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Timing is Everything? An Experimental Investigation of Incomplete Information in Centralized Matching Mechanisms

Bingbing Li

(Tsinghua University)

Stephanie W. Wang (University of Pittsburgh)

Xiaohan Zhong

(Tsinghua University)

(Preliminary and Incomplete)

Abstract In this paper, we experimentally investigate in a large game four matching mechanisms related to China’s college admission system: Boston or serial dictatorship with students submitting their preferences either before (incomplete information) or after (complete information) the score realization. We find significant differences in the proportion of truth-telling across the mechanisms, but there is no significant difference in non-truth-telling reporting strategies. Students' reporting strategies affect their matching outcome: down-reporting on the first choice school leads to worse outcome on average, while up-reporting is associated with higher risk. The Boston mechanism with incomplete information is not superior to the other three mechanisms in ex-ante efficiency and ex-ante/post fairness when the score uncertainty is small, but may be superior to others in ex-ante fairness when the score uncertainty is large. Keywords: Boston, Serial dictatorship, School choice matching, Experiment JEL classification. C78, C92, D81, I28 1. Introduction

China’s college admission system is one of the world’s largest centralized

matching systems. A variety of school choice matching mechanisms prevails among regions and years within the country which differ mainly in two dimensions: preference submission timing and matching mechanism. Preferences can be submitted before the exam, after the exam but before knowing the exam scores, or after knowing the exam scores. Also two different matching mechanisms have been adopted: Boston mechanism (known as mechanism “without parallel options” in China) and Serial Dictatorship mechanism (known as mechanism “with parallel options” in China). In spite of those disparities, one rule remained uniform: students are ranked and matched to schools by their total score earned in the College Entrance Examination (CEE). This is the rule so highly regarded as “leveling the field” for college entrance competition.

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In this paper, we experimentally investigate four matching mechanisms: Boston in complete information (BOS_C for short) and incomplete information (BOS_I) environment, Serial Dictatorship in complete information (SD_C) and incomplete information (SD_I) environment. We call the information is complete/incomplete if students submit their preference ranking after/before the score realization. We study student truth-telling and non-truth-telling behaviors (i.e., preference list reporting strategies) and the effects of different reporting strategies on matching outcome under the 4 matching mechanisms. Moreover, we investigate the efficiency, the ex-ante and ex-post fairness of the 4 mechanisms by simulation.

We find that the preference submission timing plays an important role in student

truth-telling behavior under Boston Mechanism. There is significantly higher proportion of truth-telling under BOS_C than BOS_I, but no significant difference between SD_C and SD_I. We do not find strong evidence showing that there are systematically different patterns within non-truth-telling behavior under the 4 different mechanisms. We then explore the effects of students' reporting strategies on their matching outcome and find that down-report leads to worse outcome on average, while up-report is associated with higher risk, i.e., higher variation of matching outcome.

There is theoretical literature arguing that BOS_I might be superior to other 3

mechanisms (i.e., BOS_C, SD_C/I) with regard to ex ante efficiency and fairness (See Abdulkadiroglu, et al., 2011; and Lien, et al. 2015), under some specific conditions. There is also some empirical and experimental literature trying to find evidence on BOS_I’s superiority (See Wu and Zhong, 2014; Lien, et al., 2016), but until now the outcome is at most mixed. In this paper we address this issue in a large game with some degree of preference homogeneity/heterogeneity and information asymmetry, which we believe are more close to reality than the existing literature. Our results show that within SD, SD_C outperforms SD_I in efficiency, ex-ante and ex-post fairness. Within BOS, BOS_C is also superior to BOS_I in efficiency and ex-post fairness, but is inferior in ex-ante fairness. In general, there is no evidence showing that BOS_I is superior to other 3 mechanisms in ex ante efficiency and fairness. However, if we increase the score uncertainty endowed in the experiments, we do find a trend that BOS_I may be able to excel on the ex-ante fairness.

Our paper adds to the school choice literature regarding preference submission

timing in a school choice problem. We raise doubts on the superiority of BOS_I to other mechanisms. In Abdulkadiroglu et al. (2011), one condition for BOS_I’s superiority on ex ante efficiency is homogeneity of student ordinal (but not cardinal) preference. In this paper, student preference is only partially homogeneous (homogeneous between some groups of schools), this may help to explain the vanishing of the superiority on ex-ante efficiency. As for ex ante fairness, as suggested by Lien et al. (2015), even if students have some degree of homogeneous

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preference, their competition may be so strong that the “screening” effects essential to ex-ante fairness collapse. Our paper therefore suggests that BOS_I may only have advantage on ex-ante efficiency and fairness side under a very theoretically narrow base. In reality, this advantage is likely to vanish. Yet we also suggest that if score uncertainty is large, BOS_I can be a promising mechanism as for ex-ante fairness, which is consistent with the theory suggested in Lien, et al. (2015)

The paper is also one of the first school choice experimental papers to study

students’ non-truth-telling behavior, especially in a large game. We focus on students’ first choice, and use their fair school as benchmark. By this way, we are able to study the impact of individual reporting strategies (up/fair/down-report) on individual matching outcome. Wu and Zhong (2015) use the same method to gauge students’ misreport and mismatch, yet with field data in China's college admission system. We instead link students reporting behavior with their matching outcome using the laboratory data.

The paper is organized as follows. Section 2 offers a review of related literature.

Section 3 describes the experimental design. Section 4 provides the analysis of students' behaviors and their matching outcome under different mechanisms. Section 5 presents the simulation results on matching outcome for efficiency and fairness under various mechanisms. Section 6 concludes the paper.

2. Literature Review

There is a fast growing literature on school choice problems. We review three

branches of literature which our paper is mainly related to: literature on canonic BOS and SD mechanism, literature on preference submission timing, and literature on relationship between individual preference submission behavior and individual matching outcome.

Literature on canonic BOS and SD mechanism Canonical Boston (BOS) and Serial Dictatorship (SD) mechanism have been

widely discussed in literature. Generally, canonical BOS is proved theoretically to be inferior to the SD mechanism with respect to strategy-proofness, efficiency and fairness (Abdulkadiroglu and Sonmez, 2003; Ergin and Sonmez, 2006). On the empirical side, Chen and Sonmez (2006) test the school choice matching mechanisms on subjects in the laboratory and give supportive evidence on the theory. They find higher proportion of misrepresentation of truthful preference under BOS, with approximately 80% of the subjects manipulating their true preferences under BOS, 53% under TTC (equivalent to SD mechanism), 36% under Gale-Shapely. They also present lower efficiency under BOS by using recombinant estimation technique and running simulation. Our experiment in this paper was conducted with reference to their experimental design.

