A dynamical model of risky choice

A Dynamical Model of Risky Choice Marieke M. J. W. van Rooij (vanroomm@mail.uc.edu) Luis H. Favela (favelalh@mail.uc.edu) MaryLauren Malone (malonemo@mail.uc.edu) Michael J. Richardson (richamo@ucmail.uc.edu) Center for Cognition, Action, and Perception Department of Psychology, University of Cincinnati Cincinnati, OH 45221, USA. Abstract participants do take risk (Tversky & Kahneman, 1974). In another study, Kahneman and Tversky (1979; 1983) showed that risks with low probabilities are either grossly overweighed, or completely neglected, and that there is large heterogeneity among individuals. Specifically, individuals show more variability in deciding about potential loss than potential gain (Tversky & Kahneman, 1981). These examples suggest that human decision-making behavior under uncertainty can well be described using a nonlinear, dynamic narrative; individual decision behavior is highly context-specific, unstable, and heterogeneous. The aim of this article is therefore to investigate the feasibility of extending current efforts in decision science towards a nonlinear, dynamical approach. Individuals make decisions under uncertainty every day based on incomplete information concerning the potential outcome of the choice or chance levels. The choices individuals make often deviate from the rational or mathematically objective solution. Accordingly, the dynamics of human decision- making are difficult to capture using conventional, linear mathematical models. Here, we present data from a two- choice task with variable risk between sure loss and risky loss to illustrate how a simple nonlinear dynamical system can be employed to capture the dynamics of human decision-making under uncertainty (i.e., multi-stability, bifurcations). We test the feasibility of this model quantitatively and demonstrate how the model can account for up to 86% of the observed choice behavior. The implications of using dynamical models for explaining the nonlinear complexities of human decision- making are discussed, as well as the degree to which nonlinear dynamical systems theory might offer an alternative framework for understanding human decision-making processes. Decision-Making and Multi-Stability Keywords: Decision-making; Complex Systems; Dynamical Systems Modeling; Risky Choice; Multi-stability; Phase Transitions. Introduction Decision-making is part of almost everything humans do. Decisions can be commonplace or trivial but can also have lifelong consequences. Therefore, it is important to understand how individuals make decisions and how various factors play a role in decision-making processes. One such factor is uncertainty, which occurs in situations where there is limited information, ambiguous information, or unreliable information. Another factor is risk, which is different from uncertainty and can be defined as ‘probabilized’ uncertainty (Etner, Jeleva, & Tallon, 2010). Johnson and Busemeyer (2010) distinguish three major streams of development in decision theory: normative research, descriptive research, and the computational approach. While the normative approach defines what would be the optimal decision in a given situation, descriptive research describes how humans actually decide. For example, this approach has lead to the insight that individuals are sensitive to framing. When a decision is framed in terms of potential loss, the majority of participants avoid taking risk, but when the same decision is framed in terms of potential gain, the majority of Heterogeneity, multi-stability, and context-sensitivity in general, are all strong indications that decision-making under uncertainty is characterized by nonlinear dynamics. A multi-stable system can, for the same input, settle in more than one possible internal stable state. A possible consequence of multi-stability is hysteresis, which is the phenomenon whereby a system’s immediate history influences the current state of the system. Sir James Alfred Ewing first coined the term hysteresis while observing the phenomenon in magnetic materials (Ewing, 1881). Figure 1A displays hysteresis in the magnetization and demagnetization of a magnet as a result of varying strength of the magnetic force. Depending on the direction of change of the magnetic field, the change from magnetization in one direction to the opposite direction occurs at a different moment. The system has a primitive form of memory, and remains in an existing stable state longer than expected. The opposite of hysteresis, reversed hysteresis, can also occur in multi-stable systems. Rather than remaining in the existing stable state longer (as with hysteresis), the system changes to another stable state sooner. Hysteresis and reversed hysteresis are important indications of nonlinearity (Kelso, 1995). Hysteresis in behavioral dynamics has been found in body-scaled transitions like grasping of objects (Richardson, Marsh, & Baron, 2007; Lopresti-Goodman, Turvey, & Frank, 2011), speech categorization (Tuller, Case, Ding, & Kelso, 1994), perception of whether a slanted surface supports upright