"How can applicants’ and colleges’ preferences be elicited and reconciled in assigning applicants to university places?" (Economics 1 Q1 All Souls Entrance Exam)
Every year Oxford takes in thousands of graduate & undergraduate students each of which must be assigned to one of the 38 colleges. Typical of Oxford the method they use to do this assignment gives the appearance of giving the power to the student but in reality is extremely biased in favour of the colleges.
Assigning applicants to colleges is a typical "stable marriage" problem which has been well studied in computer science and is one of the standard examples in undergraduate algorithms teaching. The stable marriage problem seeks to pair up a set of men who have a list of preferences for each of the available women with a set of women who have a list of preferences for each of the men such that foreach pairing there is no other pair where switching the coupling would make all parties happier (all would be matched with an individual higher on their preference list).
There is a known algorithmic solution to this problem by iteratively having one party (the man) propose to the highest person on their list, who hasn't declined, and the other party (the woman) provisionally accept the offer of their most preferred man and decline all others. This is repeated until everyone is paired up (or only men/only women remain if the numbers are uneven). It can be mathematically demonstrated that the final pairing is stable.
However, there can be multiple stable marriage solutions to any given set of individual preferences and each one is not necessarily optimal from the perspective of the individuals involved. It has been shown mathematically that the side (the men) which makes the proposals will obtain their most preferred match (bride) for which a stable marriage solution is possible while the receivers of the offers end up settling for their least preferred match for which a stable solution exists.
The algorithm is obviously extendable to the case when each 'man' is looking for a certain number of brides which is identical to the problem of a college seeking to admit a particular number of students.
Oxford, by letting incoming students apply to their most preferred college, appears to be optimizing the outcome in the students' favour. However, due to the inefficiency of iterative cycles of proposal and acceptance/rejection when each cycle demand e-mailing and receiving a reply from the student, after the first round the rejected students are simply dumped into a pool which the colleges fight over on a first-come first-serve basis. Since this process is independent of the students' preferences and completely opaque to most students any student who has been rejected from their first choice college is put at a severe disadvantage from then on.
Since student choices are not independent (most students would like to go to the top ranked or richest colleges) subsequent iterations are likely to have offers from less and less preferred colleges. Thus it is in the student's best interest to accept the first offer they get after being dumped in the pool as long as it isn't from the student's least preferred college. Since offers are not sent out simultaneously from all colleges (only one at a time for the student to accept or reject) a student is likely to accept an offer from a less preferred college without knowing a more preferred college would also have offered. Such a system is not guaranteed to produce a stable matching since students are not dumped into the pool simultaneously, colleges themselves must gamble on whether a more preferred student will appear later versus the risk a good candidate will accept an offer from another college first, just as the student must gamble on whether they will receive an offer from a more preferred college if they reject the current offer.
In addition, the pool system produces unnecessary stress among college staff involved in admissions and delays the assignment of a college from the initial offer of acceptance from the university. This can create difficulties for international students who require visas to study in the UK, the acquisition of which can take many months and cannot be done prior to full acceptance at the university & college. Thus it is unsurprising that every year several international students miss some of their course due to late visas.
A much better system would solicit the complete list of college preferences from the student at the time of application not just the top one, which can then be made in order to each of the colleges following the algorithm outlined above (provisional offers can be done be done behind the scenes). This would ensure the student will be assigned to their most preferred college possible, which Oxford claims it desires to achieve. In addition, since it removes the need for both colleges and students to gamble on whether they will receive a better option stress on all sides is reduce. Since it can be done entirely behind the scenes and potentially completely computationally if colleges also submit preference lists, it could be done faster reducing the risk of visa-delay problems. Finally, the result is guaranteed to be a stable matching unlike the current system thus removing all impetus to have students migrate to other colleges.
