I have three product categories, $A,B,C$. Each category has two products, $0,1$. I provide a number of different kinds of choice situations, 1) the test subject is presented a single category and made to choose a product, 2) the test subject is presented with two categories and made to choose a product from two categories, and 3) the the test subject is presented with all three categories and made to choose a product from each. I believe that product choices depend on a number of measured covariates of the individual products, the product categories presented, and the choice in the other category (if such a choice is possible).
For example, let's say that we had a product category of vinegar, with two brands. The first brand is an expensive, balsamic vinegar. The second brand is an inexpensive, store brand, apple vinegar. Now, let's say we have two other product categories: salad greens and kitchen gloves, each containing an expensive, high quality brand and a cheap, generic brand. Even if a consumer chooses the expensive vinegar when asked to choose only from the vinegar category or from the vinegar and salad category, we might still expect that he would select the inexpensive vinegar if asked to choose products from the vinegar and kitchen glove categories. We might also expect that a person who chose the inexpensive vinegar, when asked to choose from the vinegar and salad green categories, will also choose the inexpensive salad greens.
This situation is similar to the "shopping basket" problems reviewed by P.B. Seetharaman, et. al. in "Models of Multi-Category Choice Behavior". However, the models I have seen consider the incidence of a product category as a function of the consumer, often as a stage model.
How would we estimate the coefficients of the measured covariates in the case when the chooser does not choose the categories they must select from.