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The manipulability of different matching mechanisms is also analyzed by He (2012) and Chen and Kesten (2013). He (2012) uses field data from Beijing to study the strategic manipulations under the Boston mechanism and finds that parents, especially the poorer/less educated, are revealed to be overcautious and avoid top ranking the most popular school. By using laboratory data, Chen and Kesten (2013) conclude that the likelihood of truthful preference revelation is the highest under the Deferred Acceptance (DA) mechanism, followed by the Chinese parallel (modeled as a hybrid of BOS and DA) and then the Boston mechanism. Their empirical and experimental results are consistent with both the theory and prior experimental findings.

Literature on preference submission timing Inspired by China's college admission system, some researchers introduce

preference submission timing as a source of incomplete information or uncertainty into the standard BOS and SD mechanism. Lien, Zheng and Zhong (2015) develop a theoretical framework and propose that BOS mechanism which requiring students submitting their preferences before the exam (i.e., pre-exam BOS mechanism) is more likely to achieve complete ex-ante fairness, although the advantage is narrow. Wu and Zhong (2014) confirm that the pre-exam BOS mechanism can be ex-ante more fair and efficient than the mechanisms with preference being submitted after exam (i.e., post-exam BOS or SD) using field data from a top Chinese university. Lien, Zheng and Zhong (2016) test the theory in a series of laboratory experiments and claim that pre-exam BOS mechanism is superior to other mechanisms in ex-ante efficiency and can have advantages in ex-ante fairness when subjects have chance to learn with experience. Based on these studies, we further test the efficiency and the ex-ante/ex-post fairness of the mechanisms interacted with timing in a "larger-scale" game. By large scale game, some features dumb in a “small-scale” game can voice out. We would explore in the paper some characteristics or “rules” of student strategy, which might be used by students to simplify their choices under complicated games. In particular, their first choice school may become extremely important, especially under non-truth telling mechanisms like BOS mechanism.

By introducing preference submission timing, our paper also can speak to other

papers focusing on the effect of incomplete information or uncertainty on the school choice matching. In those papers, incomplete information or uncertainty often come from private information of student preferences, or some random tie-breaking rules used by schools, although in our paper the source is preference submission timing: before the exam, students could not know perfectly their score rankings used as school priorities. In spite of this major difference, a general lesson is reached: the advantages of the SD mechanism compared with BOS may no longer hold under situations with incomplete information or uncertainties. (Ehlers and Masso, 2015; Pais and Pinter, 2008; Featherstone and Niederle, 2008; Edril and Ergin, 2008; Abdulkadiroglu, Pathak, and Roth, 2009; Abdulkadiroglu, Che and Yasuda, 2011).

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Literature on individual preference submission behavior and individual matching outcome

Since students report their preferences strategically under non-truth-telling mechanisms, and there is non-truth-telling equilibrium even under truth-telling mechanisms, it is important to study individual strategic behavior under various mechanisms. Wu and Zhong (2015) show that, under BOS, if schools have acyclic priorities over students and students have homogenous preferences over schools, students’ first choice become extremely important: they should put their fair schools as their first choice in equilibrium and be admitted by first choice, except for students with lowest scores. They use field data to examine students' preference submission behavior and their matching outcome, and find that preference submission behavior does matter for students’ matching outcome. In this paper we examine students' truth-telling and their non-truth-telling behavior in reporting the first choice and test the effect of these behaviors on their matching outcome, by using the laboratory data under both BOS and SD mechanism.

Student behavior may be in turn determined by their risk attitudes. Klijn et al.

(2012) experimentally study how individual's risk preference influence their strategy, and they emphasize that higher degree of risk aversion lead to more protective strategies under the Gale–Shapley but not under the Boston mechanism. Following Tanaka et al. (2010) to elicit students' risk attitude parameters, we also investigate this issue, yet we have not found significant effect of students' risk attitude on their preference submission behavior (truth-telling or strategic reporting). 3. Experimental design

Our experiment is designed to compare student behaviors and matching

outcomes under different mechanisms with different preference submission timings. In particular, we conducted 4 different treatments, Boston and SD mechanism under either complete or incomplete information environment. For complete environment we mean preference submission being conducted after the exam and all the students’ scores being known by each other. For incomplete environment, we mean preference submission being conducted before the exam without score being known. For each treatment, we have 2 or 3 sessions, with 36 students in each session. We have 10 sessions in total and each student only participated in one session.

Ranking/Mechanism Boston Serial Dictatorship Complete 2 (N=72) 2 (N=72) Incomplete 3 (N=108) 3 (N=108)

In the complete information setting, the scores for the 36 students in each session

are independently drawn without replace from integers between 105 and 140. Each student is informed of his/her ranking. In the incomplete environment, the estimated scores for the 36 students in each session are independently drawn without replace

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from integers between 105 and 140. Each student is informed of his/her estimated ranking. The actual score for each student is drawn from a uniform distribution -10 to +10 around his/her estimated score. The actual ranking is not revealed until the preference rankings are already submitted.

In each session, there are 36 school slots across 7 schools A-G: 3 slots each at A

and B, 5 slots at C and E, 6 slots at D and F, 8 slots in G. All students prefer A and B to the other five schools, which is common knowledge. Some students prefer A to B and some prefer B to A, but the proportion is not publicly known. Except the public information mentioned above, students only know his/her own preference ranking. The payoff setting is similar with Chen and Sonmez (2006) and the payoffs obtained are symmetric, i.e., each student gets the same payoff for the same preference ranking of choice. The actual monetary revenue obtained is multiplied by 5 RMB from the payoff table. Schools always prefer student with higher (realized) scores. Payoff table is shown as below.

