Probabilistic Choice Models Research Articles

Best, worst, and best&worst choice probabilities for logit and reverse logit models

This paper builds upon the work of Professor Marley, who, since the beginning of his long research career, has proposed rigorous axiomatics in the area of probabilistic choice models. Our study concentrates on models that can be applied to best and worst choice scaling experiments. We focus on those among these models that are based on strong assumptions about the underlying ranking of the alternatives with which the individual is assumed to be endowed when making the choice. Taking advantage of an inclusion–exclusion identity that we showed a few years ago, we propose a variety of best–worst choice probability models that could be implemented in software packages that are flourishing in this field.

Value Propositions of Public Adult Hearing Rehabilitation in Denmark.

Objective: To obtain and evaluate detailed descriptions of potential value propositions as seen by adults undergoing hearing rehabilitation with hearing aids. Design: Semi-structured interviews with patients and audiologists, a literature search, and the inclusion of domain knowledge from experts and scientists were used to derive value propositions. A two-alternative forced-choice paradigm and probabilistic choice models were used to investigate hearing aid users' preferences for the value propositions through an online platform. Study sample: Twelve hearing aid users (mean age 70, range 59-70) and eleven clinicians were interviewed. A total of 173 experienced hearing aid users evaluated the value propositions. Results: Twenty-nine value propositions as described by patients, clinicians, and hearing care experts where identified, from which twenty-one value propositions were evaluated. Results of the pair-wise evaluation method show that the value propositions judged to be the most important for the hearing aid users were: "13. To solve the hearing problem you have", "09. Thorough diagnosis of the hearing", and "16. The hearing aid solution is adapted to individual needs", which are related to finding the correct hearing solution and to be considered in the process. The value propositions judged to be least important were: "04 Next of kin and others involved in the process", "26. To be in the same room as the practitioner", and "29. The practitioner's human characteristics", related to the involvement of others in the process and the proximity and personal manner of the practitioners.

Scaling preferences using probabilistic choice models: is there a ratio-scale representation of subjective liking?

In two online experiments, we tested whether preference judgments can be used to derive a valid ratio-scale representation of subjective liking across different stimulus sets. Therefore, participants were asked to indicate their preferences for all possible pairwise comparisons of 20 paintings (Experiment 1) and 20 faces (Experiment 2). Probabilistic choice models were fit to the resulting preference probabilities (requiring different degrees of stochastic transitivity), demonstrating that a ratio-scale representation of the liking of both paintings and faces can be derived consistently from the preference judgments. While the preference judgments of paintings were consistent with the highly restrictive Bradley–Terry–Luce model (Bradley and Terry, Biometrika 39:324–345, 1952; Luce, 1959), the liking of faces could be represented on a ratio scale only when accounting for face gender as an additional aspect in an elimination-by-aspects model. These ratio-scaled liking scores were then related to direct evaluative ratings of the same stimuli on a 21-point Likert scale, given both before and after the pairwise comparisons. It was found in both studies that evaluative ratings can be described accurately as a logarithmic function of the indirectly derived liking scores for both types of stimuli. The results indicate that participants are able (a) to consistently judge preferences across two heterogeneous stimulus sets, and (b) to validly report their liking in direct evaluative ratings, although the numeric labels derived from direct evaluative ratings cannot be interpreted at face value for ratio-scaled liking scores.

Sequential Competitive Facility Location: Exact and Approximate Algorithms

In “Sequential Competitive Facility Location: Exact and Approximate Algorithms,” Qi et al. consider a competitive facility location problem (CFLP), where two firms sequentially open new facilities within their budgets and maximize their market shares of demand following a probabilistic choice model. They derive an equivalent, single-level mixed-integer nonlinear program (MINLP) reformulation of the bilevel MINLP and exploit the problem structures to derive valid inequalities based on submodularity and concave overestimation. They also develop an approximation algorithm with a constant approximation guarantee. They further study several extensions of CFLP that have general facility-opening costs, outside competitors, and diverse facility-planning decisions. Their approaches significantly accelerate the computation of CFLP on large-sized instances that have not been solved optimally or even heuristically by existing methods.

