Discussion of modeling assumptions

Independence from irrelevant alternatives (IID)

The multinomial logit model (MNL) predicts that the ratio between two options does not depend on other options:

$$\frac{p_i}{p_j} = \frac{e^{\mu \, V_i}}{e^{\mu \, V_j}} \ .$$ (29.27)

There are many cases where this assumption is too strong. The maybe most famous case is the ``red bus, blue bus'' example. Assume that a traveler has the choice between taking the car, taking a blue bus, and taking a red bus. Assume that the two buses have exactly the same service characteristics; for example, assume that the traveler is the only passenger. Further assume that the probabilities to select the car, the blue bus, and the red bus are $50\%$, $25\%$, and $25\%$, respectively, corresponding to the ratios $2:1:1$. In consequence, the model predicts that the traveler will take her/his car with probability $1/2$.

Now assume that the blue bus is taken out of service. The model now predicts that the ratio between car and red bus will be $2:1$, meaning that the traveler will now take her/his car with probability $2/3$. This is rather implausible since one would assume that the availability of several colors for the bus will not affect the mode choice behavior significantly.

The reason for this behavior can be traced back to the assumption that the $\eps_i$ are all statistically independent from each other; this assumption is used when the statistical properties of $\max_j[\Delta V_{1j} + \eps_j]$ and of $\eps_* - \eps_1$ are derived. If they are not statistically independent, then other (usually more complicated) formulations result.

[[the above a little different??]]