We develop a robust estimator—the hyperbolic tangent (tanh) estimator—for overdispersed multinomial regression models of count data. The tanh estimator provides accurate estimates and reliable inferences even when the specified model is not good for as much as half of the data. Seriously ill-fitted counts—outliers—are identified as part of the estimation. A Monte Carlo sampling experiment shows that the tanh estimator produces good results at practical sample sizes even when ten percent of the data are generated by a significantly different process. The experiment shows that, with contaminated data, estimation fails using four other estimators: the nonrobust maximum likelihood estimator, the additive logistic model and two SUR models. Using the tanh estimator to analyze data from Florida for the 2000 presidential election matches well-known features of the election that the other four estimators fail to capture. In an analysis of data from the 1993 Polish parliamentary election, the tanh estimator gives sharper inferences than does a previously proposed heteroskedastic SUR model.