Travel decision determinants during and after COVID-19: The role of tourist trust, travel constraints, and attitudinal factors.

The COVID-19 pandemic has forced tourism practitioners to create efficient strategies to attract travelers. Using three theoretical frameworks, such as tourist trust (political, destination, and interactional trust), travel constraint (intrapersonal, interpersonal, and “social distancing” structural constraint), and extended theory of planned behavior (travel attitude, perceived behavioral control, subjective norm, perceived health risk, past travel experience), we develop a comprehensive framework to explain the impact of travel promoting, restricting, and attitudinal factors on travel decision during and after the pandemic. Data was obtained through an extensive survey conducted on 1451 Korean travelers and was analyzed using probabilistic choice models and count models. The results show the specific factors that determine travel decisions during the pandemic (whether to travel and frequency) and travel intention after the pandemic. This study provides important theoretical and practical insights into how to develop successful COVID-19 recovery strategies in the tourism industry.

Probabilistic discrimination of relative stimulus features in mice

During perceptual decision-making, the brain encodes the upcoming decision and the stimulus information in a mixed representation. Paradigms suitable for studying decision computations in isolation rely on stimulus comparisons, with choices depending on relative rather than absolute properties of the stimuli. The adoption of tasks requiring relative perceptual judgments in mice would be advantageous in view of the powerful tools available for the dissection of brain circuits. However, whether and how mice can perform a relative visual discrimination task has not yet been fully established. Here, we show that mice can solve a complex orientation discrimination task in which the choices are decoupled from the orientation of individual stimuli. Moreover, we demonstrate a typical discrimination acuity of 9°, challenging the common belief that mice are poor visual discriminators. We reached these conclusions by introducing a probabilistic choice model that explained behavioral strategies in 40 mice and demonstrated that the circularity of the stimulus space is an additional source of choice variability for trials with fixed difficulty. Furthermore, history biases in the model changed with task engagement, demonstrating behavioral sensitivity to the availability of cognitive resources. In conclusion, our results reveal that mice adopt a diverse set of strategies in a task that decouples decision-relevant information from stimulus-specific information, thus demonstrating their usefulness as an animal model for studying neural representations of relative categories in perceptual decision-making research.

The potential of urban distributed solar energy in transition economies: The case of Beirut city

In this paper, using Lebanon's capital, Beirut, as a case study, a methodology is proposed to assess the potential for solar photovoltaics (PV) in urban areas incorporating both economic and non-economic factors. Utilizing a rich spatial dataset of solar irradiation augmented with electricity bills at the building level, the cost and benefit of installing rooftop PV systems for each building is estimated. Additionally, incentives and barriers for adopting those systems are investigated using a probabilistic choice model. The results show that Beirut city has a potential for distributed rooftop solar PV to be between 195 and 295 MWp. However, adoption rates are low at 0.49% and 1.23% for residential and commercial buildings, respectively, reflecting the limitation of financial incentives alone to promote the deployment of distributed renewable energy systems in transition economies. The impact of different incentive policy instruments and the role of solar PV in today's economic crisis in Lebanon is analyzed. The biggest impact was achieved through removing (or lowering) electricity tariff subsidy, although this option remains highly constrained by political calculus. We argue that the Lebanese government should fast-track and implement the required legal framework to facilitate and incentivize distributed power generation from renewable sources to promote both green energy and its financial resilience. The proposed modeling framework together with the results obtained in this study will have important implications for energy policy makers in Lebanon and other transition economies.