Preference Ranking

Payoff 1 2 3 4 5 6 7

16 13 11 9 7 5 2 We ran paper-and-pencil experiment at Tsinghua University in China on May

24th (2 sessions), June 1st (4 sessions) and June 2nd (4 sessions) of 2012. Each session lasted approximately an hour and the participation-fee is 20 RMB. All sessions were conducted in Tsinghua University, School of Economics and Managements' Experimental Economics laboratory (ESPEL).

We also collected student background information, including their risk attitude,

experience (province, whether or not taking college entrance exam, year of college entrance exam), gender, age, major, etc. We follow Tanaka et al. (2010) using three series of paired lotteries to elicit the three prospect theory parameters (curvature, probability weighting, and loss aversion) and the risk attitude test is incentivized. 4. Individual Behavior and Individual Mismatch

In this section, we report two groups of results on evaluating the four

mechanisms. The first group of result is about individual behavior. We report both the likelihood of truth-telling (or naive) behavior and the pattern of non-truth-telling (or strategic, sophisticated) behavior. In our second group of result, we link individual behavior with individual matching outcome under different mechanisms.

We use some shorthand notations when presenting the results in the following parts. Let X>Y denote that a measure of efficiency or fairness under mechanism X is greater than (or superior to) the corresponding measure under Y. Let X≈Y denote that a measure under X is not significantly different from the corresponding measure under Y. The significant level (p-value) is shown under each of these notations.

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4.1 Individual Behavior: Truth-telling and Non-truth-telling We first consider whether a student play truth-telling behavior. Table 1 presents

the proportion of truth-telling under different mechanisms. Truth-telling requires that the entire reported preference ranking of a student is identical to his or her true preference ranking. The significance of t-test of truth-telling between any two mechanism ranking next to each other is shown in panel B of Table 1(for brevity, we omit those tests between mechanisms not ranking next to each other).

The results show that within BOS, there is significantly less truth-telling under incomplete environment than under complete environment (p=0.001). Actually, there are almost no students reporting truthfully under BOS with incomplete environment. However, within SD, there is no significant difference in truth-telling between incomplete environment and complete environment.

We summarize our findings as the following: Result 1: Under different mechanisms, the proportion of students' truth-telling

behavior is ranked as: SD_C ≈ SD_I> BOS_C > BOS_I.

Insert Table 1 here Table 2 reports determinants of truth-telling by using a Logit model. In each

specification, the dependent variable is truth-telling, which equals one if a student reports her/his preference ordering truthfully and zero otherwise. The independent variables are: (1) Mechanism dummies: a dummy for BOS, a dummy for incomplete environment, and one for their interaction term. (2) Individual score rank: we use student expected score ranking for incomplete environment and actual score ranking for complete environment. (3) Student’s SD fair school slots: the SD fair school for each individual is the school matched to the student by SD mechanism according to their expected/actual score in incomplete/complete information environment. A fair school with less slots implies a higher competitiveness of the school, therefore students may try to avoid to list such a school, or a school he prefer to this school, as his or her first choice. (4) Student personal characteristics: including gender, age and major dummies. (5) Students risk attitude: we follow Tanaka et al. (2010) by using the three prospect theory parameters σ, α, λ to represent the curvature, probability weighting, and loss aversion.

The result shows that while matching mechanisms are significant factors, all

other variables are mostly insignificant. The effects of mechanisms on truth-telling are consistent with what we have in Table 1.

Among risk attitude parameters, we find that students who are more risk averse (with higher σ) are less likely to be truth-telling, while the probability weighting (α) and loss aversion (λ) parameters are insignificant. This result is easy to explain: since truth-telling implies putting the most competitive school at one’s first choice, it is indeed a risky choice which a risk averse student would try to avoid.

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Insert Table 2 here From Table 1, we also know that a large proportion of students play

non-truth-telling strategies. An important question is what kind of strategy students would adopt if they decided not truth telling? As pointed out by Wu and Zhong (2015), a student’s top (first) choice might be important, especially when students have some degree of homogenous preferences. We now focus on how students choose their first choice school, especially for those non-truth-telling players. Wu and Zhong (2015) proved that when students have (completely) homogeneous preferences, under BOS mechanisms with complete information environment, they should put their (ex-post) fair schools as their first choice. We then use one’s (ex-ante or ex-post) fair school as a benchmark for his or her first choice.1We call a student up-reporting his or her first choice when he or she chooses a school preferred to his or her fair school. By the same token, down-reporting, or fair-reporting can also be defined. However, note that students may not be able to figure out their fair school, due to asymmetric information on preferences. Therefore our measure of up- or down-report only captures some intentions of students to up or down-report their first choice under an imperfect information environment.

In particular, we use the gap between the preference ranking of the first choice and the SD fair school to measure the degree of misreport for non-truth-telling students. The sign can be negative or positive, with negative indicating down–reporting, positive indicating up-reporting, and zero for fair-reporting. Figure 1 shows the distribution of the degree of misreport. There seem no systematic differences on misreporting behaviors among different mechanisms. Table 3 reports the proportions of different non-truth-telling reporting strategies under each of the four mechanisms in panel A, and the t-test result of the significance of these differences in panel B. Students are less likely to up-report their first choice under BOS mechanisms than under SD mechanisms. Yet as a whole, when we use the measure of degree of misreport (i.e., the rank of one’s fair school minus the rank of his/her first choice school, averaged among students within a mechanism), the results show that there is almost no significant difference in non-truth-telling reporting strategies among different mechanisms.

Insert Table 3 here Table 4 reports the regression results of degree of misreport. We find no

significant evidence on the effect of four mechanisms on misreport, which is consistent with what we find in Table 3.Among other determinants, students who are ranked lower are more likely to down-ward report their first choice. This is interesting because the measure of the degree of misreport has already taken into account one’s 1Whenmeasuringmisreportandmismatch,weuseex-anteSDfairschoolforBOS_IandSD_Imechanism,ex-postSDfairschoolforBOS_CandSD_C2Whenmeasuringmismatch,weuseex-anteSDfairschoolforBOS_IandSD_I,ex-postSDfairschoolforBOS_CandSD_C3 Thet-testresultsofefficiencyindifferentscenariosunderthefourmechanismsare:scenario 1>scenario 4≈

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relative ranking. The result implies that students ranked lower are somehow discouraged or over-pessimistic when they report their preferences. The result also shows that if a student's SD fair school has more slots, he or she is more likely to up-report, using his or her fair school as a safer backup. Another interesting result is that students majored in economics are more likely to up- reporting even when we control the risk attitude parameters. This might be because economics is one of the most competitive major in the university we conduct this experiments, so those students seem more confident than others. All the other personal characteristics and risk attitude parameters are all insignificant.

