A Bayesian method for outlier robust estimation of multinomial choice models is presented. The method can be used for both correlated as well as uncorrelated choice alternatives and guarantees robustness towards outliers in the dependent as well as independent variables. To account for outliers in the response direction, the fat tailed multivariate Laplace distribution is used. Moreover, to account for leverage points the likelihood is adapted using a shrinkage procedure. By exploiting the scale mixture of normals representation of the multivariate Laplace distribution, an efficient Gibbs sampling algorithm is developed. A simulation study shows that estimation of the model parameters is less influenced by outliers compared to non-robust alternatives. An analysis of margarine scanner data shows how our method can be used for better pricing decisions.