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# Relative Utility Modelling

## General Formulation

To represent context dependencies in a comprehensive and systematic way, Zhang et al. (2004) defined three types of relative utility (Unit): (3.1) (3.2) (3.3)

where n (or n') i (or j), and t (or t') refer to individual, alternative, and time, respectively, is an error term, and represent the influences of attributes of alternatives i and j in the choice set, respectively.

Eqs. (3.1)-(3.3) define alternative-oriented relative utility, time-oriented relative utility and decision-maker-oriented relative utility, respectively. The three types of relative utility reflect the influence of different types of context dependencies. In Eq. (3.1), the relative utility of alternative i is defined by referring to the existence of other alternatives in choice set; in Eq. (3.2), the reference point for the relative utility is the alternative that was chosen in the past and/or will be chosen in the future; and the alternatives chosen by other persons serve as reference points (i.e. social reference group) for the relative utility in Eq. (3.3). With the above definitions, the principle of relative utility maximisation was proposed to replace the conventional random utility maximisation principle. The principle of relative utility maximisation argues that a decision-maker chooses an alternative with the highest relative utility from his/her choice set.

## Alternative-Oriented Relative Utility

This chapter mainly focuses on alternative-oriented relative utility. Note that suffix t is omitted for simplicity. (3.4)

Here, is the deterministic term of is a relative interest parameter for accommodating the idea that people may not equally evaluate different alternatives in a choice set. By using the term , observed similarities among alternatives in choice set are explicitly incorporated into the utility function.

Assuming that eni follows an identical and independent Weibull distribution, one can obtain the following relative utility-based multinomial logit (r_MNL) model, which is clearly a non-IIA choice model. (3.5)

In theory, can take on any real value. For parameter identification, one of the must be fixed during model estimation. For ease of interpreting the estimation results, one can assume that, (3.6)

When determining the relative utility of an alternative, an individual may not equally deal with comparative alternatives in the choice set. To reflect such a decision-making mechanism, Eq. (3.4) can be expanded as, (3.7) (3.8)

where is a weight parameter reflecting the influence of alternative j on the choice of i.

It is obvious that the influence of alternative j on the choice of alternative i is expressed as (see Eq. (3.7)), which are summarised in Table 3.1 for all alternatives (Zhang & Fujiwara, 2004). For example, the influence of alternative i on alternative 1 is , but the opposite is , implying that the mutual influence is not symmetric. Thus, Eq. (3.7) accommodates not only unequal but also asymmetric choice structures.

Since different individuals may show different interests in alternatives and attach different weights on alternatives when comparing with each other, these two types of parameters can be defined as a function of individual attributes and/or other factors, respectively, to reflect the influence of heterogeneity, as follows: (3.9a)  (3.9b)

where are explanatory variables with parameters and , respectively.

To estimate relative interest and weight parameters, the corresponding parameter of one reference alternative should be fixed during model estimation. If and Table 3.1: Matrix of mutual influences among alternatives.

 Alternative in choice set Target alternative 1 i I 1   :   I   :   I   are all the same across alternatives, then the r_MNL model based on Eq. (3.7) collapses into the conventional MNL model.

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