Mice Can Learn a Stimulus-Invariant Orientation Discrimination Task

Understanding how the brain computes choice from sensory information is a central question of perceptual decision-making. Relevant behavioral tasks condition choice on abstract or invariant properties of the stimuli, thus decoupling stimulus-specific information from the decision variable. Among visual tasks, orientation discrimination is a gold standard; however, it is not clear if a mouse – a recently popular animal model in visual decision-making research – can learn an invariant orientation discrimination task and what choice strategies it would use. Here we show that mice can solve a discrimination task where choices are decoupled from the orientation of individual stimuli, depending instead on a measure of relative orientation. Mice learned this task, reaching an upper bound for discrimination acuity of 6 degrees and relying on decision-making strategies that balanced cognitive resources with history-dependent biases. We analyzed behavioral data from n=40 animals with the help of a novel probabilistic choice model that we used to interpret individual biases and behavioral strategies. The model explained variation in performance with task difficulty and identified unreported dimensions of variation associated with the circularity of the stimulus space. Furthermore, it showed a larger effect of history biases on animals’ choices during periods of lower engagement. Our results demonstrate that mice can learn invariant perceptual representations by combining decision-relevant stimulus information decoupled from low-level visual features, with the computation of the decision variable dependent on the cognitive state.

Dual choice axiom and probabilistic choice

A decision maker chooses in a probabilistic manner if she does not necessarily prefer the same choice alternative when repeatedly presented with the same choice set. Probabilistic choice may occur for a variety of reasons such as unobserved attributes of choice alternatives, imprecision of preferences, or random errors/noise in decisions. The Luce choice model (also known as strict utility or multinomial logit) is derived from the choice axiom (also known as the independence from irrelevant alternatives). This axiom postulates that the relative likelihood of choosing one choice alternative A over another choice alternative B is not affected by the presence or absence of other choice alternatives in the choice set. This paper presents a dual choice axiom: the relative probability of NOT choosing A over the probability of NOT choosing B is independent from irrelevant alternatives. A new model of probabilistic choice is derived from this dual axiom. This model coincides with Luce’s choice model only in the case of a binary choice. The new model has similar properties as the Luce choice model: the higher is the utility of a choice alternative, the higher is the probability that a decision maker chooses this alternative and the lower is the probability that he or she chooses any other alternative. The new model differs from the Luce choice model in two aspects: utility of choice alternatives is bounded (from above and below) and choice probabilities are more sensitive to differences in utility of choice alternatives.

Mice tune out not in: violation of prediction drives auditory saliency.

Successful navigation in complex acoustic scenes requires focusing on relevant sounds while ignoring irrelevant distractors. It has been argued that the ability to track stimulus statistics and generate predictions supports the choice of what to attend and what to ignore. However, the role of these predictions about future auditory events in drafting decisions remains elusive. While most psychophysical studies in humans indicate that expected stimuli are more easily detected, most work studying physiological auditory processing in animals highlights the detection of unexpected, surprising stimuli. Here, we tested whether in the mouse, high target probability results in enhanced detectability or whether detection is biased towards low-probability deviants using an auditory detection task. We implemented a probabilistic choice model to investigate whether a possible dependence on stimulus statistics arises from short-term serial correlations or from integration over longer periods. Our results demonstrate that target detectability in mice decreases with increasing probability, contrary to humans. We suggest that mice indeed track probability over a timescale of at least several minutes but do not use this information in the same way as humans do: instead of maximizing reward by focusing on high-probability targets, the saliency of a target is determined by surprise.

Probabilistic independence axiom

One of the most well-known theories of decision making under risk is expected utility theory based on the independence axiom. The independence axiom postulates that decision maker’s preferences between two lotteries are not affected by mixing both lotteries with the same third lottery (in identical proportions). The probabilistic independence axiom (also known as the cancelation axiom) extends this classic independence axiom to situations when a decision maker chooses in a probabilistic manner (i.e., she does not necessarily prefer the same choice alternative when repeatedly presented with the same choice set). Probabilistic choice may occur for a variety of reasons such as unobserved attributes of choice alternatives, imprecision of preferences, random errors/noise in decisions. According to probabilistic independence axiom, the probability that a decision maker chooses one lottery over another does not change when both lotteries are mixed with the same third lottery (in identical proportions). This paper presents a model of probabilistic binary choice under risk based on this probabilistic independence axiom. The presented model generalizes an incremental expected utility advantage model of Fishburn (Int Econ Rev 19(3):633–646, 1978) and stronger utility model of Blavatskyy (Theory Decis 76(2):265–286, 2014).

