Cornell Sorority Rush as a Matching Mechanism (with Game Theory)
https://stanford.edu/~alroth/sorority.html
An interesting academic paper by Susan Mongell and Alvin E. Roth considers sorority “rush,” or the process by which american college students join sororities, as a two-sided matching mechanism. The authors attempt to study the mechanics of the rush process because it is often unstable and/or too competitive. The conclusion of the paper was that “analysis of the rules of the match, and of preference lists from twenty-one matches, shows unstable matching procedure that gives agents incentives to behave strategically, how the agents act on these incentives, and how the resulting strategic behavior has contributed to the longevity of the matching system, and to the stability of the resulting matches.”
Although the study examined data from 21 rushes on 4 campuses, I will be considering the rush process at only Cornell due to personal experience and knowledge. I will also be creating my own criteria and conditionals for the matching algorithm, as I do not know exactly how it works (Cornell uses an app called “PNM Companion”). I plan to examine the rush process as a matching mechanism, and examine how there are incentives to use Game Theory and behave strategically in the process.
The mechanics of rush change from campus to campus and the rules are determined by eachs Campus Panhellenic Conference (CPC), but the National Panhellenic Conference (NPC) has a general format for the process that can be summarized as a matching process determined by both the preferences of people looking to join a house (Potential New Members, or PNMs) and the sorority houses.
At Cornell, the rush process is broken up into four rounds, with the two-sided matching algorithm (one side being the PNM, and the other side being the sorority house) applied after each round. The rounds are as follows: on the first and second day of “rush week,” PNMs visit all 13 sororities at Cornell. After that, each PNM ranks the top 9 houses they would like to consider “rushing,” and then chooses a bottom four, ranked 2-5. Additionally, each sorority house ranks PNMs to decide which they would like to “re-invite” for the subsequent round. According to the NPC handbook, “Panhellenic strongly urges each sorority to re-invite… only those rushees they are seriously considering for membership. This will enable both the rushee and the sororities to know `how they stand’ early in the formal rush period.” Every house decides how they would like to rank and invite back PNMs differently (some out of 5, some out of 10, etc) and to my knowledge and research there is no uniform method of doing this, so I am going to make the assumption that sorority houses rank out of 5, with 1 being the highest, and 5 the lowest. So, after Round 1 the rankings of a PNM at Cornell may look like this:
| Houses | PNM’s Rank of House (PNMR) | House’s Rank of PNM (HR) |
| AAA | 1 | 1 |
| BBB | 1 | 1 |
| CCC | 1 | 2 |
| DDD | 1 | 5 |
| EEE | 1 | 1 |
| FFF | 1 | 1 |
| GGG | 1 | 3 |
| HHH | 1 | 1 |
| III | 1 | 5 |
| JJJ | 2 | 1 |
| OOO | 3 | 1 |
| LLL | 4 | 1 |
| MMM | 5 | 2 |
After both the PNMs and the sororities have ranked their choices, the matching algorithm decides which houses the PNM will continue rushing in the second round. In this case, we do not have a bipartite graph and there are no perfect matches because there are only 13 sorority houses and over 800 PNMs: each house invites back many more than one PNM. However, a variation of a “perfect match” might be that every house that the PNM gave a “1” also ranked her highly and wanted to invite her back. Therefore in our example, the PNM would return to houses AAA-III for Round 2.
However, that is rarely the case and many PNMs either get “cut” from their top choices and must return to houses they did not like. Many also attend far fewer than the maximum number of houses they are able to each round based off the matching algorithm.
Since I am not privy to the exact mechanics of the algorithm, I will make the assumption that if PNMR + HR <= 5, the PNM will return to that house for the second round. The matching algorithm will also work in the PNM’s favor this round: the algorithm will fill up the 9 spots in order of the PNM’s preference, so if more than 9 houses have a value <= 5 it will weigh the HR less than PNMR. So in our example, this is the matching outcome for the PNM’s second round:
| Houses | PNM’s Rank of House (PNMR) | House’s Rank of PNM (HR) | PNMR + HR | Is PNM returning for Round 2? |
| AAA | 1 | 1 | 2 | YES |
| BBB | 1 | 1 | 2 | YES |
| CCC | 1 | 2 | 3 | YES |
| DDD | 1 | 5 | 6 | NO |
| EEE | 1 | 1 | 1 | YES |
| FFF | 1 | 1 | 2 | YES |
| GGG | 1 | 3 | 4 | YES |
| HHH | 1 | 1 | 2 | YES |
| III | 1 | 5 | 6 | NO |
| JJJ | 2 | 1 | 4 | YES |
| OOO | 3 | 1 | 4 | YES |
| LLL | 4 | 1 | 5 | NO |
| MMM | 5 | 2 | 7 | NO |
In this case, our PNM was fairly successful. Only two of her top houses did not invite her back (DDD and III) and she did not have to return to her bottom two houses (LLL and MMM). It should be noted that she did not have to return to LLL even though PNMR + HR = 5 because she already had 9 houses that met the matching criteria. However, since it was not a “perfect match” she did have to return to two houses she gave a lower rank to, JJJ and KKK.
