## Abstract

We investigate the relative merits of the Boston and Serial Dictatorship mechanisms when the timing of students’ preference submission over schools varies within the structure of the mechanism. Despite the well-documented disadvantages of the Boston mechanism Abdulkadiroglu and Sonmez (American Economic Review 93:729–747 2003), we hypothesize that a Boston mechanism where students are required to submit their preferences *before* the realization of their exam scores, can in fact have fairness and efficiency advantages compared to the often favored Serial Dictatorship mechanism. We test these hypotheses in a series of laboratory experiments which vary by the class of mechanism implemented, and the preference submission timing by students, reflective of actual policy changes which have occurred in China. Our experimental findings confirm the efficiency hypothesis straightforwardly, and lend support to the fairness hypothesis when subjects have the chance to learn with experience. The results have important policy implications for school choice mechanism design when students’ relative rankings by schools are initially uncertain.

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## Notes

Another example is Featherstone and Niederle (2008). They found that when preferences are private information, and all the schools have equal quotas and all the students’ preferences are randomly drawn from a uniform distribution of all possible preference orderings, truth-telling can be an equilibrium in BOS, and BOS can first-order stochastically dominate Deferred Acceptance in terms of efficiency, both in theory and in the laboratory.

Our study is also related to the theoretical work of Chiu and Weng (2009). They describe a model in which schools may pre-commit admissions slots to students (ie. early admissions), and endogenously derive the strategic motives for schools in adding such a feature to their admission process. As in their work, our primary variable of interest involves the timing of events occurring within the mechanism.

An implicit additional benefit is that for a given number of subjects in our entire study, we can have a relatively large sample of matching outcomes so our comparison of welfare consequences can be statistically valid given the stochastic exam score generation process. This is particularly valuable for cases of testing ex-ante fairness and efficiency, which requires a large enough sample for different realizations of exam score rankings across students.

We omit ‘ex-post’ efficiency (which conditions on score realizations) from our analysis. Note that in the case of the Pareto dominance criteria, for any given score realization, no matching is a Pareto improvement over another.

Since risk attitudes were a factor we wished to control for, but were not the main purpose of our study, we incentivized the Tanaka, Camerer, Nguyen test on a randomly selected subject in each session as follows: Subjects filled out the risk attitude form shown in the Appendix, and we informed subjects ahead of time that one randomly selected subject from each session would be chosen to have the risk attitude test implemented according to their answers. Once a subject had been randomly chosen in each session, we randomly drew a row number and implemented the lottery from that row. Then selected subject was paid according to his stated preference in the risk attitude form.

The complete instruction manual of session “B-a-1” is in Online Appendix. Instruction manuals of session “S-a-1”, “B-p-1” are also summarized in appendix by showing how they differ from session “B-a-1”. Instructions of sessions “B-a-2”, “S-a-2” and “B-p-2” are not reported, since they only differ from the corresponding session indexed by number “1” in title, in terms of the sequences of conducting “pre-exam submission” and “post-exam submission” procedure. The instruction manual of session “S-p-1/2” is a combination of “S-a-1/2” and “B-p-1/2” in an easily understandable way.

For the SD-before mechanism, the finding that type-3 students deviate from its truth-telling equilibrium a lot may also help to explain why SD-before mechanism achieves a degree of ex-ante fairness higher than the theoretical prediction. If type-3 students deviate from (A, B, C) to (B, *, *), students with higher abilities (students 1&2) are more likely to match with the best school (A).

There are very few demographics and experience variables showing significant effects, the exceptions being age and the experience of taking the College Entrance Exam, in the case of student 3 under the preference-wise design, and for student 2 under the ability-wise design, the loss aversion parameter.

In the case of more than one subject within a group having the same quiz score, ties are broken randomly.

The result using the measure of ex-ante blocking pair is similar, except in the Earnings Change treatment. There, the average number of ex-ante blocking pairs is significantly fewer than in the original design. This is consistent with our prediction that a threat of being admitted by a very bad school can push student 2 back to its equilibrium (or conservative) strategy.

The exception is in period 10 where the proportion of equilibrium strategies played by Student 2 dropped again. We speculate that this could be due to subjects’ knowledge that it is the final round of play, and the desire to try non-equilibrium strategies in their last opportunity.

In China, student preference orderings over schools are largely homogenous. Students also have a good estimation of their relative rankings of expected and realized scores. So it is possible at least for students to know what schools they are a good match for. In fact, they are usually advised by teachers, parents, and even consulting firms on these matters.

