Models that are robust to aberrant choice behaviour have received limited attention in dis-crete choice analysis. In this paper, we analyse two robust alternatives to the multinomialprobit (MNP) model. Both alternative models belong to the family of robit models, whosekernel error distributions are heavy-tailed t-distributions. The first model is the multino-mial robit (MNR) model in which a generic degrees of freedom parameter controls theheavy-tailedness of the kernel error distribution. The second alternative, the generalisedmultinomial robit (Gen-MNR) model, has not been studied in the literature before and ismore flexible than MNR, as it allows for alternative-specific marginal heavy-tailednessof the kernel error distribution. For both models, we devise scalable and gradient-freeBayes estimators. We compare MNP, MNR and Gen-MNR in a simulation study and acase study on transport mode choice behaviour. We find that both MNR and Gen-MNRdeliver significantly better in-sample fit and out-of-sample predictive accuracy than MNP.Gen-MNR outperforms MNR due to its more flexible kernel error distribution. Also,Gen-MNR gives more reasonable elasticity estimates than MNP and MNR, in particularregarding the demand for under-represented alternatives in a class-imbalanced dataset.