An information-theoretic 2-stage, 2-decision rule, choice model

Recent articles in choice modeling advocate two-stage decision processes in which the decision maker uses the first stage to screen down the number of feasible alternatives, and the second stage to make a final choice. A key question in the analysis of two-stage decision processes is whether the probabilities estimated in the first stage strongly influence the final second stage probabilities, or is the first stage used only to reduce the number of alternatives? This question has not been addressed in the literature, primarily because the second stage model is generally a sinusoidal simultaneous compensatory model such as logit, probit, tobit, … using maximum likelihood estimation (MLE) which cannot directly incorporate first stage probability estimates as the dependent variable. This paper formulates a minimum discrimination information (MDI) approach to intertwine the two stages of a decision process. The MDI methodology discussed in this paper is capable of using proportions directly as input for estimation of the simultaneous compensatory models which makes it potentially useful for a variety of other applications. For a wide class of models the MDI estimates exist and the MDI estimate of logit model is unique. The performance of the MDI method is compared with the MLE in a Monte Carlo simulation and in application to a real world data set. For the real world data set analyzed, the first stage probabilities appear to strongly influence the final choice probabilities.