Exact MCMC for Choices from Menus -- Measuring Substitution and Complementarity among Menu Items

Choice environments in practice are often more complicated than the well studied case of choice between perfect substitutes. Consumers choosing from menus or configuring products face choice sets that consist of substitutes, complements and independent items, and the utility maximizing choice corresponds to a particular item combination out of a potentially huge number of possible combinations. This reality is mirrored in menu-based choice experiments. The inferential challenge posed by data from such choices is in the calibration of utility functions that accommodate a mix of substitutes, complements, and independent items. We develop a model that not only accounts for heterogeneity in preferences but also in what consumers perceive to be substitutes and complements and show how to perform Bayesian inference for this model based on the exact likelihood, despite its practically intractable normalizing constant. We characterize the model from first principles and show how it structurally improves on the multivariate probit model (Liechty et al., 2001) and on models that include cross-price effects in the utility function (Orme, 2010). We find empirical support for our model in a menu-based discrete choice experiment investigating demand for game consoles and accessories. Finally, we illustrate substantial implications from modeling substitution and complementarity for optimal pricing