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.
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.
Van Cranenburgh, S., Prato, C.G. & Chorus, C.G. “Accounting for variation in choice set size in Random Regret Minimization models” (working paper).