How does matching theory apply to university admissions?

Contrary to the beliefs of many politicians around the world, not all free markets can be reduced to demand and supply.  For example, the markets for university acceptance are unique in that participants must choose and be chosen; the price isn’t the key allocating mechanism. In this two-sided market, each individual and university forms a matching configuration until all valid matches of students and universities have been formed.

A cursory glance at game theory reveals the concept of a Nash Equilibrium, where an allocation is only stable if neither player is able to unilaterally deviate and achieve a higher payoff. If we apply this logic to the market for university admissions, we can say that a stable allocation is one where there is no potential pairing where a university and student would rather be matched. This means that we can have students that would rather be at certain universities as long as those universities would not rather have those students, and vice versa. Theoretically, this allocation is maximising as many preferences as possible; there are slight technicalities involving who proposes to whom, but we’ll ignore them for the sake of simplicity.

The tricky part is howwe decide to reach a final allocation. Do we have universities immediately accept students as they apply, or do we force them to tentatively hold positions? The former is what is known as Immediate Acceptance whereas the latter is known as Deferred Acceptance. It turns out that the Immediate Acceptance algorithm does notresult in the optimal outcome, as outlined above. Let’s illustrate this with an example.

Suppose an individual applies to their preferred university and is, unfortunately, rejected. Using the Immediate Acceptance algorithm, it is entirely possible that the individual’s second, third and even fourth preferences are already filled by other students who applied to those university’s as their first preference. Even if that student and one of those universities would prefer to be matched, they can’t be, as that university has already accepted another student; the welfare of students and universities is therefore not maximised. The Deferred Acceptance algorithm, on the other hand, avoids this problem and thus potentially increases the welfare of both students and universities.

Thanks to economists, use of the Immediate Acceptance algorithm is widely discredited and even banned in places such as the United Kingdom. Exceptions to the rule of perfectly competitive markets have been studied for the past few decades, yet it seems that there is still a stigma associated with government intervention. The key lesson from this example is that one must also be aware of the rules that shape how markets function, rather than assuming that they will inevitably reach some perfect outcome by themselves.