In the second round of rush, the PNM now ranks her top 6 choices “1” and her bottom three 2-4. Because this round is more competitive, let’s up the ante by changing our matching criteria: now the PNM will only continue rushing those houses that meet the criteria PNMR + HR <= 4. However, if more that 6 houses do meet that criteria the matching algorithm will again work in the PNM’s favor.
| Houses | PNM’s Rank of House (PNMR) | House’s Rank of PNM (HR) | PNMR + HR | Is PNM returning for Round 3? |
| AAA | 1 | 3 | 4 | YES |
| BBB | 1 | 2 | 3 | YES |
| CCC | 1 | 1 | 2 | YES |
| EEE | 1 | 5 | 6 | NO |
| FFF | 1 | 5 | 6 | NO |
| GGG | 1 | 4 | 5 | NO |
| HHH | 2 | 1 | 3 | YES |
| JJJ | 4 | 1 | 5 | NO |
| OOO | 3 | 1 | 4 | YES |
In this round, only 5 houses had PNMR + HR <= 4. Therefore our PNM only returned to 5 houses in Round 3, even though the maximum number is 6. As I said before, this is very common. Unfortunately for her, only three of her top choices invited her back.
Notice how now we can also consider the strategy in ranking. The study concluded that there are incentives to behave strategically. Based off intuition and personal experience, I hypothesize that this will be the case in my example as well. Say our PNM likes house HHH, but doesn’t have enough space to give it a “1.” If she is confident that HHH really liked her and would rank her highly, she could use Game Theory to rank it lower. In the example above, the PNM’s strategy worked and she was able to return for Round 3 despite giving it a lower rank. Game Theory can also be used from the house’s point of view. For example, if house EEE really did not want to invite back our PNM but suspected that the PNM would give EEE a “1,” EEE can rank our PNM a “5” in order to ensure that she will not return.
To prove this hypothesis, consider this Game Theory scenario. I will assume that both the PNM’s rankings and the house’s ranking are representative of their true evaluations; that is, neither are trying to “beat the system” and use strategy. Therefore I will make the assumption that a PNM will have a “win” (1) if she attends a house she ranked a “1,” and a “loss” (0) if she attends a house ranked below 1. I will also assume that a house will have a “win” (1) if it ranked a PNM either 1 or 2 and she returns, and a “loss” (0) if not. For a PNM to return to a house in the next round, PNMR + HR <= 4.
| 1 | 2 | 3 | 4 | 5 | |
| 1 | (1,1) | (1,1) | (1,0) | (0,1) | (0,1) |
| 2 | (0,1) | (0,0) | (1,1) | (1,1) | (1,1) |
| 3 | (0,1) | (1,0) | (1,1) | (1,1) | (1,1) |
| 4 | (1, 0) | (1,0) | (1,1) | (1,1) | (1,1) |
Rows: PNM
Columns: Sorority House
Outcome: (PNM, SH)
1 = win, 0 = loss
We can see from this chart that in Round 3, it is in the sorority houses’ best interest (aka its Dominant Strategy) to give every PNM they do not like a 4 or 5 in this round: either ranking would ensure the PNM is unable to return even if the PNM ranked them “1.” However, sadly the PNM does not have a dominant strategy in Round 3, and therefore there is no Nash Equilibrium for this matching algorithm.
Because there is no Nash Equilibrium, it is in each party’s best interest to use strategy to ensure they will have the best outcomes – especially in Round 4, where a PNM must choose only her top 2 houses. Houses do not want to give girls they know will not rank them highly high ranks, because then the PNM may not return and the house would be losing girls that would like to return. The opposite case is also true: if a PNM highly ranks a house that does not want her, she is losing the opportunity to be matched with some of her second, or maybe even third or fourth, choice houses. Let’s see an example where this occurs and a PNM is sadly left with no houses in Round 4. The criteria for this round is still PNMR + HR <= 4 for PNM to return for Round 4:
| Houses | PNM’s Rank of House (PNMR) | House’s Rank of PNM (HR) | PNMR + HR | Is PNM returning for Round 4? |
| AAA | 1 | 4 | 5 | NO |
| BBB | 1 | 4 | 5 | NO |
| CCC | 2 | 3 | 5 | NO |
| HHH | 3 | 2 | 5 | NO |
| OOO | 4 | 2 | 6 | NO |
In this case, the PNM decided to rank her top choices AAA and BBB. However, she didn’t know that her ranks there were 3 and 2, respectively. For this round AAA and BBB decided to rank her lower, and now she will be unable to return for Round 4. Furthermore, the houses she chose to give low ratings to also decided to rank her slightly lower (perhaps because they thought she wasn’t interested?). Nevertheless, our PNM is left with no options for final round!
However, perhaps if our PNM strategized with Game Theory and decided to rank houses that she suspected she was highly ranked in higher, she now has two good options, House CCC and House HHH.
| Houses | PNM’s Rank of House (PNMR) | House’s Rank of PNM (HR) | PNMR + HR | Is PNM returning for Round 4? |
| AAA | 2 | 4 | 6 | NO |
| BBB | 3 | 4 | 7 | NO |
| CCC | 1 | 2 | 3 | YES |
| HHH | 1 | 2 | 3 | YES |
| OOO | 4 | 2 | 6 | NO |
In Round 4, the final round, a PNM “prefs” her remaining two houses by ranking them #1 and #2. Hopefully all PNMs end up in the houses they “pref,” although unfortunately this isn’t always the case. I hope it is obvious from my analysis of the matching mechanisms as to why!
In the end, because I made up the matching algorithm I cannot claim that there is a correct way to rush at Cornell. However, analyzing the process through matching mechanisms and game theory is incredibly interesting to think about!