Accounting for variation in choice set size in RRM models
In many choice situations the choice set size, i.e. the number of
alternatives which are available to the decision-makers, varies
across choice observations. In RRM models such variation in choice
set size is consequential for the model predictions. Since the
overall regret level of an alternative equals the sum of all
pairwise regrets arising from bilateral comparisons with competing
alternatives, overall regret levels rise with an increase of the
choice set size. The rise of regret levels with choice set size
predicted implies that regret differences between alternatives
also tend to grow with the increase of the choice set size. This
in turn means that when there is variation in choice set sizes in
the data, RRM models predict larger differences in regret levels
and therefore more deterministic choice behaviour in observations
where the choice set is relatively large, and smaller differences
in regret levels and hence more random choice behaviour in
observations where the choice set is relatively small. It goes
without saying that such variation in choice consistency is
behaviourally unrealistic and deteriorates the model performance
of RRM models in the context of data sets with varying choice set
sizes.
To account for variation in the choice set size when estimating RRM models a simple, but effective correction factor can be used, see the equation below. The overall regret level is scaled by the choice set size of the observation, denoted Jn. Note that the numerator in the correction factor is set equal to 2. Setting the numerator to 2 results in a natural parametrization as under this setting RRM models and the linear-additive RUM model yield exactly the same parameters in the context of binary choice data.
To account for variation in the choice set size when estimating RRM models a simple, but effective correction factor can be used, see the equation below. The overall regret level is scaled by the choice set size of the observation, denoted Jn. Note that the numerator in the correction factor is set equal to 2. Setting the numerator to 2 results in a natural parametrization as under this setting RRM models and the linear-additive RUM model yield exactly the same parameters in the context of binary choice data.
$$ \widetilde{R}_{in} = \frac{2}{J_n}R_{in} $$
However it is important to realize that while for some types of RRM
models the choice of the numerator is inconsequential for the
behaviour imposed by the model, for other types it is. More
specifically, for the μRRM and P-RRM model the choice of the
numerator is inconsequential. After all, in the μRRM model the scale
parameter μ is estimated. Since the numerator is perfectly
confounded with μ, setting the numerator to a large value will
merely result in estimating a small scale parameter μ, and vice
versa. Therefore, in a μRRM model the imposed behaviour is not
affected by the choice of the numerator, implying that the numerator
can freely be chosen by the choice modeller. Likewise, the choice of
the numerator is inconsequential for the behaviour imposed by the
P-RRM model. Since the attribute level regret function of the P-RRM
model is scale-invariant, it always imposes the same degree of
regret minimizing behaviour, irrespective of the choice of the
numerator. In contrast, the size of the numerator is consequential
for the classical RRM model and for the G-RRM model. Since the
attribute level regret functions of these two RRM models are not
scale-invariant, a different choice of the numerator leads to
different degrees of regret minimizing behaviour imposed by the
model, and hence a different empirical performance.