Insert Table 4 here We summarize our findings on non-truth-telling behaviors among different

mechanisms as the following: Result 2. There is no difference on non-truth-telling behaviors among four

mechanisms. Although different mechanisms differ a lot on their strategy-proofness, i.e., on

their incentives provided for truth-telling, students who choose to play strategically do not play systematically differently among those mechanisms. In fact, for BOS and SD with complete information environment, one equilibrium is for all the students to choose fair school as their first choice. For BOS and SD under incomplete information environment, such a strategy is also a useful benchmark even if it may not be an equilibrium strategy (see Lien, Zheng and Zhong, 2015). Therefore for those students not truth-telling, their behaviors might converge to such a “focal point”.

4.2 Linking Individual Matching Outcome with Individual Behavior After knowing different strategies students may choose to play, we are interested

in how those different strategies may be linked with different matching outcome, i.e., how different misreport behaviors may be linked with different mismatch. We use the gap between the preference ranking of a student's matched school and his/her SD fair school2 to measure the degree of mismatch. Same with the measurement of misreport, the sign of the degree of mismatch can be negative or positive, indicating down-matching or up-matching respectively.

Panel A of Table 5 presents the summary statistics of the degree of mismatch for different misreport behaviors under different mechanisms. The t-test results are provided in panel B. Overall, the higher preference rank a student’s reported school is, the higher rank his or her matched school will be, although sometimes the difference is insignificant. Note that a truth-telling strategy would always be equal to an “up-report” strategy as “up” as possible, since it requires putting the most competitive school at the first choice. In particular, we can see that down-reporting students are

2Whenmeasuringmismatch,weuseex-anteSDfairschoolforBOS_IandSD_I,ex-postSDfairschoolforBOS_CandSD_C

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significantly more down-matched than the other three strategies under all the four mechanisms.

Insert Table 5 here

Table 6 shows the OLS regression of the degree of mismatch under different mechanisms in column (1)-(4). Compared to fair-reporting (the default value), down-reporting will significantly lead to worse matching outcome, for any given mechanism. All the other reporting strategies (truth-telling or up-report) show insignificant differences from fair reporting.

Insert Table 6 here We also run regression to analyze the effect of different reporting strategies on

the variance of the degree of mismatch and the results are shown in column(5)-(8) of Table 6. To measure the variance of degree of mismatch, we run regression by the following steps. First we run OLS regression of the degree of mismatch, with all the control variables shown in Table 6. After the regression we save the residuals and calculate the square of the residual for each observation. Finally we run OLS regression of the squared residuals on all the control variables and get the coefficients. The results show that up-reporting students face significantly larger variance of degree of mismatch (implying higher risk) under all the four mechanisms, compared to fair-reporting.

We summarize our result with regard to the linkage between misreport and mismatch as the following:

Result 3. Among non-truth-telling behaviors, down-reporting leads to worse

matching outcome and up-reporting leads to larger variation (or risk) of matching outcome.

Why mismatch are so closely correlated with misreport at the first choice? The

fact can be explained by the high first choice admission rate. The probability of top choice admission can be found in Table 7.There are very high top choice admission rate under Boston mechanism. Even under SD mechanism, where truth-telling is a weakly-dominant equilibrium, top choice admission is still pretty high among students playing non-truth telling behavior. In fact, it can be proved in theory when students have homogenous preferences, in equilibrium (or one of equilibria) under the BOS and SD with complete information, students would be admitted by their first choice. The equilibrium can even be unique under the BOS mechanism (Wu and Zhong, 2015).

Insert Table 7 here

5. Evaluate Matching Quality by Simulation

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We now turn to evaluate matching quality under four different mechanisms. For

doing so, we first introduce our simulation method which generate statistically meaningful matching outcomes for our comparison. Then we define various welfare criteria for evaluations. Finally we present our results.

5.1 Simulation Method The initial condition of our game is that students only know their distribution of

scores used as school priorities. Under the BOS and SD with incomplete information environment, students have to base their strategy on such score uncertainties. Yet under the BOS and SD with complete information environments, students know their realized scores first, and then submit their preference rankings. To make the mechanism with incomplete and complete information environment comparable, we need to compare their matching outcome by only giving their distributions of score. For the mechanisms under incomplete information, we only need to realize the matching outcome many times according to the score distributions and in each time utilize strategies of all students, which is known and fixed among different score realizations, to figure out the matching outcome. Yet for the mechanisms under complete information, we have to consider the counter facts, i.e., how the students (or subjects) would response to all the possible score realizations we could not observe. One conjecture would be that students only respond to their realized score ranking, so for any given realized score ranking, students would play the same strategy as the one we really observe in the lab. Yet since students have many other personal characters, and students may choose to play mixed strategy, it would be naive to assume realized score ranking is the only determinant of strategy played and the observed strategy (or action) is the only “true” strategy. We therefore simulate student behaviors under each realized score under complete-information mechanism (SD/BOS) by alternative scenarios, each summarizing some behavior patterns we found from the experimental data. To facilitate the comparison between mechanisms with or without complete information, we also simulate preference submission behaviors for mechanisms under incomplete information by considering the parallel scenarios with those under complete information.

We first describe our simulation process in general, and then we will describe four scenarios we use to simulate preference submission behaviors.

The simulation steps are as follows. Step1.Randomly draw all the students’ realized score rankings from the given

score distribution. Step2. Simulate students’ preference submission for each ex-ante/realized score

distribution under incomplete/complete-information mechanism, through each of the four scenarios we will describe below

Step3. Match according to the realized scores and simulated preferences submission behaviors of all the students, according to the BOS or SD mechanism.

Step4. Simulate step 1-3 200 times.

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We now describe the four scenarios we propose to capture the behavior patterns we draw from the observed actions of students under the four mechanisms.

Scenario1 For Boston- and SD-before, just do matching according to each realized score

rankings, each time under the same observed preference submission behavior of each student.