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## Acknowledgments

We thank the National Institute for Fiscal Studies at Tsinghua University for funding the SEM Economic Science and Policy Experimental Laboratory (ESPEL). All experimental treatments were run using Z-tree software (Fischbacher 2007). GE Zhigang, LI Qin, MA Lin, MA Mingming, QIAO Yuanbo, WANG Jinming, XU Haoyi, and ZHAO Shanke provided excellent assistance in running the experiments. For helpful comments we thank Yan Chen, Vincent Crawford, Pinghan Liang, Tracy Xiao Liu, Dawen Meng, Charles Noussair, Stephanie Wang, Lijia Wei, Maoliang Ye, and Yi Zhang, as well as participants at the Chinese Economist Society Meeting, Chengdu (2013), China Meeting of the Econometric Society, Beijing (2013), Economic Science Association World Meeting, Zurich (2013), Gezhi Economics Conference, Shanghai (2013), Tsinghua Conference on Theoretical and Behavioral Economics, Beijing (2013). We are especially grateful to our Editor, David Cooper, and two anonymous referees for providing detailed comments and helpful advice which greatly improved the paper. This research was funded in part by National Natural Science Foundation of China, Projects No. 71173127, No. 71203112 and No. 71303127, Tsinghua University (Project Nos. 2012z02182 and 2012z02181) and Ministry of Education, China (No. 20130002120030). All errors are our own.

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## Appendices

### Appendix 1: Proof of propositions

### 1.1 Proof of proposition 1

Given the score distribution of all the three students (uniformly distributed over high, normal and low scores), the score-rank distribution is the following:

Ranking of students by score | Probability of occurrence |
---|---|

(1, 2, 3) | 10/27 |

(1, 3, 2) | 7/27 |

(2, 1, 3) | 7/27 |

(2, 3, 1) | 1/27 |

(3, 1, 2) | 1/27 |

(3, 2, 1) | 1/27 |

Consider the first choice school of each of the three students. Note that in equilibrium each of the students must be admitted to one of the three schools. We first show the necessary part, that is all Nash equilibria of the game must have student 1 listing A as her first choice and students 2 and 3 listing B as their first choice.

###
**Claim 1.1**

*There in no equilibrium where some student chooses school C as her first choice*.

###
*Proof*

Note that every student has a positive probability of being ranked first. Thus, listing C as her first choice, is strictly dominated in expectation, compared to listing either B or A as her first choice.

###
**Claim 1.2**

*There is no equilibrium where all three students choose school A as their first choice.*

###
*Proof*

Suppose that there is an equilibrium where students 1, 2 and 3, each choose school A as their first choice. All students will then have incentive to choose school B, instead of C, as their second choice, in order to maximize their expected payoffs. Consider the expected payoff of student 3 in this case: 30 × 2/27+25 × 8/27+15 × 17/27 = 515/27, whereas by choosing B as her first choice, her expected payoff is 25 with certainty, which is greater than 515/27.

###
**Claim 1.3**

*In any equilibrium, at least one student chooses school A as her first choice.*

###
*Proof*

Suppose there is an equilibrium where no students choose school A as their first choice. Then for student i (i=1,2,3), her payoff would be 30 for sure if she choose school A as her first choice instead of playing the equilibrium strategy, which yields an expected payoff strictly less than 30.

###
**Claim 1.4**

*In any equilibrium, student 1 chooses school A as her first choice.*

###
*Proof*

Suppose there is an equilibrium where student 1 does not choose school A as her first choice. Then by Claim 1.1, she must choose school B as her first choice. By Claim 1.3, there are in total three cases.

###
**Case 1**

Both of the other two students choose school A as their first choice. Then student 1 will get 25 by choosing school B as her first choice. But if she chooses school A as her first choice and school B as her second choice, she will get in expectation: 30 × 17/27+25 × 8/27+15 × 2/27=740/27 > 25.

###
**Case 2**

Student 2 chooses school A as her first choice while student 3 chooses school B as her first choice. Then by choosing school B as her first choice, student 1’s expected payoff is: 25 × 24/27+15 × 3/27=215/9. By choosing school A as her first choice, student 1’s expected payoff is: 30 × 18/27+15 × 9/27=25 > 215/9.

