International Choice Modelling Conference, International Choice Modelling Conference 2017

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Estimating the reference frame: a smooth twice-differentiable utility function for non-compensatory loss-averse decision-making
Francisco J. Bahamonde-Birke

Last modified: 28 March 2017


Since the introduction of prospect theory in 1979 (Kahneman and Tversky, 1979), it has gained numerous supporters, while concepts such as loss-aversion and reference dependence are now common in the behavioral economics literature (Barberis, 2012; Karle et al., 2015). Moreover, it laid the theoretical foundation for the development of non-compensatory models challenging the classic economic assumptions about utility and its impact is felt in fields from marketing (Hardie et al., 1993; Ho et al., 2006) to labor economics (Dunn, 1996; Fehr et al., 2008), as well as medicine (Rizzo and Zeckhauser, 2003; Sokol-Hessner et al., 2012), safety (Flügel et al., 2015) and transportation (de Borger and Fosgerau, 2008; Dixit et al., 2015), among many others.

Even though prospect theory was originally developed for addressing choices under risk, it is straightforward to extend its principles to any kind of choice situation involving expected utilities. Basically, prospect theory sustains the existence of reference frames and that the gains and losses relative to these reference points are valued differently by decision makers. Thus, in order to correctly assess the behavior, it is necessary to know the referential frame.

The majority of the literature contributions dealing with loss aversion assumes that the referential points are known a priori (e.g. de Borger and Fosgerau, 2008; Flügel et al., 2015), which allows for clearly differentiating gains from losses and using different functional forms to include them in the expected utility functions. Normally, the reference points are assumed to be equal to the status quo. At first glance, it appears to be a sound assumption, when it is possible to identify the current conditions (for instance, when dealing with repeated choices). Nevertheless, as Kőszegi and Rabin (2006) correctly point out, references do not depend exclusively on the status quo, but also on the individuals’ previous expectations. Moreover, even if it were possible for the modeler to identify for certain these previous expectations (which is not), the model would still exhibit shortcomings, as empirical evidence indicates that reference frames are also affected by the choice-sets offered to the individuals (Zeelenberg and Pieters, 2007), which is clearly illustrated by the well-known decoy-effect (Fukushi and Guevara, 2015).

A possible way to address the aforementioned problems would be for the modeler to estimate directly the reference points as a part of the decision model. This way, the reference would appear as a parameter of the model, representing a change or an inflection point in the utility provided by a given attribute of the decision. Nevertheless, under the usual assumptions for the utility functions - see Maggi (2004) for a good overview on S-shaped utility functions - this approach is not feasible, as they are defined piecewise and are not twice differentiable around zero, which is a critical point for estimating the reference frames (nevertheless Kőszegi and Rabin, 2006,2007 show that it is possible to derive a functional model, although it still exhibits discontinuities and a degree of complexity that made it impractical for most discrete choice modeling applications)

This paper introduces an S-shaped utility function that is continuous and twice-differentiable around zero, while still satisfying the main properties of loss-averse utility functions and microeconomic theory. This paper presents an extensive analysis of its properties, such as non-satiation, decreasing marginal utilities, axial asymmetry (which can be calibrated), etc. Such a representation offers multiple possibilities when dealing with loss-averse decision-making as it not only allows estimating the reference frames, but also how frames are affected by the individuals’ characteristics and/or the offered choice-sets. The function exhibits a simple structure, which allows for an easy implementation in discrete choice models. Finally, the approach is tested relying on a study case. It shows that in the context of semi-compensatory lost-averted decision-making reference frames may diverge from the status quo. In fact, the study case shows that the reference point for travel expenses was 8.42€ below the status quo and, thus, even paying the current fare would be perceived as a loss. Previous experiences as well as personal appreciations provide plausible explanations for the phenomenon. Along this line, this example shows that simply considering the reference frame to be equal to the status quo may lead to biased results.

Keywords: loss-aversion, utility function, discrete choice modeling, non-compensatory models



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