For Boston- and SD-after, assume a student's behavior is only determined by his realized ranking, and extract the preference submission behavior (for all the seven choices) from the lab data for each score ranking. Since we have two sessions for each mechanism, we randomly choose from one of the two sessions the students' behavior as the behavior of students with the same ranking.

The matching quality (the evaluation criteria will be detailed below) for each mechanism is the average of the matching quality of the several sessions under that mechanism.

Scenario 2 In this scenario we extract from the lab data the distribution of truth-telling and

up/down/fair reporting strategies for the first choice in simulating student choices. In particular, within each session, we calculate the proportion of truth-telling

(x%). For the non-truth-telling students, calculate the distribution of the reporting strategies by looking at the degree of misreport (w.r.t. ex ante/realized scores for incomplete/complete-information mechanisms).

We then randomly choose x% students in that session to be truth-telling. For the non-truth-telling students, randomly choose them to up /fair/down report their first choice to the degree as collected above. Note that we use the calculated ex-ante/ex-post SD-fair school according to the ex-ante/ realized student scores when simulating the up/down/fair reporting choices.

For the 2nd-7th choice of each non-truth-telling student, randomly assign the rest schools to these choices.

Scenario 3 The same as Scenario 2, except that for non-truth-telling students, we also

simulate their second choice in their preference ranking list. When simulating the second choice, we first find the distribution of the degree of

misreporting for the second choice from the lab data, then randomly choose students to match such a distribution. We then randomly assign the 3rd-7th choice for each student.

Scenario 4 The same as Scenario 2, except that for non-truth-telling students, we also

simulate their second choice and do this by considering its relation with their first choice. In particular, when simulating the second choice for the non-truth-telling students, we first find the distribution of the gap between the preference ranking of

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the first and second choice from the lab data within each session. Then we randomly assign the preference ranking to students’ second choice according to that distribution. We finally randomly assign the 3rd-7th choice.

5.2 Welfare Criteria The welfare consequences of the matching outcome under a mechanism can be

evaluated by both efficiency and fairness criteria, as discussed in various literatures. The following is the welfare criteria we would consider and the measures we would use.

(Ex-ante) Efficiency. The (ex-ante) efficiency is calculated as the payoffs per capita, averaged among different sessions and simulations under a mechanism.

Ex post Fairness. In literature, fairness is measured by the number of blocking pairs. A blocking pair is a pair of student-school in which the student prefers the school to his own matched school, and the school either has an empty seat or gives a higher priority to this student than some students it admits. Here the school priority is solely determined by student realized scores. In this paper, we adopt two measures of fairness, which share the same spirit as the idea of blocking pairs.

(i) Ex-post Fairness by average absolute value of degree of mismatch. We calculate the gap between a student’s preference ranking of his or her matched school and his/her ex-post SD fair school, i.e. a student’s degree of mismatch. The ex-post fairness for each mechanism is measured by the absolute value of the degree of mismatch, averaged across all students, sessions and simulations within a mechanism. A higher degree of mismatch implies a lower level of fairness of the matching outcome.

(ii) Ex-post Fairness by number of blocking pairs. This measurement of fairness corresponds to the notion of stability. Define (i,S) as a blocking pair for student if student i prefers school S to his matched school and his score is above the minimum score of all the students matched to school S. The fairness for the whole matching is calculated as the number of blocking pairs averaged across all students, sessions and simulations within a mechanism. The lower the average number of blocking pairs, the more fair the matching outcome is.

Note that although the two fairness measures share the same spirit as the measure of fairness, they may not be consistent with each other all the time. For example, consider 3 students, student 1, 2 and 3, are matched to 3 schools, A, B, C. All the students have the same preference: A>B>C, and all the schools have the same priorities: 1>2>3. It is easy to calculate that among all the 6 possible matching outcomes, matching outcomes has the highest “degree” of unfairness are (3=A, 2=B, 1=C), (2=A. 3=B, 1=C), and (3=A, 1=B, 2=C) for the measure of degree of mismatch, with the value as 4/3. While for the measure of number of blocking pairs, only (3=A, 2=B, 1=C) has the highest degree of unfairness, with the average number of blocking pairs as 1. The other two matching outcomes have the average number of blocking pairs as 2/3.Yet intuitively, it would be hard to see whether (2=A, 3=B, 1=C) or (3=A, 1=B, 2=C) is less unfair than (3=A, 2=B, 1=C) or not. In the last case, at least student 2 is matched to his fair school, while the first and the second match have all students

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mismatched. Ex-ante Fairness. Ex-ante fairness is defined similarly to ex-post fairness, except

that all the fairness measures are defined based on expected student scores instead of realized scores. For example, a fair match is defined as a student being matched with his/her ex-ante SD fair school.

5.3 Simulation Result We report in this part the simulation result for comparing the welfare result of

different mechanisms under each of the four scenarios. Table 8 reports results for the (ex-ante) efficiency of the four mechanisms. Panel A reports statistic summary and Panel B reports t-test on difference among four mechanisms, as we did for truth-telling behavior (Table 1). We can conclude that for all the scenarios, for ex-ante efficiency: SD_C>SD_I>BOS_C>BOS_I, all significant at above 5% significant level.

Insert Table 8 here These findings are contradicted with the experimental findings from Lien, et al.

(2016), where BOS_I mechanism has the highest efficiency among all the four mechanisms. The possible reasons for different findings may come from the fact that in our environment, uncertainties are not only with scores, but also with student preferences. In fact, contrasted with Lien et al. (2016), here students cannot figure out their ex-ante or ex-post fair schools. Under such a circumstance, a non-truth-telling mechanism may require a high-level sophisticated strategy to achieve the equilibrium outcome, which may in theory be a more efficient outcome. In addition, the degree of preference homogeneity may also matter. As suggested by Abdulkadiroglu et al. (2011), student preference homogeneity in an ordinal sense may be essential for the efficiency superiority of BOS_I mechanism. While in Lien et al. (2016), preference homogeneity is held, which make their setup perfectly fit with the theoretical assumption of the Abdulkadiroglu et al. (2011), in this paper students only share a very limited level of preference homogeneity.

