Testing for violations of the Independence from Irrelevant Alternatives

The MNL is based on a fairly restrictive IIA property that assumes that the ratios of probabilities of the alternatives remain unchanged. This is true only for the choice set (set of all the alternatives) that does not share any common characteristic or, put differently, the alternatives are not correlated.

The most famous example of the IIA violation is the red bus/blue bus paradox. Consider a situation where you are choosing between traveling by car, train, or blue bus. For the sake of simplicity, we assume that the probability of selecting each of the options is equal to 1/3. Under IIA, if we added a red bus to the choice set, the ratio of the probabilities of the remaining options would remain constant so the probabilities would now equal 1/4.

However, in reality, does the color of the bus matter that much?! Let's, for the sake of argument, assume that it does not, and in effect we are still selecting between the car...