###
**Case 3**

Student 3 chooses school A as her first choice, while student 2 chooses school B as her first choice. Then by choosing school B as her first choice, student 1’s expected payoff is: 25 × 18/27+15 × 9/27=65/3. By choosing school A as her first choice, student 1’s expected payoff is: 30 × 24/27+15 × 3/27=85/3 > 65/3.

So in any of the three possible cases, student 1 prefers choosing school A as her first choice.

###
**Claim 1.5**

*In any equilibrium, student 2 chooses school B as her first choice.*

###
*Proof*

Suppose there is an equilibrium where student 2 chooses school A as her first choice.

Then in equilibrium, student 3 must choose school B as her first choice by Claims 1.3 and 1.4. Then student 2’s expected payoff is: 30 × 9/27+15 × 18/27 = 20. But if student 2 chooses school B as her first choice, her expected payoff is 25 × 18/27+15 × 9/27 = 65/3, which is greater than 20.

###
**Claim 1.6**

*In any equilibrium, student 3 chooses school B as her first choice*.

###
*Proof*

By Claims 1.4 and 1.5, in equilibrium student 1 chooses school A as her first choice, and student 2 chooses school B as her first choice. Given this, if student 3 chooses school A as her first choice, her payoff is: 30 × 3/27+15 × 24/27 = 50/3. If she chooses school B as her first choice, her payoff is: 25 × 9/27+15 × 18/27 = 55/3 > 50/3.

The necessary result is immediate from Claims 1.4–1.6. For sufficiency, it is easy to verify that any such choice profile constitutes a Nash equilibrium.

Given the first choices of all the three students in equilibrium, the resulting outcome must be that student 1 is admitted to school A, student 2 is admitted to either school B or C, depending on her score relative to student 3. Student 3 goes to the remaining school. Thus, the equilibrium matching result is ex-post uniquely determined.

### 1.2 Proof of Proposition 2

As we assume, each student has the same ability. Thus each of them has the same score distribution which implies an equal probability of getting into each of the three schools. We first show the necessary part, that is all Nash equilibria of the game must have students 1 and 2 submitting (A, B, C) and students 3 listing B as her first choice.

###
**Claim 2.1**

*There is no equilibrium where all three students choose school A as their first choice.*

###
*Proof*

Suppose that there is an equilibrium where students 1, 2 and 3, each choose school A as their first choice. All students will then have incentive to choose school B, instead of C, as their second choice, in order to maximize their expected payoffs. Consider the expected payoff of student 3 in this case: 25 × 1/3+ 22 × 1/3+18 × 1/3 = 65/3, whereas by choosing B as her first choice, her expected payoff is 22 with certainty, which is greater than 65/3.

###
**Claim 2.2**

*In any equilibrium, at least one student chooses school A as her first choice.*

###
*Proof*

Suppose there is an equilibrium where no students choose school A as their first choice. Then for student i (i = 1,2,3), her payoff would be u*(i) = 31 (i = 1,2) or u*(i) = 25 (i = 3) for sure if she choose school A as her first choice instead of playing the equilibrium strategy, which yields an expected payoff strictly less than u*(i).

###
**Claim 2.3**

*In any equilibrium, both students 1 and 2 choose school A as their first choice.*

###
*Proof*

Consider student i, i = 1,2. Suppose there exists an equilibrium where student i does not choose school A as her top choice. Then by Claim 2.2, at least one of the other two students would choose school A as their first choice, so student i’s highest possible payoff will be 22. If she deviates by choosing school A as her first choice, her expected payoff will be at least 31 × (1/3)+18 × (2/3) = 67/3, which is greater than 22. So such an equilibrium cannot exist.

###
**Claim 2.4**

*In any equilibrium, student 3 chooses school B as her first choice.*

###
*Proof*

By Claims 2.1 and 2.3, in equilibrium student 3’s first choice must be either school B or school C. In either case student 3 will be admitted to her first choice school with certainty. Comparing the payoffs (22 vs. 18), student 3 must choose school B as her first choice.

###
**Claim 2.5**

*In any equilibrium, both students 1 and 2 choose school B as their second choice.*

###
*Proof*

Consider student i, i = 1,2. Suppose there exists an equilibrium where student i does not choose school B as her second choice. Then by Claims 2.3 and 2.4, it must be the following case: student i’s strategy is (A, C, B); student 3’s first choice is school B; the other student (denoted by j)’s first choice is school A. Thus student 3’s payoff is 22. However, if student 3 deviates by playing (A, B, C), her expected payoff will be either 25 × 1/3+22 × 1/2+18 × 1/6 = 67/3 [when student j’s strategy is (A, B, C)) or 25 × 1/3+22 × 2/3 = 23 (when student j’s strategy is (A, C, B)]. In either case, her payoff will be greater than 22. So such an equilibrium cannot exist.