We also note that the calculated value of ex ante efficiency (as well as ex-post and ex-ante fairness in analysis later on) differ a lot among different scenarios for any given mechanism.3 Therefore, we are not sure which scenario is more close to reality. But, since our simulation covers a very wide range, from simulating only the students' first choice in scenario 2 to simulating all the students' seven choices in scenario 1, and when we compare the ex-ante efficiency and fairness among different mechanisms, the results in different scenarios are consistent, we can safely say that our results are close to the real relative performance (in large sample) of different

3 Thet-testresultsofefficiencyindifferentscenariosunderthefourmechanismsare:scenario 1>scenario 4≈ scenario 3> scenario 2, at 1% significant level. The t-test result of ex-post fairness are: scenario 4 < scenario 3< scenario 1 ≈scenario 2 for BOS_C at 10% significant level; scenario 1 < scenario 4≈scenario 3 < scenario 2 for the other 3 mechanisms at 1% significant level. The t-test result of ex-ante fairness are: scenario 1 < scenario 4≈scenario 3 < scenario 2 at 1% significant level.

15

mechanisms. Table 9 and 10 present results for the ex-post fairness measured by average

number of blocking pairs and average absolute value of degree of mismatch under different mechanisms. Within SD mechanisms, SD_C consistently outperforms SD_I. Within BOS mechanisms, BOS_C generally performs better than BOS_I, except for the result measured by the blocking pairs in scenario1.Therefore, in general, it can be concluded that for ex-post fairness: SD_C> SD_I>BOS_C>BOS_I.

Insert Table 9 and Table 10 here The results for ex-ante fairness by two different measures are shown in Table 11

and Table 12 respectively. Within BOS mechanism, by the measure of number of blocking pairs, it is unclear whether BOS_I outperforms BOS_C, or the opposite (See Table 11). Yet when we measure the ex-ante fairness by the absolute value of degree of mismatch, BOS_I is significantly superior to BOS_C(see Table 12). Within SD, we can tentatively say that SD_C performs better than SD_I, or is at least not inferior to SD_I in ex-ante fairness.

Insert Table 11 and 12 here In order to further test the effect of the mechanisms on (ex-ante) efficiency,

ex-post fairness and ex-ante fairness, we run regressions for each scenario. The regression result of efficiency measured by payoff per capita is shown in Table A1 and the results for ex-post and ex-ante fairness are reported in Table A2 and A3. There are 2 groups of regressions in both Table A2 and A3, where the dependent variables are the absolute value of degree of ex-post/ante mismatch for each student and the number of ex-post/ante blocking pairs for each student. The independent variables includes the mechanism dummies, students score rank，their SD fair school slots. In our simulation, student preference submission behavior is simulated according to the summarized statistics in each session or according to the recombinant of students actual behaviors in each session, thus the individual's personal characteristics should not influence the matching outcome and are not controlled in the regression model .

The results are consistent with the t-test result before. Within BOS mechanisms, BOS_I has significantly lower efficiency, higher (absolute value of) degree of ex-post mismatch and more ex-post blocking pairs than BOS_C in scenario2-4 in Table A1 and A2, while the coefficients of the regressions in scenario1 are insignificant. Furthermore, it is more likely that BOS_I leads to less degree of ex-ante mismatch than BOS_C, although the effect is not always significant in the regression of ex-ante fairness measured by blocking pairs in Table A3. Our results also show that SD_I in general leads to lower efficiency, larger degree of ex-ante/post mismatch and more ex-ante/post blocking pairs than SD_C, while these effects are not always significant.

We summarize our findings as the following:

16

Result 4: (1) For the ex-ante efficiency, we have: SD_C>SD_I>BOS_C>BOS_I.

(2) For the ex-post fairness, we have: SD_C>SD_I>BOS_C>BOS_I. (3) For the ex-ante fairness, we have: SD_C≧SD_I≧BOS_I≧BOS_C.

5.4 The Effects of Increasing Score Uncertainty on Matching Quality In the above simulation, the actual score for each student is drawn from a

uniform distribution -10 to +10 around his/her estimated score. We may change the score distribution to examine the effects of the score uncertainty on matching outcome. Assuming that students' strategies simulated from the experiments are not sensitive to score spans, we change the score distribution by change the score span under uncertainty. Table A4-A8 show the ex-ante efficiency and fairness result of the simulation when score is drawn from a uniform distribution -20 to +20 around the estimated score for each student.

We can conclude that for ex ante efficiency and ex-post fairness, BOS_I<BOS_C<SD_I<SD_C, which is consistent with the simulation result before. Within SD, we have SD_C>SD_I in ex-ante fairness generally. We see improvement in ex-ante fairness under BOS_I when score uncertainty increases. In Table A8 where ex-ante fairness is measured by the absolute value of degree of mismatch, BOS_I even performs better than SD mechanisms, although the result is inconsistent when the ex-ante fairness is measured by the number of blocking pairs in Table A7.4

6. Conclusion

In this paper, we introduce preference submission timing, i.e., preference

submission before or after exam, into the two classic school choice mechanisms (BOS and SD), and conduct laboratory experiments to compare the four mechanisms. Compared with the similar research of Lien et al. (2016), our game size is much larger, with much more uncertainties, asymmetric information, and only partially homogenous student preference, which provide us more opportunities to look at student strategic behavior closer to a real world situation. We also look at the first choice of students, which might be important when students have roughly homogeneous preferences, a feature prevailing in China’s college admission system.

Within SD or within BOS, we see significantly less truth-telling in incomplete information environment. However, we find no evidence for the effect of mechanisms on various non-truth-telling strategies. By linking individual behavior with individual matching outcome, we find that down-reporting leads to more down-matches than other strategies (fair reporting, up-reporting and truth-telling), under any mechanism. There is also larger outcome variance for up-report (higher risk) under all mechanisms. The results show trade-offs between risk and return.

We run simulation to evaluate the welfare consequences of matching outcomes. 4 We further examine this effect by expanding and narrowing the score fluctuation range, to reset the fluctuation range from -30 to +30, and -5 to +5. We find that the result of the former is close to that with the score fluctuated from -20 to +20 and the latter is quite similar to the result with the score uniformly distributed -10 to +10 around the estimated score.

17

The results show that within BOS/SD mechanism, efficiency and ex-post fairness are almost always higher under complete information than under incomplete information. Within BOS mechanisms, we find some evidence for the improvement of ex ante fairness from complete information to incomplete information, while within the two SD mechanism, the results are opposite. But even in this case, two SD mechanisms are still better than two BOS mechanisms.

