A Mathematical Formalization of Fuzzy Trace Theory

A Mathematical Formalization of Fuzzy Trace Theory David Andre Broniatowski (Broniatowski@Gwu.Edu) The George Washington University, Department of Engineering Management and Systems Engineering, 1776 G Street NW Washington, DC 20052 USA Valerie F. Reyna (vr53@Cornell.Edu) Human Neuroscience Institute, Cornell University Ithaca, NY 14853 USA Abstract stimulus retains its surface form. Examples include memory representations of exact words, numbers, and pictures. Even though verbatim representations reproduce the details of a given stimulus, they are also symbolic representations. Rivers et al. (2008) illustrate the differences between gist and verbatim representations using the following scenario: Consider an adolescent who must decide between attending a party where alcohol will be served to minors (which the adolescent perceives as a fun but risky option) and attending a friend’s sleepover where alcohol is not served (which the adolescent perceives as a fun but “safe” option in the sense that there is no risk of getting in trouble for underage drinking). Suppose that the adolescent thinks that the party will be more fun than the sleepover; however, the adolescent faces a small risk (e.g., a 10% chance) of being caught drinking at the party. An expected-value framework might characterize the options as follows: 1. A 100% chance of an amount of fun at the sleepover. 2. A 90% chance of twice as much fun and a 10% chance of no fun (getting caught) at the party. A verbatim representation of these two options would be what is described above, that is, a precise description of outcomes and their probabilities. Outcomes and probabilities need not be fully explicit for a mental representation to be “verbatim”; verbatim representations encode the literal content of information or experience, however limited that might be. Option 2 is preferable based on explicit outcomes and probabilities; in many instances, the odds are with the risk-taking adolescent. However, a categorical gist representation of these two options is: 1. Some fun with certainty at the sleepover. 2. Some chance of some fun and some chance of no fun at the party. The gist representation encourages risk avoiding (option 1) because the possibility of “no fun” is confined to the risky option (option 2). Research on risky choices suggests that decision makers represent decision options in both ways simultaneously – i.e., in terms of specific verbatim outcomes and probabilities (when those are known or estimated) and as qualitative gist representations. Figure 1a shows a visual representation of this choice in a two- dimensional Euclidean space (the “decision space”), whereas Figures 1b and 1c shows how points in this space are mapped to gists, represented as curves within the decision space (“constraints”). Finally, Figure 1d indicates which gist will be chosen given multiple interpretations. In this paper, we develop a novel formalization of Fuzzy Trace Theory (FTT), a leading theory of qualitative risky decision-making. Our model is the first to explicitly formalize and integrate the concepts of gist and the gist hierarchy. Domain knowledge constrains the space of possible decision problems, explaining which gists are chosen in which contexts. We test our model against risky-choice framing and Allais paradox problems, and manipulations of these problems. Our results also confirm new predictions regarding how problem manipulations can enhance or attenuate framing effects. Keywords: Decision-making; mathematical model; risky choice; framing effect; Allais paradox; gist Introduction In this article, we introduce a mathematical model of Fuzzy Trace Theory (FTT), a leading theory of decision- making under risk, which assumes that decision-makers use a qualitative “gist” representation of a stimulus, in parallel with a precise verbatim representation (Reyna, 2012). We integrate memory and decision-making research to formalize how decision options and probabilities are mentally represented. By “formalization,” we mean a mathematical description, and extension, of a verbal theory. We focus our analysis on risky choice tasks. We explain experimental evidence from several classic decision problems and experimental manipulations of these problems (e.g., Allais, 1953; Kuhberger & Tanner, 2010; Peters & Levin, 2008; Reyna, 2012; Tversky & Kahneman, 1981). We then use our mathematical theory to make and test novel predictions. This paper provides the first explicit formalization of the concepts of gist, the gist hierarchy, and the fuzzy-processing preference (described below). Gist and Verbatim The central tenet of Fuzzy Trace Theory (FTT) is that people encode, store, retrieve, and forget memories that are characterized by different levels of detail. We refer to these levels as “gist” and “verbatim. ” These representations are encoded separately and roughly in parallel (Brainerd et al., 2009). A gist representation captures the meaning, or essence, of a stimulus, and is therefore a symbolic mental representation. Gists representations are simple, qualitative (for reviews, see Reyna, 2012) and are grounded in experience. In contrast, a verbatim representation of a