The necessary result is immediate from Claims 2.3–2.5. For sufficiency, it is easy to verify that any such choice profile constitutes a Nash equilibrium. It is also easy to check that the equilibrium matching result described in Proposition 2 is ex-post uniquely determined.

### 1.3 Proof of proposition 4

For sufficiency, note that in SD-mechanisms, truth-telling is a weakly dominant strategy for every student, so the truth-telling strategy profile [(A, B, C), (A, B, C), (A, B, C)] indeed constitutes a Nash equilibrium for the SD-before mechanism. In the following, we show the uniqueness result (the necessary part) for the ability-wise design case and the preference-wise design case respectively.

The uniqueness result for the ability-wise design case:

###
**Claim 4.1A**

*In any equilibrium, student 1 chooses school A as her first choice. In addition, student 1 chooses school B as her second choice if another student lists A as her first choice*.

###
*Proof*

By playing (B, *, *), student 1’s expected payoff is at most 30 × 10/27+25 × 17/27 = 725/27; By playing (C, *, *), her expected payoff is at most 30 × 10/27+15 × 17/27 = 555/27. However, note that student 1’s expected payoff by playing (A, B, C) is at least 30 × 17/27+25 × 8/27+15*2/27 = 740/27, which is greater than both 725/27 and 555/27. This implies (B, *, *) and (C, *, *) are both strictly dominated by (A, B, C), and hence can never be equilibrium play for student 1.

In addition, note that if student 2 or 3 chooses A as their first choice, student 1 will be strictly worse off by playing (A, C, B) instead of (A, B, C). This is because student 1 has a positive probability of being ranked 2nd meanwhile school A is taken by the 1st ranked student. In that case student 1 will be admitted by school C if playing (A, C, B) but by school B if playing (A, B, C).

###
**Claim 4.2A**

*In any equilibrium, student 2 chooses school A or B as her first choice.*

###
*Proof*

Since student 1 plays (A, *, *) in equilibrium (by Claim 4.1A), by playing (C, *, *), student 2’s expected payoff is at most 30 × 1/27+25 × 8/27+15 × 18/27 = 500/27. However, note that student 2’s expected payoff by playing (A, B, C) is at least 30 × 8/27+25 × 11/27+15 × 8/27 = 635/27 > 500/27. This implies (C, *, *) can never be equilibrium play for student 2.

###
**Claim 4.3A**

*In any equilibrium where student 2 chooses school B as her first choice, student 3 must play* (*A, B, C*).

###
*Proof*

Given that student 1 plays (A, *, *) (by Claim 4.1A) and student 2 plays (B, *, *) in equilibrium, student 3’ expected payoff is at most 30 × 1/27+25 × 9/27+15 × 17/27 = 510/27 by playing (B, *, *); Her payoff is 15 for sure by playing (C,*,*); Her expected payoff is 30 × 3/27+15 × 24/27 = 450/27 by playing (A, C, B) and it is 30 × 3/27+25 × 7/27+15 × 17/27 = 520/27 by playing (A, B, C). Based on the comparisons of the payoffs by playing different strategies, (A, B, C) is student 3’s best response when student 1 chooses school A as her first choice and student 2 chooses school B as her first choice.

###
**Claim 4.4A**

*In any equilibrium, student 1 plays (A, B, C).*

###
*Proof*

By Claims 4.2A and 4.3A, in equilibrium either student 2 or 3 (or both) will choose school A as her first choice. This together with Claim 4.1A implies immediately that student 1 plays (A, B, C).

###
**Claim 4.5A**

*In any equilibrium where student 3 plays (A, B, C), student 2 plays (A, B, C)*.

###
*Proof*

Given that student 1 plays (A, B, C) (by Claim 4.4A) and student 3 plays (A, B, C) in equilibrium, student 2’ expected payoff is 25 × 19/27+15 × 8/27 = 595/27 by playing (B, *, *); It is 30*8/27+15*19/27 = 525/27 by playing (A, C, B); It is 30 × 8/27+25 × 11/27+15 × 8/27 = 635/27 by playing (A, B, C). By Claim 4.2A, based on the comparisons of the payoffs by playing different strategies, (A, B, C) is student 2’s best response when both students 1 and 3 play (A, B, C).

