Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation
Stephane Hess and Caspar G. Chorus
Purpose – This chapter proposes a new mixture model which allows for heterogeneity in sensitivities and decision rules across decision makers and attributes.
Theory – A new mixture model is put forward in which the different latent classes make use of different decision rules, where the use of generalised random regret minimisation kernel allows for within class heterogeneity in the decision rules applied across attributes.
Findings – Our theoretical developments are supported by the findings of an empirical application using data from a typical stated choice survey.
Originality and value – Existing work has looked at heterogeneity in decision rules and sensitivities across respondents. Other work has focused on the possibility that different decision rules apply to different attributes. This chapter puts forward a model that combines these two directions of research and does so in a way that lets the optimal specification be driven by the data rather than being imposed by the analyst.
Keywords: Generalised random regret model; compromise effect; mixture model; random utility model; route choice
Recent years have witnessed a rapidly growing number of studies that aim to incorporate decision rules other than the conventional linear-additive random utility maximisation (RUM) rule in discrete choice models of travel behaviour. Motivated by the wish to increase the behavioural realism of travel demand models, as well as their empirical performance, decision rules like symmetric relative advantage maximisation (e.g. Leong & Hensher, 2014a), reference dependent utility maximisation (e.g. Stathopoulos & Hess, 2012), random regret minimisation (RRM, e.g. Chorus, 2010) and decision tree approaches (e.g. Arentze & Timmermans, 2007) have been proposed and applied in recent travel behaviour studies. Earlier work tended to focus on comparing the properties and empirical performance of models based on these alternative decision rules with those of models based on the conventional linear-additive utility maximisation rule. While such comparisons have generated interesting insights, more recent work has adopted a different perspective and puts more emphasis on capturing potential heterogeneity in applied decision rules.
Two main strands of this more recent research can be distinguished: first, it is increasingly being acknowledged that it is unrealistic to assume that every individual applies the same decision rule; rather it makes more sense to allow for different groups of individuals to use different decision rules. This conceptual idea can be operationalised using mixture models of the type put forward by Hess, Stathopoulos, and Daly (2012) – see Hess and Stathopoulos (2013) and Boeri, Scarpa, and Chorus (2014) for other recent examples. These models incorporate different latent classes, each with their own decision rule (and set of taste parameters). Estimation results show large improvements in fit compared to models that assume one and the same decision rule for the entire population, where the work by Hess and Stathopoulos (2013) also provides further insights into who might be making choices in what way.
A second and related body of papers questions the assumption that every attribute is processed using the same decision rule. The resulting (often called hybrid) model structures allow for different attributes to be processed using different decision rules; an example being the hybrid utility-regret model put forward by Chorus, Rose, and Hensher (2013) and employed by, amongst others, Leong and Hensher (2014b); this model allows for some attributes to be processed in a regret minimisation fashion while others are processed in a utility maximisation fashion.
An obvious next step would be to combine these two approaches; that is, allowing for different individuals to use different decision rules, for different attributes. In principle, a combined hybrid-discrete mixture model would be able to accommodate for both types of decision rule heterogeneity simultaneously. However, there is an important caveat that so far has hampered progress in this direction: when the number of attributes is nontrivial, a large number of latent classes is needed to capture all possible combinations of decision rules. For example, hybrid utility maximisation-regret minimisation models for a four-attribute choice context would already result in 16 latent classes. Resulting mixture models are generally not well behaved due to the large number of classes, precluding successful estimation on empirical data.
This chapter builds on a recent advance in regret minimisation modelling to circumvent this combinatorial explosion. The recently proposed generalised RRM model (Chorus, 2014) estimates a regret-weight for each attribute; if equal to 1, conventional regret minimisation behaviour – that is as in Chorus (2010) – is obtained. If equal to 0, conventional utility maximisation behaviour is obtained. Values between 0 and 1 imply regret minimisation behaviour, but with a smaller degree of nonlinearity in the regret function than is the case in conventional regret minimisation models. Since the regret-weight can be estimated on empirical data, this provides an opportunity to infer from the data, for each attribute, if it is processed using a utility maximisation or regret minimisation rule. That is, rather than having to estimate all possible hybrid utility-regret combinations to find the optimal constellation, the generalised RRM model allows one to directly infer the best fitting decision rule for every attribute.
When estimating hybrid mixture models of the type discussed above, the generalised random regret model provides a clear conceptual and operational advantage over previously used hybrid utility-regret models: conceptually, it is much more elegant to estimate attribute-specific decision rules from the data as opposed to these rules being imposed by the researcher. Operationally, the generalised approach limits the number of latent classes, as one no longer needs to allow for every possible combination of attribute-specific decision rules. This results in better behaved and more manageable models. Nevertheless, the model still clearly allows for the possibility of different classes making use of a different combination of RUM/RRM parameters. This chapter is the first to combine the generalised RRM model and the discrete mixture paradigm; as such, it is the first attempt that we know of, to simultaneously allow for heterogeneity in decision rules across individuals and attributes in a discrete choice model of traveller behaviour.
The remainder of this chapter is structured as follows. Section 2.2 presents the model structure put forward in our chapter. Section 2.3 presents empirical analysis based on a stated route choice dataset. Conclusions and directions for further research are put forward in Section 2.4.