QTest 2.1: Quantitative testing of theories of binary choice using Bayesian inference

This stand-alone tutorial gives an introduction to the QTest 2.1 public domain software package for the specification and statistical analysis of certain order-constrained probabilistic choice models. Like its predecessors, QTest 2.1 allows a user to specify a variety of probabilistic models of binary responses and to carry out state-of-the-art frequentist order-constrained hypothesis tests within a Graphical User Interface (GUI). QTest 2.1 automatizes the mathematical characterization of so-called “random preference models”, adds some parallel computing capabilities, and, most importantly, adds tools for Bayesian inference and model selection. In this tutorial, we provide an in-depth introduction to the Bayesian features: We review order-constrained Bayesian p-values, DIC and Bayes factors, building on the data, models, and prior QTest based frequentist data analyses of an earlier (frequentist) tutorial by Regenwetter et al. (2014).

Product optimization with the improved marginal moment model

ABSTRACT The probabilistic choice model is an important theory foundation for investigating consumer choice behavior in product optimization. In most studies, probabilistic choice behavior is simulated by the multinomial logit (MNL) model and a product always obtains some market share even though the utility a consumer obtains from buying the product is negative. However, the MNL model has a very restrictive substitution pattern – the independence of irrelevant alternatives (IIA). This paper investigates a product optimization problem based on the marginal moment model (MMM). Residual utility is involved in the MMM and negative utility is considered as well. The optimization model of product line design, based on the improved MMM, is established to maximize total profit through three types of problems. The established model fits reality better because the MMM does not have the IIA problem and has good statistical performance. Numerical experiments are carried out to evaluate the feasibility of the proposed model. Meanwhile, the relationships are explored between the optimal solutions and several factors, including the competitive products’ prices, utility variance, rate of cost reduction, and utility of competitive products.

Extended formulations for order polytopes through network flows

Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and convex polyhedral set, or polytope. Polyhedral combinatorics have thus played a key role in studying the mathematical structure of these models. However, standard methods for characterizing the polytopes of such models are subject to a combinatorial explosion in complexity as the number of choice alternatives increases. Specifically, this is the case for random preference models based on linear, weak, semi- and interval orders. For these, a complete, linear description of the polytope is currently known only for, at most, 5–8 choice alternatives. We leverage the method of extended formulations to break through those boundaries. For each of the four types of preferences, we build an appropriate network, and show that the associated network flow polytope provides an extended formulation of the polytope of the choice model. This extended formulation has a simple linear description that is more parsimonious than descriptions obtained by standard methods for large numbers of choice alternatives. The result is a computationally less demanding way of testing the probabilistic choice model on data. We sketch how the latter interfaces with recent developments in contemporary statistics.

Price or performance? A probabilistic choice analysis of the intention to buy electric vehicles in European countries

Traditional choice models perform poorly in understanding the determinants of the adoption of new products. First, data on pioneers and early adopters are biased towards specific performance characteristics of the product and the socio-demographic characteristics of the consumers. Second, surveys on the intention to buy underperform in detecting movements of those who do not intend to buy, who are the majority in the case of new products. Probabilistic choice models try to overcome these issues. By using survey data on electric vehicles, we theoretically contribute to this stream of literature and empirically estimate the impact of specific performance improvements and price reduction on the probability of consumers switching from non-intention to buy to intention to buy. Results show that price reduction is the most important triggering factor for the diffusion of electric vehicles, as it determines more than other factors the transition of consumers from the non-intention to the intention to buy an electric vehicle. The improvement in the driving range constitutes the second most important factor for low initial values of the stated intention to buy, while the possibility of recharging at home matters significantly more for consumers with high initial values of the stated intention to buy.

