For whatever reason, I’m on a “data is complicated kick.”
So, this story is one of many today discussing the gender gap in wages in ‘Merica. In a nutshell, President Obama pointed out “that women make, on average, only 77 cents for every dollar that a man earns.” Critics (most notably the American Enterprise Institute) immediately pointed out that “the median salary for female employees is $65,000 — nearly $9,000 less than the median for men.”
There are LOTS of angles on this thorny issue. I want to raise the specter of social choice theory as a mechanism by which we can understand why this debate goes around and around. The basic idea is that aggregation of data involves simplification, which involves assumptions. Because there are various assumptions one can make (properly driven by the goal of one’s aggregation), one can aggregate the same data and reach different conclusions/prescriptions.
To keep it really simple, consider the following toy example. Suppose that a manager currently has one employee, who happens to be a man, who makes $65,000/year, and the manager has to fill three positions, A, B, and C. Furthermore, suppose that the manager has a unique pair of equally qualified male and female applicants for each of these three positions. Finally, suppose that position A is paid $70,000/year, position B is paid $60,000/year, and position C is paid $45,000/year.
Now consider two criteria:
(1) eliminate/minimize the gender gap in terms of average wages, and
(2) minimize the difference between proportions of male and female employees.
How would the manager most faithfully fulfill criteria (1)? Well, if you hire the woman for position B and the two men for positions A and C, then the average wage of women (i.e., the woman’s wage) is $60K, and the average of the three men’s (the existing employee and the two new employees) wages is $60,000. This is clearly the minimum achievable.
How about criteria (2)? Well, obviously, given that one man is already employed, the manager should hire two women. If the manager satisfies criteria (2) with an eye toward criteria (1), then the manager will hire a man for position B and women for positions A and C.
Note that the two criteria, each of which has been and will be used as benchmarks for equality in the workplace (and elsewhere), suggest exactly and inextricably opposed prescriptions for the manager.
In other words, the manager is between a rock and a hard place: if the manager faithfully pursues one of the criteria, the manager will inherently be subject to criticism/attack based on the other.
Note that this is not “chaos”: the manager, if faithful, must hire no more than 2 of either gender: hiring three men or three women is incompatible with either of these criteria. But the fact remains—and this is a “theory meets data” point—one can easily (so easily, in fact, that one might not even realize it) impose an impossible goal on an agent if one uses what I’ll call “data reduction techniques/criteria” to evaluate the agent’s performance.
In other words: real world politics is inherently multidimensional. When we ask for simple orderings of multidimensional phenomena (however defined, and of whatever phenomena), we are in the realm of Arrow’s Impossibility Theorem.
 This argument is made in a more general way in my forthcoming book with Maggie Penn, available soon (really!) here: Social Choice and Legitimacy: The Possibilities of Impossibility.
 Here, by “average,” I mean arithmetic mean. Because this example is so small, there is no real difference between mean, median, and mode in terms of how one measures the gender gap. If these differ in practice, then the problem highlighted here is merely (and sometimes boldly) exacerbated.
 To be clear, I am setting aside the issue of “how much does a gender make if none of that gender is employed?” While technically undefined, I think $0 is the most common sense answer, and I’ll leave it at that.
 Of course, as Maggie Penn and I discuss in our aforementioned book, there are many criteria. Our argument, and that presented in this post, is actually strengthened by arbitrarily delimiting the scope of admissible criteria.