###
**Claim 4.6A**

*In any equilibrium, student 2 plays (A, B, C)*.

###
*Proof*

By Claims 4.2A, 4.3A and 4.5A, student 2 must play (A, *, *) in equilibrium. Given that student 1 plays (A, B, C) (by Claim 4.4A) in equilibrium, by playing (A, C, B), student 2’ expected payoff is at most 30 × 9/27+25 × 8/27+15 × 10/27 = 620/27, and it is at least 30 × 8/27+25 × 11/27+15 × 8/27 = 635/27 by playing (A, B, C).

By Claims 4.4A and 4.6A, in any equilibrium students 1 and 2 must play (A, B, C). It is straightforward to verify that given students 1 and 2’s strategy, student 3’s best response is (A, B, C).

The uniqueness result for the preference-wise design case:

###
**Claim 4.1P**

*In any equilibrium, at least one student chooses school A as her first choice*.

###
*Proof*

Suppose not. That is, there exists an equilibrium where none of the three students choose school A as their first choice. Under this assumption, there are in total two cases: (1) both students 2 and 3 choose school A as their last choice; (2) at least one of students 2 and 3 choose school A as their second choice. In case (1), student 1’ expected payoff is at most 31 × 2/3+22 × 1/3 = 84/3 by playing (B, *, *) and it is at most 31 × 2/3+18 × 1/3 = 80/3 by playing (C, *, *), both of which are smaller than 31, student 1’s payoff if she played (A, *, *). Thus case (1) cannot occur in equilibrium. In case (2), student 1’ expected payoff is at most 31 × 1/2+22 × 1/2 = 53/2 by playing (B, *, *) or at most 31 × 1/2+22 × 1/6+18 × 1/3 = 151/6 by playing (C, *, *), either of which is less than 31*2/3+18*1/3 = 80/3, student 1’s lowest possible expected payoff if she played (A, *, *). Thus case (2) cannot occur in equilibrium, either. This implies that it is impossible that none of the three students choose school A as their first choice in equilibrium.

###
**Claim 4.2P**

*In any equilibrium, student 3 will not choose school C as her first choice.*

###
*Proof*

Suppose not. That is, there exists an equilibrium where student 3 plays (C, *, *). By Claim 4.1P, either student 1 or 2 or both will play (A, *, *). So student 3’s expected payoff by playing (C, *, *) is at most 25 × 1/6+22 × 1/3+18 × ½ = 123/6, which is smaller than 25 × 1/3+22 × 1/3+18 × 1/3 = 65/3, student 3’s lowest possible expected payoff if she played (A, B, C). This implies that (C, *, *) cannot be the equilibrium play for student 3.

###
**Claim 4.3P**

*In any equilibrium where student 3 chooses school B as her first choice and student 1 chooses school A as her first choice, student 2 chooses school A as her first choice.*

###
*Proof*

Given that student 3 plays (B, *, *) and student 1 plays (A, *, *), student 2’s expected payoff is at most 31 × 1/6+22 × 1/2+18 × 1/3 = 133/6 by playing (B, *, *) or 18 for sure by playing (C, *, *), either of which is smaller than 31 × 1/2+22 × 1/6+18 × 1/3 = 151/6, student 2’s expected payoff by playing (A, B, C). This implies that either (B, *, *) or (C, *, *) cannot be student 2’s best response when student 3 plays (B, *, *) and student 1 plays (A, *, *).

###
**Claim 4.4P**

*In any equilibrium where both students 1 and 2 choose school A as their first choice, student 3 plays (A, B, C).*

###
*Proof*

By Claim 4.2P, student 3 will play either (A, *, *) or (B, *, *) in equilibrium. If both students 1 and 2 play (A, C, B), student 3’s payoff is 22 by playing (B, *, *); Her expected payoff is 25 × 1/3+22 × 1/3+18 × 1/3 = 65/3 by playing (A, C, B) and it is 25 × 1/3+22 × 2/3 = 23 by playing (A, B, C). If either student 1 or 2 (or both) play (A, B, C), student 3’s expected payoff is at most 22 × 5/6+18 × 1/6 = 64/3 by playing (B, *, *); It is at most 25 × 1/3+18 × 1/2+18 × 1/6 = 122/6 by playing (A, C, B) and it is at least 25 × 1/3+22 × 1/3+18 × 1/3 = 65/3 by playing (A, B, C). Therefore, (A, B, C) is student 3’s best response when both students 1 and 2 play (A, *, *).

