggested that approaches that directly investigate individual differences are likely to be useful alternatives.     

Intensity and number of alternatives in hue identification: Piéron’s law and choice reaction time

Two experiments investigated the effects of intensity and the number of response alternatives in hue identification. Three questions were addressed. First, what is the relationship between these effects? If intensity and number of alternatives affect different processes in series, these factors would have additive effects on reaction time (RT). Other models predict a multiplicative interaction. These models assume that when a stimulus is presented, evidence accrues over time for each response alternative, and that when the evidence for a particular alternative exceeds its criterion, the response is made. A natural auxiliary assumption is that intensity affects the rate of evidence accrual, while the number of alternatives affects the criterion. The second question addressed is, how does choice RT change as a function of intensity? Piéron used a power function to describe the effect of intensity on simple RT. This paper considers three candidates for intensity’s effect on choice RT: the power, logistic, and Michaelis functions. Each candidate function was tested in an additive model and a multiplicative model. The best account was given by the power function in an additive model. The last question is, does each process produce an output that can be classified as correct or incorrect? Schweickert (1985) proposed that factors that selectively influenced such processes would have additive effects on log percent correct. Instead, an interaction was found. Perhaps the assumption that the process affected by intensity produces a single output is wrong.     

Component processes of phased decision strategies

It is often assumed that decision makers apply mixtures of strategies including heuristic processes in complex choice tasks. In support of this assumption the component processes involved in phased strategies were investigated. Expectations for an experimental investigation were generated by simulations within the effort-quality framework. Effort-quality trade-off functions for tasks of different complexity favor compensatory dimension-based strategies for binary choice. For multialternative choice, a combination of initial dimension-based elimination procedures followed by alternative-based final processing would be expected. These expectations were tested in two experiments. While choosing between two, four, or eight alternatives, subjects could successively retrieve parts of the available information. Depending on their menu selection, either all features of one alternative were displayed together (alternative-based retrieval), or all features on one dimension were displayed together (dimension-based retrieval). Vincent averaging was used to analyze the retrieval sequences with respect to their stationarity. While the stationary sequences for binary choice are compatible with single processing strategies, non-stationarity of the sequences obtained for multialternative choice indicates the use of phased strategies. In particular, non-stationarity due to decreasing dimension-based retrievals over time suggests that subjects first applied dimensional elimination and later derived their choices by more thorough, alternative-based processing of the remaining alternatives.     

Discrete dynamic choice: An extension of the choice models of thurstone and luce

The present paper deals with the extension of two well-known static discrete choice theories to the dynamic situation in which individuals make choices at several points in (continuous) time. A dynamic version of Luce's Axiom, “independence from irrelevant alternatives”, is proposed and some of its implications are derived. In the static case Yellott (J. Math. Psych. 1977, 15, 109–146) and others have demonstrated that an independent random utility model generated from the extreme value distribution exp(?e^?ax?b) becomes equivalent to Luce's Axiom. Yellott also introduced an axiom called “invariance under uniform expansions of the choice set”, and he proved that within the class of random utility models with independent identically distributed utilities (apart from a location shift) this axiom is equivalent to Luce's Axiom. These results are extended to the dynamic situation and it is shown that if the utility processes are expressed by so-called extremal processes the corresponding choice model is Markovian. A nonstationary generalization is proposed which is a substantial interest in applications where the parameters of the choice process are influenced by previous choice experience or by time-varying exogenous variables. In particular, it is demonstrated that the nonstationary model is Markovian if and only if the joint choice probabilities at two points in time have a particular form. Thus, the paper provides a rationale for applying a specific class of Markov models as the point of departure when modelling mobility processes that involve individual discrete decisions over time.     

On adaptation, maximization, and reinforcement learning among cognitive strategies

Analysis of binary choice behavior in iterated tasks with immediate feedback reveals robust deviations from maximization that can be described as indications of 3 effects: (a) a payoff variability effect, in which high payoff variability seems to move choice behavior toward random choice; (b) underweighting of rare events, in which alternatives that yield the best payoffs most of the time are attractive even when they are associated with a lower expected return; and (c) loss aversion, in which alternatives that minimize the probability of losses can be more attractive than those that maximize expected payoffs. The results are closer to probability matching than to maximization. Best approximation is provided with a model of reinforcement learning among cognitive strategies (RELACS). This model captures the 3 deviations, the learning curves, and the effect of information on uncertainty avoidance. It outperforms other models in fitting the data and in predicting behavior in other experiments.     

Choice: A local analysis

Analyses of free-operant choice usually employ one of two general procedures: the simple concurrent procedure (i.e., a concurrent variable-interval variable-interval schedule) or the concurrent-chains procedure (i.e., concurrently available initial links, each leading to an exclusively available terminal link). Theories about choice usually focus on only one of the two procedures. For example, maximization theories, which assert that behavior is distributed between two alternatives in such a way that overall rate of reinforcement is maximized, have been applied only to the simple concurrent procedure. In the present paper, a form of the pairing hypothesis (according to which pairings between one stimulus and another affect the value of the first, and pairings between responses and reinforcers affect the value of the former) is developed in a way that allows it to make qualitative predictions with regard to choice in a variety of simple concurrent and concurrent-chains procedures. The predictions include matching on concurrent variable-interval variable-interval schedules, preference reversal in the self-control paradigm, and preference for tandem over chained terminal links.     

The effect of normative models on individual and group choice

Studied the effect of a normative model (maximization of expected value) upon group and individual choice. 109 MBA students in a Lain square research design chose between two alternatives differing in expected value and in range of outcomes. Group choices were significantly (p < .05) closer to those predicted by the normative model than were individual choices. This difference was not due only to information about the presence and applicability of the normative model but rather it was due to the persuasiveness of the model in a group as a cogent and correct solution to the choice dilemma. Task instructions emphasizing the rewards from risk taking produced significantly more choices (p < .05) of the riskier alternative, particularly by individuals as opposed to groups, than did instructions emphasizing the penalties of risk taking. ‘Risky’ and ‘conservative’ shifts in choice between groups and individuals were explicable through knowledge of the influence of the normative model in individual and group choice.