The role of independence and stationarity in probabilistic models of binary choice.

After more then 50 years of probabilistic choice modeling in Economics, Marketing, Political Science, Psychology, and related disciplines, theoretical and computational advances give scholars access to a sophisticated array of modeling and inference resources. We review some important, but perhaps often overlooked, properties of major classes of probabilistic choice models. For within-respondent applications, we discuss which models require repeated choices by an individual to be independent and response probabilities to be stationary. We show how some model classes, but not others, are invariant over variable preferences, variable utilities, or variable choice probabilities. These models, but not others, accommodate pooling of responses or averaging of choice proportions within participant when underlying parameters vary across observations. These, but not others, permit pooling/averaging across respondents in the presence of individual differences. We also review the role of independence and stationarity in statistical inference, including for probabilistic choice models that, themselves, do not require those properties.

Boundedly Rational Expected Utility Theory

We build a satisficing model of probabilistic choice under risk which embeds Expected Utility Theory (EUT) into a boundedly rational deliberation process. The decision maker accumulates evidence for and against alternative options by repeatedly sampling from her underlying set of EU preferences until the evidence favouring one option satisfies her desired level of confidence. Notwithstanding its EUT core, the model produces patterns of behaviour that violate standard axioms, while at the same time capturing the systematic relationship between choice probabilities, response times and confidence judgments, which is beyond the scope of theories that do not take deliberation into account.

Head-to-head comparison of health-state values derived by a probabilistic choice model and scores on a visual analogue scale

BackgroundHealth states were quantified based on discrete choice (DC) modeling and visual analogue scale (VAS) values using the five-level version of the EQ-5D (EQ-5D-5L). The aim of this study was to determine the extent of the relationship between DC derived values (indirect method) and VAS values (direct method).MethodsData were collected in Canada, the United Kingdom, the Netherlands, and the United States. Respondents were asked to perform paired comparisons between two EQ-5D-5L health states for DC. In total, 400 different EQ-5D-5L states were included. After each DC task, respondents were prompted to score the two states one after another on a VAS. Intraclass correlation coefficients were calculated between DC and VAS values and illuminating graphs were designed.ResultsApproximately 400 respondents participated from each country. High similarity [individual intraclass correlation coefficients (ICC) >0.85] of DC and moderate correspondence of VAS values were observed for the four countries. Cross-country comparison of DC values shows a nonlinear relationship to the VAS values.ConclusionEQ-5D-5L derived DC and VAS values show a close but nonlinear relationship. Given the obvious biases associated with the VAS, DC methods based on ordinal responses may be a better alternative.

Forecasting for Spatial Patterns of Empty Houses in Rural Area Using Probabilistic Choice Model

Forecasting for Spatial Patterns of Empty Houses in Rural Area Using Probabilistic Choice Model

Efficiency of the coordinate-exchange algorithm in constructing exact optimal discrete choice experiments

ABSTRACTThe use of discrete choice experiments (DCEs) for modeling real marketplace choices, in both fundamental and applied research, has gained much attention recently. To improve the quality of designing DCEs, most researchers have drawn on optimal design theory. Because of the nonlinearity of the probabilistic choice models, to construct a proper choice design, one needs the help of efficient search algorithms, among which the coordinate-exchange algorithm (CEA) has shown itself to work very well under the widely used multinomial logit discrete choice model. However, due to the discrete nature of the choice design, there are no computationally feasible ways to verify that the resulting design is indeed optimal or efficient. In this article, an approach of evaluating the performance of the CEA for Bayesian optimal designs is proposed. This approach gives a lower bound of the efficiency of the resulting design under the continuous/approximate optimal design framework where well-established mathematical ...