Experimental Design
(For a cleaner version of math notations, please refer to section 2.3. docs & material, pre-registration.pdf)
1. General Environment
There are three schools: A, B, and C. The number of seats in schools A, B, C is 2k, 4k, and 6k, respectively, where k = 1 and k = 10 depending on the size of the economy. Each student has a priority at each school. Let ri(s) denote the priority of student i at school s. We assume that ri(s) ∼ U [0, 1] for each student i and school s. Each student’s priority at each school is drawn from the same distribution to capture the idea that students are ex-ante symmetric from the perspective of each school.
For students, there are six possible preference orderings over the three schools, e.g., A ≻ B ≻ C, A ≻ C ≻ B, and so on. We consider an economy where the preference orderings are uniform. That each preference ordering is equally common captures the idea that schools are of the same quality. Schools differ only in the number of seats.
There are two possible ways to implement the uniform preference orderings in an experiment: (1) exogenously assign the same number of students to each preference ordering, and (2) for each student, independently draw their preference from the uniform distribution over the six preference orderings. We will do both cases. The first one is the no aggregate uncertainty treatment, and the second one is the aggregate uncertainty treatment.
The small market consists of 12 students. Thus, in the small market × no aggregate uncertainty treatments, there are two students for each of six preference orderings, while small market × aggregate uncertainty treatments require drawing 12 students’ preferences independently with an equal chance for each preference ordering. In the case of the large market, there are 120 students. Similarly, there are 20 students for each preference ordering for no aggregate uncertainty treatments and we draw preferences 120 times independently for aggregate uncertainty treatments. Priority and preference profiles are the same between TTC and CE treatments.
For no aggregate uncertainty, we consider 10 large markets (LM1, LM2,..., LM10) and 10 small markets (SM1, SM2,...,SM10). The subjects’ priorities in the large market are generated randomly. To construct SM1, for each preference profile, take the first two students in LM1 with that preference profile, and then add them to SM1. This makes a market with 12 students. Repeat for SM2 and LM2, and so on.
For aggregate uncertainty, we again consider 10 large markets (LM1, LM2, ..., LM10) and 100 small markets (SM1, SM2, ..., SM10). The subjects’ preferences and priorities in an uncertain large market (ULM) are generated randomly. To construct USM1, take the first 12 students from ULM1 (regardless of preference profile). Repeat for USM2 and ULM2, and so on.
2. TTC Treatment
In the TTC treatment, subjects are informed of (1) the number of seats in each school, (2) the number of students (size of the economy), (3) their earnings from each school (their own preferences), (4) their priority at each school, and (5) distribution of preferences (whether there’s no aggregate uncertainty or not). Given the information, subjects submit a full preference ranking.
They make choices twice (Round 1 and Round 2). All other things remain the same between the two rounds, except for priorities. Allocations follow the traditional TTC mechanism.
3. CE Treatment
The key insight we take from Leshno and Lo (2021) is that the TTC allocation can be replicated using priority prices. In other words, each student is assigned a “budget set” of schools they can apply to based on their priorities. The TTC allocation can then be interpreted as assigning each student to their best school in their budget set. Because priorities determine the budget set, they also play a role in CE prices (or cutoffs) of schools in a decentralized economy. These cutoffs—and the resulting budget sets—can be computed directly from the distribution of preferences and priorities in a continuum economy.
We calculate the CE prices under the continuum economy where Schools A, B, and C have a measure 2/12, 4/12, and 6/12, of seats respectively, and there is a measure 2/12 of students with each preference profile. Then, we use these prices to form subjects’ budget sets.
Subjects are informed of (1) the number of seats in each school, (2) the number of students (size of the economy), (3) their earnings from each school (their own preference), (4) their own budget set, (5) odds of getting into a school they choose conditioning on all others choose their highest-earning school from their budget set, and (6) distribution of preferences (whether there’s no aggregate uncertainty or not). Given the information, subjects choose a school from their budget sets.
Subjects make choices twice (Round 1 and Round 2). All other things remain the same between the two rounds, except for priorities. If a school has an excess demand, we randomly select which students will be assigned to it. Students who are not selected will be assigned to a school with available seats.