Ordinal scale responses capture qualitative user feedback which can be used to model individual choice preference, or are employed in traffic accident analysis to evaluate accident severity. We present a new choice model for ordinal scale responses in choice tasks that combines a Multinomial Logit model with a Poisson probability mass function. The Poisson distribution, which is suitable for modelling the occurrence of the number of events in a fixed time frame, independent of previous events, can be adapted into the unobserved error distribution of a standard MNL model to capture the natural ordering of the choices by imposing a unimodal constraint on the a posteriori choice probability. In this paper we describe the theoretical framework and the specification of the Unimodal Logit model. We apply our model to evaluate accident severity concerning road collisions. Our results are compared against the traditional ordered logit model and the MNL model.