###
**Claim 4.5P**

*In any equilibrium, student 3 chooses school A as her first choice*.

###
*Proof*

By Claim 4.2P, student 3 will play either (A, *, *) or (B, *, *) in equilibrium. If student 3 plays (B, *, *) in equilibrium, by Claim 4.1P either student 1 or 2 (or both) will play (A, *, *). Then by the symmetry of students 1 and 2 and Claim 4.3P, both students 1 and 2 will play (A, *, *). However, by Claim 4.4P, both students 1 and 2 playing (A, *, *) implies that student 3 plays (A, B, C), contradicting the assumption that student 3 plays (B, *, *). Thus student 3 must play (A, *, *) in any equilibrium.

###
**Claim 4.6P**

*In any equilibrium where student 1 chooses school B as her first choice, student 2 plays (A, B, C)*.

###
*Proof*

Given that student 1 plays (B, *, *) and student 3 plays (A, *, *) (by Claim 4.5P) in equilibrium, student 2’ expected payoff is at most 31 × 1/6+22 × 1/2+18 × 1/3 = 133/6 by playing (B, *, *); Her payoff is 18 for sure by playing (C, *, *); Her expected payoff is 31 × 1/2+18 × 1/2 = 49/2 by playing (A, C, B) and it is 31 × 1/2+22 × 1/6+18 × 1/3 = 151/6 by playing (A, B, C). Based on the comparisons of the payoffs by playing different strategies, (A, B, C) is student 2’s best response when student 1 chooses school B as her first choice and student 3 chooses school A as her first choice.

###
**Claim 4.7P**

*In any equilibrium, either student 1 or 2 plays (A, B, C).*

###
*Proof*

First note that (C, *, *) can never be an equilibrium play for student 1. Thus in equilibrium, student 1 plays either (A, B, C), (A, C, B), or (B, *, *). If student 1 plays (A, C, B) in equilibrium, given that student 3 plays (A, *, *) (by Claim 4.5P), student 2’ expected payoff is at most 22 by playing (B, *, *) and it is at most 22 × 1/3+18 × 2/3 = 58/3 by playing (C, *, *); Her expected payoff is at most 31 × 1/3+22 × 1/3+18 × 1/3 = 71/3 by playing (A, C, B) and it is at least 31 × 1/3+22 × 1/2+18 × 1/6 = 146/6 by playing (A, B, C). Based on the comparisons of the payoffs by playing different strategies, (A, B, C) is student 2’s best response when student 1 plays (A, C, B) and student 3 plays (A, *, *). If student 1 plays (B, *, *) in equilibrium, by Claim 4.6P, student 2 plays (A, B, C). Thus, in any case either student 1 or 2 plays (A, B, C) in equilibrium.

###
**Claim 4.8P**

*In any equilibrium, both students 1 and 2 play (A, B, C)*.

###
*Proof*

By Claim 4.7P and the symmetry of students 1 and 2, suppose student 1 plays (A, B, C). Given that student 1 plays (A, B, C) and student 3 plays (A, *, *) (by Claim 4.5P) in equilibrium, student 2’ expected payoff is at most 22 × 5/6+18 × 1/6 = 128/6 by playing (B, *, *) and it is at most 22 × 1/6+18 × 5/6=56/3 by playing (C, *, *); Her expected payoff is at most 31 × 1/3+22 × 1/6+18 × 1/2 = 23 by playing (A, C, B) and it is at least 31 × 1/3+22 × 1/3+18 × 1/3 = 71/3 by playing (A, B, C). Based on the comparisons of the payoffs by playing different strategies, (A, B, C) is student 2’s best response when student 1 plays (A, B, C) and student 3 plays (A, *, *).

By Claims 4.4P and 4.8P, all students play (A, B, C) in any equilibrium.

Based on the uniqueness of the equilibrium strategy and the features of the SD-before mechanism, it follows immediately that the equilibrium matching result is [(i, A), (j, B), (k, C)] and it is ex-post uniquely determined.

### Appendix 2: Tables

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Lien, J.W., Zheng, J. & Zhong, X. Preference submission timing in school choice matching: testing fairness and efficiency in the laboratory.
*Exp Econ* **19**, 116–150 (2016). https://doi.org/10.1007/s10683-015-9430-7

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DOI: https://doi.org/10.1007/s10683-015-9430-7