## Test your intuition 43: Distribution According to Areas in Top Departments.

In the community of mamathetitians in a certain country there are mamathetitians in two areas: Anabra (fraction p of the mamathetitians) and Algasis (fraction 1-p of  mamathetitians.) There are ten universities with 50 faculty members in each mamathetics department and there is a clear ranking of these universities from best to worse. (So a better university pays higher salary and gives better working conditions.)  Every year 40 fresh Ph D’s are in the job market and each university  hires two of the fresh Ph. D.’s as new faculty members replacing retired faculty members.

The objective value v(x) of every mamathetitian x is a random variable uniformly distributed in the interval [0,1]The subjective value of x in the eyes of another mamathetitian y is v(x)+t if they belong to the same area and v(x)-t if they belong to different areas, where t is a small real number.

The top ranked university hires the best two fresh Ph. D.’s  according to average value in the eyes of the faculty, the second university hires the second best two students etc.

Let’s assume that to start is p=0.25, so 25% of all faculty are anabraists, and 75% are algasisits and this is the ratio in every university. Let us also assume that among the students for ever 25% are anabraists, and 75% are algasisits.

### Test your intuition (and/or, programming skills) 43: What will be the distribution of anabraists and algasisits in the departments as time advances (depending on t), what will be the stationary distribution across departments?

Further test you intuition, imagination and programming skill: How does the answer change if you change the model in various ways. Change the value of p? Make the initial distribution random across departments?  Add noise to the subjective value? Make the bias in subjective evaluation asymmetric? make different assumptions on the fresh Ph. D.s? etc. etc.

References to similar models considered in the literature are most welcome.

Disclaimer: I do not know the answers.

### Outcomes by Joshua Paik

Here are very interesting outcomes by Joshua Paik. We see a sort of Zebra-strips patterns between Algasis dominated departments and Anabra dominated departments.

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### 9 Responses to Test your intuition 43: Distribution According to Areas in Top Departments.

1. Ori says:

Well, there are a lot of parameters here, so it’s hard to see the forest.

Let’s look at just the top university. This has the advantage that it’s not influenced by the other universities. At the beginning, there’s a positive drift in the number of algasists there, which will rise roughly until it gets to the equilibrium point – until the percentage is such that the expected number of hires matches this percentage. Now, for 100% algasists we still get that the probability of hiring algasists is less then 100%, so the equilibrium is not at 100% algasists, but rather at some 0.75<A<1. Of course, we still have random fluctuations, so there's a small chance (exponentially small in the number of faculty) that there's a streak of hiring anabraists, which might cause them to become to majority, or even completely take over the department. When this happens, the value of t becomes very important – if it's small (say, less than 1/100 or so) – then there's still better chances for hiring algasists, so they will again become to majority after a short while. If t is large enough (say, 1/10), then the anabraists majority will be semi-stable (meaning it will go on for a exponentially long time). This is all due to your assumption that the new students are still 75% algasists and 25% anabraists regardless of the faculty and that their values are i.i.d.

I'm guessing that something similar also happen when looking at the entire group of universities. If you change the model slightly so that the number of students in each area is proportional to the number of faculty in this area, then the only stable states are when everyone is of the same area.

Regarding similar models in literature, I am reminded of work by Noga Alon, Michal Feldman, Yishay Mansour, Sigal Oren, Moshe Tennenholtz on "Dynamics of Evolving Social Groups" (https://arxiv.org/abs/1605.09548).

• Gil Kalai says:

Thanks for the comment, Ori!

2. Tom Hutchcroft says:

Here’s my guess, having not done any simulations: If 50 t << 1/40 then the subjective differences mostly do not affect the ranking of candidates, and we get approximately the usual distribution. Otherwise, the top few universities will have a substantial bias towards the algasisist majority. This bias will decrease sharply as we move down the table. For most of the midtable, the proportion will be very close to the actual 25/75 proportion. Finally, at the very bottom of the table there will be a bias towards the anabraist minority.

• Tom Hutchcroft says:

“Finally, at the very bottom of the table there will be a bias towards the anabraist minority.” Actually I don’t think this will happen since in the set up the number of candidates is greater than the number of places.

• Gil Kalai says:

Thanks for the comment, Tom!

3. Joshua Paik says:

I have run some experiments, and it seems that Ori was pretty spot on. My original hypothesis was that they all became Algasists, which was incorrect. I ran the code with 10 departments, each of size 52, and each “faculty member,” with a Field and an Age, with age uniformly chosen between
[0,1]. The “t-parameter” was 0.1. Each PhD student was given an “objective value,” uniformly chosen from [0,1]. I then ran a for loop 100 times representing 100 years. In each year, I iterated from the 0th department (“best”) to the 9th department (“worst”), calculating the “subjective value” for each student. After adding the top two students to the ith department, I dropped them from the PhD pool and then dropped the oldest two faculty members (I suppose their last decision before becoming happily emeritus). I then added 1 year to the age of each faculty member. After playing around with a few set.seeds, one sees a pattern that 70-80% of the departments become Algasis dominated, which I believe supports a steady state argument. I record these experiments in this google colab notebook, https://colab.research.google.com/drive/1Wo0rC0-zkmsMMNfDuSwXm6jcRTTUJUC2 , so it shouldn’t be too hard to play around further and in case my verbal description was not adequate.

• Gil Kalai says:

Great job, Joshua! I will add a table with your outcomes to the post itself.
GK: Done- see the post.

• Gil Kalai says:

Joshua, I think that in the above model even if everybody add t to the majority area Algasists will fully occupy the top departments but then will keep their ratio in other departments. I really like the zebra-strip pattern in your outcomes. There are several variations of the model where I could imagine would lead to different outcomes.

4. Gil Kalai says:

I wonder if the following somewhat similar model is familiar. you have n particles some black and some white with a certain proportion and their weights are uniform at random. So by gravity they will be arranged from heavier to lighter. But in addition assume that particle of the same color attracts each other. What will happen? Will they form strips? (This is essentially 1D model.)