Flexible Utility Function Approximation via Cubic Bezier Splines
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EXIBLE UTILITY FUNCTION APPROXIMATION VIA CUBIC BEZIER SPLINES
Sangil Lee , Chris M. Glaze, Eric T. Bradlow and Joseph W. Kable UNIVERSITY OF PENNSYLVANIA
In intertemporal and risky choice decisions, parametric utility models are widely used for predicting choice and measuring individuals’ impulsivity and risk aversion. However, parametric utility models cannot describe data deviating from their assumed functional form. We propose a novel method using cubic Bezier splines (CBS) to flexibly model smooth and monotonic utility functions that can be fit to any dataset. CBS shows higher descriptive and predictive accuracy over extant parametric models and can identify common yet novel patterns of behavior that are inconsistent with extant parametric models. Furthermore, CBS provides measures of impulsivity and risk aversion that do not depend on parametric model assumptions. Key words: flexible modeling, heterogeneity, intertemporal choice, risky choice, generalized utility functions.
1. Introduction Intertemporal choices (ITCs) and risky choices (RCs) are heavily studied across many disciplines. ITCs are decisions regarding outcomes that occur at different times: for example, deciding between spending money now versus saving and investing that money for later, smoking now versus having better health later, or whether to pay an additional price for expedited shipping in order to receive a package earlier. RCs are decisions made regarding outcomes that occur probabilistically: for example, buying lottery tickets, investing in stock markets, or gambling. ITCs and RCs are studied both in basic and applied research. In basic research, researchers are interested in how people make ITCs or RCs and have generated many different proposals for the cognitive processes that underlie these choices. In applied research, researchers are often interested in how individual differences in ITC and RC relate to real-world behaviors such as pathological gambling, smoking, susceptibility to mental illness, drug and alcohol abuse, education level and financial status (Alessi and Petry 2003; Anderson and Mellor 2008; Brañas-Garza et al. 2007; Kirby et al. 1999; Krain et al. 2008; Lejuez et al. 2003, 2005; Lempert et al. 2019; Schepis et al. 2011; Shamosh and Gray 2008). ITC and RC data are usually modeled using one of three ways: parametric, structured nonparametric, or fully non-parametric approaches (Fig. 1). The most popular approach uses parametric utility models to describe choice (e.g., Table 1). Its popularity is driven by two factors. First, parametric utility models can distill complex patterns of behavior into one or two interpretable parameters. For example, the discount rate parameter in ITC models represents the rate at which the value of future options declines with time delay (parameter k in Table 1); the risk-aversion parameter in RC models (often substituted by the value function curvature parameter: parameter α in Table 1) captures the deviation of utilities from risk-neutral expected value. These parameters are especially
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