Best-Worst Scaling in analytical closed-form solution

Best-Worst Scaling (BWS), sometimes also called Maximum Difference (MaxDiff), is a discrete choice modeling method widely used for finding utilities and choice probabilities among multiple alternatives. It can be seen as an extension of the paired comparison techniques for the simultaneous presentation of several items together to respondents. A respondent identifies the best and the worst ones and estimation of utilities is performed using a multinomial-logit (MNL) model in numerical nonlinear estimations. The main contribution of this paper consists in finding an analytical closed-form solution producing an approximation of the results for utilities and choice probabilities that are obtained using MNL models. The analytical formulae permit the inference of the characteristics of the model׳s quality, including standard errors of the utilities and choice probabilities, the residual deviance and pseudo-R2. This approach enriches the BWS methods and is useful for theoretical descriptions and practical applications.