Today we hosted Dame Professor Celia Hoyles who gave a talk at the launch event of Cardiff’s new Mathematics Education Research Group (MERG). Amongst many interesting things the topic of the gender imbalance in Mathematics came up. In this post I’ll use a game theoretic model of evolution to describe one aspect of lack of diversity: the fallacy that is a meritocracy.
One definition of a meritocracy is: “a society governed by people selected according to merit”. In other words, the idea is that people should be selected based on their ability and their ability alone. For example, a paper should be published in a mathematics journal based purely on the content of the paper and not the gender of the authors.
Often, mathematicians are keen to say that mathematics is a meritocracy: the “purity” of the subject being clear and the only thing that matters.
One anecdotal piece of evidence showing that this isn’t the case is the fact that the late Maryam Mirzakhani was the first women to win a Fields Medal (out of 56!). It would be difficult to defend that this represents a meritocracy, unless you believed that women were naturally less able to do Mathematics than men (in which case please stop reading and go elsewhere).
So despite our best intentions why does the idea of a meritocracy not really work? (Although it is of course a lovely idea!)
I am not an expert in these things but I believe there are at least two reasons. First of all: privilege, simply by being a white male I’ve started life with a bit of a head start. Here’s a nice video about privilege:
Another reason is the well researched phenomena that with all things being equal we are more likely to be receptive/enthusiastic/positive with/to things/people who are like us. Here is an interesting piece about this: sites.psu.edu/aspsy/2015/04/17/similar-to-me-effect-in-the-workplace/.
I know I’m guilty of this myself, for example when reading some of my research students work my initial reaction is that something they’ve written is not good. However, upon reflecting a bit more and distancing myself from my own bias I usually realise that it’s not written how I would have written it. This particular bias is one type of what is often referred to as unconscious bias. Here is a nice video by the Royal Society about this:
I’m now going to represent these ideas in a well studied mathematical model of population dynamics. (Disclaimer: In the interest of keeping this post short I won’t go in to the background of the model more than I need to.)
Let us consider a normalised population of individuals of men and woman (although of course this applies to other underrepresented groups) represented by a vector \(x=(x_1, x_2)\) where \(x_1 + x_2 = 1\) so that \(x_1\) represents the proportion of men and \(x_2\) the proportion of women.
Let us assume that at any moment in time the rate of increase of a particular type (gender) can be represented by the following matrix:
With \(r > 1\): this is essential to our assumption regarding the “similar to me effect”.
The “fitness” (related to the idea of “merit”) of both types is then given by:
- For men: \(f_1 = rx_1 + x_2\)
- For women: \(f_2 = x_1 + rx_2\)
To ensure that \(x_1 + x_2 = 1\) we need to adjust the total increase rate by taking away the average fitness of our individuals (I’m skipping some details here but this is a standard model of evolutionary game theory).
Using this we have the differential equation in matrix form:
We can solve this differential equation numerically to get an idea of what is happening. I’ll use Python to do this.
First some functions to set up creating the matrix \(A\) and defining the differential equation:
>>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.integrate import odeint >>> t = np.linspace(0, 200, 200) # Obtain 200 time points >>> def create_A(r=1.05): ... A = np.array([[r, 1], [1, r]]) ... return A >>> def dx(x, t, A): ... """ ... Define the derivate of x. ... """ ... f = np.dot(A, x) ... phi = np.dot(f, x) ... return x * (f - phi)
Let us assume \(r = 1.05\) (very close to a meritocracy) and let us also assume that we start with (only) slightly more men than women: \(x = (.55, .45)\).
>>> xs = odeint(func=dx, y0=[.55, .45], t=t, args=(create_A(),)) >>> plt.plot(xs)
We see that very quickly the proportion of women in our population decreases. This is simply due to the evolutionary fact that one of the types of individual (the men) is creating an environment which is more accommodating to that type of individual (men).
Note that if we run this with a perfect initial population then we get:
>>> xs = odeint(func=dx, y0=[.5, .5], t=t, args=(create_A(),)) >>> plt.plot(xs)
So indeed, even with a slight bias (\(r=1.05\) when there is a perfect initial situation then the gender balance remains. I won’t go in to it here but this would fail to be evolutionarily stable (ie it would not be stable given a small random mutation).
Of course, this is a relatively simple model and there are far more dynamics to consider but I’m hoping to have demonstrated one of the problems with a meritocracy which is it makes sense in an ideal world. Sadly, we all have too many unconscious biases for this to hold.
That doesn’t mean all is lost. Just as is described in that Royal Society video on unconscious bias, it’s important to be aware of our biases and attempt to keep them in mind. Ensuring diversity on committees and in teaching is one approach to creating an environment in which everyone can strive.
For example, when I read my research students’ work, if my initial reaction is that it’s “not good” but that that’s because it’s actually “different to how I would have done it”. This doesn’t make me a terrible person, it’s a normal unconscious bias and because I try to be aware of them I correct myself and realise that in fact the diversity offered by the writing of my students is not only good but far better than what I would have written myself.
There are (at least) two reasons to actively strive for diversity:
- It’s the right thing to do;
- It actually ensures things are better for everyone.
Whilst we are (very) far from where we need to be in Mathematics (and a number of other subjects of course) it is something that many are trying to improve. Recognising our biases is an important requirement and this includes acknowledging the fallacy of a meritocracy.