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Symposium: Deterministic and Probabilistic Models of Choice

Elizabeth Pettit

Ms. Yu Huang

Joe Johnson

Michel Regenwetter

Context effects, wherein the introduction of a third option can seemingly alter the preference relation between two other choice options, are pervasive in decision making. While decades of research have supported their existence, there has been some difference in claims regarding exactly which occur, and under which circumstances. Two specific limitations of previous work may be preventing a fuller understanding of these behaviors: studies tend to use a few characteristic stimuli to test each effect, and they analyze results based on aggregate choice proportions. The former was remedied in a recent study by Dumbalska et al. (2020) in PNAS, which used 32 stimuli spanning a two-dimensional attribute space. We address the latter concern here. In particular, we demonstrate a novel and powerful way of investigating context effects, within subjects, based on relatively simple assumptions about individual choice patterns. The framework translates hypotheses about preference or indifference on each choice problem into probabilistic models characterized by inequality constraints on binary choice probabilities. While ostensibly these models form convex polytopes in a 32-dimensional space, which would seem computationally unwieldy, it turns out to be computationally efficient to calculate Bayes factors for their empirical performance by treating them as cross-products of line-segments. We offer an easy-to-use and openly available web application allowing other researchers to test, virtually instantaneously, their own sets of hypotheses on the data from Dumbalska et al. (2020).

This is an in-person presentation on **July 19, 2023**
(09:00 ~
09:20 UTC).

Kota Saito

Random ordering models (Block and Marschak, 1960) explain various data sets from a common assumption: the answers of the subject(s) are guided by a latent probability distribution on the set of all orderings of the alternatives (here, ordering means linear ordering). For instance, there are such models for binary choice, multiple choice and best/worst choice. Characterizing a probabilistic model means describing its set of predictions (see for instance Doignon, Heller and Stefanutti, 2018). In many cases, the predicted points of a random ordering model form a convex polytope. The vertices of the polytope are known (they bijectively correspond to the orderings). The characterization problem is then turned into the search of a description of the polytope by a system of equalities and inequalities. Obtaining an efficient characterization of the binary choice model is deemed infeasible (if P ≠ NP, see Fiorini, 2006; note that the binary choice polytope is known in operations research as the linear ordering polytope). To the contrary, a remarkable achievement of Falmagne (1978) provides an explicit characterization of the multiple choice model. Falmagne generalizes inequalities formulated by Block and Marschak (1960), and he then shows by recurrence on the number of alternatives that the resulting inequalities together with obvious equalities determine the multiple choice polytope (MCP). To derive an enlightening proof of Falmagne's Theorem, Fiorini (2004) assimilates the MCP with the flow polytope of some acyclic network. We further exploit Fiorini technique, and extend its applicability. Apart from a recognition of the facets by Suck (2002), the geometric structure of the MCP was apparently not much investigated. We describe the adjacency of vertices and the adjacency of facets (Doignon and Saito, 2023). Our description of the edges of the MCP helps understand recent findings in economics papers such as Chang, Narita and Saito (2022) and Turansick (2022). As a matter of fact, Doignon and Saito describe the two adjacencies in the more general setting of flow polytopes of acyclic networks (the MCP is just a particular case). So, the results apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of random models with other types of orderings, see for instance Marley and Regenwetter, 2017): in each of the three cases, they rely on a specific, acyclic network. We also show how Fiorini technique helps in the analysis of further random ordering models, in particular we improve the results of Barberá and Pattanaik (1986) on the multiple choice model with latent weak orders (this is on-going joint work with Kota Saito). However, there is no reason that any random ordering model could be characterized in terms of the flows on an acyclic network. For instance, the best/worst choice model apparently requires other techniques. This is another story, still to be written (for a prologue, see the other presentation by the speaker at this meeting). Symposium DETERMINISTIC AND PROBABILISTIC MODELS OF CHOICE Daniel Cavagnaro Jean-Paul Doignon Marc Jekel Tim Pleskac Mike Regenwetter Reinhard Suck

This is an in-person presentation on **July 19, 2023**
(09:20 ~
09:40 UTC).