As a whole, our paper finds evidence against any superiority of BOS mechanism under incomplete information over other mechanisms either on ex ante efficiency or fairness. Although in theory these superiorities are possible (Abdulkadiroglu, et al., 2011; Lien, et al., 2015), it seems that the prerequisite is too restricted to be satisfied in the real world, at least in situations similar to our experimental set-up. The only promising result for BOS mechanism under incomeplete information emerges when we enlarge the score uncertainty, then we have this mechanism outperform all the others by one of the two measures of ex ante fairness we adopt.

Any future research either in theory or empirics should try to identify conditions for the superiority or inferiority of the BOS mechanism under incomplete information, which would give us more insights over all those mechanisms, and advise more wisely to policy makers on whether and when they can use preference submission timing as a tool to improve matching quality.

18

References [1] Abdulkadiroglu, Atila, Yeon-Koo Che, and Yosuke Yasuda, Resolving

Conflicting Preferences in School Choice: the “Boston” Mechanism Reconsidered, American Economic Review, 101(2011):399-410.

[2] Abdulkadiroglu, Atila, Parag A. Pathak, and Alvin E. Roth, Strategy-proofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match, American Economic Review, 99:5 (2009), pp. 1954–1978.

[3] Abdulkadiroglu, Atlia, and Tayfun Sonmez, School Choice: A Mechanism Design Approach, American Economic Review, 93:3 (2003), pp. 729-747.

[4] Chen, Yan, and Onur Kesten, From Boston to Chinese Parallel to Deferred Acceptance: Theory and Experiments on a Family of School Choice Mechanisms, Working paper, January 2013.

[5] Chen, Yan, and Tayfun Sonmez, School choice: an experimental study, Journal of Economic Theory, 127 (2006), pp. 202-231

[6] Edril, Aytek, and Haluk Ergin, What’s the Matter with Tie-Breaking? Improving Efficiency in School Choice, American Economic Review, 98:3 (2008), pp. 669-689.

[7] Ehlers, Lars, and Jordi Masso, Matching Markets under (In)complete Information, Journal of Economic Theory,157(2015), p.295-314.

[8] Ergin, Haluk, and Tayfun Sonmez, Games of School Choice under the Boston Mechanism, Journal of Public Economics, 90(2006), pp. 215-237.

[9] Featherstone, Clayton, and Muriel Niederle, Improving on Strategy-proof School Choice Mechanisms: An Experimental Investigation, Working paper, July 2013.

[10] He, Yinghua, Gaming the Boston School Choice Mechanism in Beijing, 2012. Manuscript, Toulouse School of Economics.

[11] Klijn, Flip, Joana Pais, and Marc Vorsatz, Preference intensities and risk aversion in school choice: A laboratory experiment, Experimental Economics, 16(2013), p.1-22.

[12] Lien, Jaimie W., Jie Zheng, and Xiaohan Zhong, Preference Submission Timing in School Choice Matching: Testing Fairness and Efficiency in the Laboratory. Experimental Economics, 19 (2016), p. 116 - 150.

[13] Lien, Jaimie W., Jie Zheng, and Xiaohan Zhong, Ex-ante Fairness in the Boston and Serial Dictatorship Mechanisms under Pre-exam and Post-exam Preference Submission, Working paper, 2015.

[14] Pais, Joana, and Agnes Pinter, School Choice and Information: An Experimental Study on Matching Mechanisms, Games and Economic Behavior, 64(2008), pp. 303-328.

[15] Tanaka, Tomomi, Colin F. Camerer, and Quang Nguyen, Risk and Time Preferences: Linking Experimental and Household Survey Data from Vietnam, American Economic Review, 100:1 (2010), pp. 557-571.

[16] Wu, Binzhen, and Xiaohan Zhong, Matching Mechanisms and Matching Quality: Evidence from a Top University in China, Games and Economic Behavior, 84(2014), pp. 196-215.

[17] Wu, Binzhen, and Xiaohan Zhong, Mismatch in China’s College Admission,

19

Working paper, 2015.

20

Figures

Figure1 Distribution of Degree of Misreport

020

4060

800

2040

6080

-4 -2 0 2 4 -4 -2 0 2 4

BOS_I BOS_C

SD_I SD_C

Perc

ent

d_reportGraphs by treatment

21

Tables Table 1:Truth-tellingunder Different Mechanisms

Panel A: Proportion of truth-telling

Mechanism % of Truth-telling # of Observations

BOS_I 0.93 108 BOS_C 11.11 72 SD_I 64.81 108 SD_C 69.44 72

Panel B: T-test result of difference in proportion of truth-telling

SD_C ≈SD_I > BOS_C > BOS_I (p=0.2606) (p=0.0000) (p=0.0010)

22

Table 2: Determinants of Truth-telling: Logit Model

Independent Variable Dependent Variable:

Truth-telling

(1) (2) (3)

Boston -0.5833*** -0.5823*** -0.5949*** (BOS_C-SD_C) (0.0646) (0.0646) (0.0626)

Incomplete -0.0463 -0.03539 -0.0475 (SD_I-SD_C) (0.0696) (0.0730) (0.0720)

Boston*Incomplete -0.05556 -0.0629 -0.0471 (BOS_I-BOS_C)-(SD_I-SD_C) (0.0792) (0.0805) (0.0793)

Rank -0.00344 -0.00336 -0.00263 (0.00237) (0.00239) (0.00240)

SD-Fair School Slots -0.0181 -0.0191 -0.0241 (0.0156) (0.0158) (0.0162)

Female -0.000912 0.00214 (0.0405) (0.0405)

Age -0.00154 0.000496 (0.0147) (0.0145)

Econ 0.0153 0.0245 (0.0550) (0.0548)

Engineer 0.0350 0.0506 (0.0572) (0.0574)

Science -0.00267 -0.00205 (0.0734) (0.0735)

Sigma 0.208*** (0.0763)

Alpha -0.0926 (0.0810)

Lambda 0.00997 (0.0124)

Observations 360 360 360 Pseudo R-squared 0.4022 0.4034 0.4186

BOS_I-BOS_C -0.1019*** -0.0982*** -0.0946*** (0.0377) (0.0374) (0.0362)

