Independence of Irrelevant Possibilities and The 2016 US Presidential Election

The multinomial logit model depends on making the assumption of independence of irrelevant alternatives (IIA).  While there are different formulations, the general idea behind this principle is that when comparing two alternatives X and Y with a certain probability of preference in relationship to each other, the addition of an additional alternative Z shouldn't change the original relationship between X and Y.  That is, if X is favorable over Y, then adding Z should not change the fact that X is favorable over Y.  Z is supposed to be an irrelevant alternative to the consideration of X vs. Y.  See the Wikipedia article for more details (https://en.wikipedia.org/wiki/Independence_of_irrelevant_alternatives).

The IIA suffers from several criticisms (e.g., Red Bus - Blue Bus problem), which usually stem from having "alternatives" that are too similar to be considered alternatives.  What most interested me in reading about this principle is its applicability (or inapplicability) to voting.  In reading about various voting strategies, the problem seems to arise when one has 3 (or more) candidates and no candidate is preferred as a majority to all other candidates.  That is, some percentage less than 50% prefers A to B, some percentage less than 50% prefers B to C, and some percentage less than 50% prefers B to C.  This means that, for any candidate elected, there could be a majority that would have preferred a different candidate to be elected (again, see the Wikipedia article for more details).

When reading about this, what immediately came to mind was the 2016 US Presidential election.  It seems that, generally speaking, everyone was dissatisfied with the two main candidates, Trump and Clinton, and that a third candidate (Johnson), while preferred by very few outright, was seen as a legitimate alternative by many as opposed to the possibility of either Trump or Clinton being elected.  The final popular vote percentages were as follows (from http://www.presidency.ucsb.edu/showelection.php?year=2016):

• Clinton: 48.2%
• Trump: 46.1%
• Johnson: 3.3%

Suppose, for the sake of argument, that:

• 48.2% preferred Clinton to Johnson, and Johnson to Trump
• 46.1% preferred Trump to Johnson, and Johnson to Clinton
• 3.3% preferred Johnson to Trump, and Trump to Clinton

Then we have the interesting result that:

• 51.5% prefer Johnson to Trump, who did in fact win.

Or suppose we change the last line and make it:

• 3.3% preferred Johnson to Clinton, and Clinton to Trump

Then we get that:

• 49.4% preferred Johnson to Clinton, which is higher than the popular vote for either Clinton or Trump.

If the above is close to people's actual preferences (that is, people preferred Johnson to the other party's candidate), then one can make the argument that Johnson "should" have been elected as the compromise candidate that most, or even a majority of, people would prefer if their primary candidate wasn't chosen.  Perhaps this isn't actually how people would have ranked the candidates, but it is certainly a reasonable possibility.

To go back to the multinomial model, what this means is that in many cases, perhaps even the most important cases where we want to predict in a multinomial way (i.e., elections), the assumption of IIA which is necessary will almost certainly be violated.  Often, our most important decisions involve choices that are too similar to each other, containing overlapping positives and negatives that are not clearly weighed against each other or decisive in leading to a choice.  We have many conditional choices that in fact do depend on the presence or absence of alternatives that we would not choose.  While there are other models that one can choose that do not rely on IIA (which have their own challenges), this problem certainly presents a challenge to those doing multinomial prediction.