Daniel Cavagnaro

Tim Pleskac

Michel Regenwetter

The decision-making literature has explored the idea of a Description-Experience Gap: that people overweight rare events when those events are described and underweight rare events when they are experienced. Some work establishing this gap has ignored the variability in choices between and within individuals, limiting the conclusions that can be drawn about preference differences between described vs. experienced gambles. Regenwetter and Robinson (2017) established how QTest could address these limitations, allowing users to implement a set of probabilistic choice models to conduct order-constrained hypothesis tests on the description-experience gap. However, their initial implementation ignored one key source of heterogeneity: experience. Here, we extend these probabilistic choice models to account for the heterogeneity of experience. We show how choice models that treat experienced proportions of outcomes as best guesses of their probabilities ---i.e., of the objective probabilities that determine the likelihoods of experienced outcomes---can be more parsimonious than models that use those objective probabilities directly. We use this more extensive set of models to test for the description-experience gap and to identify its source.

This is an in-person presentation on **July 19, 2023**
(09:40 ~
10:00 UTC).

Ms. Brittney Currie

Ms. Yu Huang

Dr. Bart Smeulders

Ms. Anna Carlson

Chaotic responses to Covid-19, political polarization, pervasive misinformation, and social unrest raise the question whether some or many individuals exercise irrational moral judgment. We provide the first mathematically correct direct test for transitivity of moral preferences. Transitivity, a core rationality criterion, is conceptually, mathematically, and statistically difficult to evaluate. We tested three parsimonious, order-constrained, probabilistic characterizations. Among 28 individuals, everyone satisfied the weak utility model, according to which an individual’s choices are noisy reflections of a single transitive preference. Tightening the bounds on error rates in noisy responses yielded a poorly performing model. Everyone obeyed the general random utility hypothesis, according to which individuals’ choices reveal uncertain, but transitive, moral preferences. Bayesian model selection favored such probabilistic transitive preferences, hence also the equivalent random utility hypothesis. The findings suggest that there is some order underlying the apparent chaos: Rather than presume widespread disregard for moral principles, policy makers may build on navigating and reconciling extreme heterogeneity compounded with individual uncertainty. Symposium submission for DETERMINISTIC AND PROBABILISTIC MODELS OF CHOICE Daniel Cavagnaro Jean-Paul Doignon Marc Jekel Tim Pleskac Michel Regenwetter Reinhard Suck

This is an in-person presentation on **July 19, 2023**
(10:00 ~
10:20 UTC).

The conditions of completeness and transitivity and their violations are important in theories of preferential choice. NaP (necessary and possible) preferences try to disentangle their interplay. They are related to so called Richter Peleg representations of partial orders, where the order is represented by a vector of numerical functions instead of a single one. They consist of splitting a quasiorder (preorder) into two nested relations. Recently, a series of papers (e.g. Gialotta & Watson (2018, 2020)) gave generalizations of this concept. It turns out that these results are rather straightforward consequences of (generalizations of) Szpilrajn’s theorem on the linear extensions of a partial order. In the present paper we investigate the set of linear extensions in order to clarify what is behind NaP preferences and their generalizations. In doing so a natural extension on probabilistic NaP preferences is suggested.

This is an in-person presentation on **July 19, 2023**
(10:20 ~
10:40 UTC).

Ms. Meichai Chen

Ms. Emily Line

Michel Regenwetter

Numerous deterministic and probabilistic choice models are available in academic literature. To promote cumulative science and inclusivity, enhancing the accessibility of these models for researchers, including those without formal modelling training, is essential. To this end, I will introduce an R-Shiny app specifically designed to assist researchers in translating both deterministic and probabilistic binary choice models into a unified mathematical representation. This unified representation is accomplished by deriving a minimal set of equalities and inequalities that embody the model's predicted relationship between choice probabilities. This representation enables researchers to evaluate a model's quality based on factors like logical consistency and parsimony before conducting lab studies and allocating resources. Moreover, the app offers methods to bridge the gap between deterministic and probabilistic models. It allows researchers to explore various probabilistic versions of deterministic models by incorporating psychologically meaningful sources of variability in choice probabilities. The app further simplifies the research process by automatically generating input files for in-depth model comparisons, eliminating the need for programming skills. Researchers can use these files to calculate Bayes factors and frequentist p-values in model comparisons, ensuring a thorough evaluation of competing models. Throughout the presentation, I will underscore the significance of providing user-friendly modelling tools in strengthening the replicability of research findings in the behavioral, social, and cognitive sciences.

This is an in-person presentation on **July 19, 2023**
(10:40 ~
11:00 UTC).

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