SD_I-BOS_C 0.5370*** 0.5469*** 0.5475*** (0.0579) (0.0594) (0.0583)

Note: Coefficients report average marginal effects. Delta-method standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1.λis the average value of its lower and upper bound. We also run regressions with the lower bound and upper bound of λ, and the results are similar

23

Table 3: Non-truth-telling Reporting Strategies under Different Mechanisms Panel A: Proportion of different reporting strategies and average degree of misreport

Mechanism

# % Fair-reporting Up-reporting Down-reporting Degree of misreport

# % # % # % Mean BOS_I 107 99.07 65 60.8 24 22.4 18 16.8 0.1215 BOS_C 64 88.89 45 70.3 8 12.5 11 17.2 -0.0313

SD_I 38 35.19 18 47.4 11 29 9 23.7 0.1579 SD_C 22 30.56 14 63.6 5 22.7 3 13.6 0.0454 Panel B: Significance test of difference in proportion of different reporting strategies and average

degree of misreport Reporting Strategy

Fair-reporting BOS_C≈ BOS_I ≈ SD_C ≈ SD_I (p=0.2087) (p=0.8019) (p=0.2305)

Up-reporting SD_I≈ SD_C ≈ BOS_I> BOS_C (p=0.6068) (p=0.9760) (p=0.0542)

Down-reporting SD_I≈ BOS_C ≈ BOS_I≈ SD_C (p=0.4293) (p=0.9513) (p=0.7150)

Degree of misreport SD_I≈ BOS_I ≈ SD_C≈ BOS_C (p=0.8531) (p=0.7627) (p=0.7555)

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Table 4: Regression of Degree of Misreport for Non-truth-telling Students

Independent Variable

Dependent Variable: Degree of Misreport

OLS Ordered Probit

(1) (2) (3) (4) (5) (6)

Boston -0.0720 -0.0706 -0.153 -0.131 -0.130 -0.220

(BOS_C-SD_C) (0.250) (0.248) (0.250) (0.274) (0.274) (0.278)

Incomplete 0.116 0.128 0.0985 0.0354 0.0553 0.0114

(SD_I-SD_C) (0.271) (0.269) (0.279) (0.293) (0.294) (0.307)

Boston*Incomplete 0.0126 0.0115 0.101 0.0941 0.0864 0.176

(BOS_I-BOS_C)-(SD_I-SD_C) (0.315) (0.312) (0.317) (0.341) (0.342) (0.349)

Rank -0.0171** -0.0272*** -0.0264*** -0.0202*** -0.0331*** -0.0329***

(0.00678) (0.00838) (0.00855) (0.00754) (0.00943) (0.00968)

SD-Fair School Slots 0.112** 0.104* 0.140** 0.134**

(0.0550) (0.0556) (0.0608) (0.0620)

Female -0.0949 -0.0808

(0.144) (0.158)

Age 0.00718 0.00723

(0.0480) (0.0530)

Econ 0.425** 0.427**

(0.189) (0.210)

Engineer 0.326 0.330

(0.198) (0.219)

Science 0.0645 -0.0626

(0.291) (0.318)

Sigma 0.0227 0.103

(0.300) (0.330)

Alpha 0.0209 -0.0181

(0.321) (0.353)

Lambda -0.0663 -0.0737

(0.0468) (0.0514)

Observations 231 231 231 231 231 231

(Pseudo)R-squared 0.032 0.050 0.091 0.0141 0.0233 0.0394

BOS_I-BOS_C 0.1282 0.1394 0.1997 0.1295 0.1417 0.1870

(0.1600) (0.1590) (0.1670) (0.1743) (0.1748) (0.1852)

BOS_C-SD_I -0.1875 -0.1985 -0.2513 -0.1665 -0.1854 -0.2317

(0.2070) (0.2056) (0.2178) (0.2231) (0.2238) (0.2395)

Note: Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1.λis the average value of its lower and upper bound. We also run regression with the lower bound and upper bound of λ, and the results are similar.

25

Table 5: Summary Statistics for the Degree of Mismatch under Different Reporting Strategies

Panel A: Summary statistics

Mechanism Report Strategy Degree of mismatch

# of Obs. Mean SD Min Max

BOS_I

Truth-telling 1 0 . 0 0 Up-reporting 24 -0.29 1.57 -4 2 Fair-reporting 65 -0.02 0.12 -1 0

Down-reporting 18 -1.22 0.73 -3 0

BOS_C

Truth-telling 8 0.13 0.83 -1 2 Up-reporting 8 0.50 1.31 -2 2 Fair-reporting 45 -0.02 0.15 -1 0

Down-reporting 11 -1.55 0.69 -3 -1

SD_I

Truth-telling 70 0.00 0.24 -1 1 Up-reporting 11 0.09 0.70 -1 2 Fair-reporting 18 -0.06 0.24 -1 0

Down-reporting 9 -1.33 0.87 -3 0

SD_C

Truth-telling 50 0.02 0.14 0 1 Up-reporting 5 0.40 0.89 0 2 Fair-reporting 14 0.00 0.00 0 0

Down-reporting 3 -1.67 0.58 -2 -1

Panel B: Significance Test of differences in the degree of mismatch

Mechanism BOS_I Truth-telling ~ Up-reporting ≈ Fair-reporting>Down-reporting

(n.a.) (p=0.9257) (p=0.0000) BOS_C Up-reporting≈Truth-telling > Fair-reporting>Down-reporting

(p=0.7973) (p=0.0411) (p=0.0000) SD_I Up-reporting≈Truth-telling ≈ Fair-reporting>Down-reporting

(P=1.000) (p=0.4270) (p=0.0000) SD_C Up-reporting≈Truth-telling > Fair-reporting>Down-reporting

(p=0.9154) (p=0.0416) (p=0.0000) All Mechanisms Up-reporting≈Truth-telling > Fair-reporting>Down-reporting

(p=0.8995) (p=0.0851) (p=0.0000)

26

Table 6: OLS Regression of the Degree of Mismatch on Misreport

Independent

Variable

Dependent Variable

Degree of Mismatch Variance of Degree of Mismatch

BOS_I BOS_C SD_I SD_C BOS_I BOS_C SD_I SD_C

(1) (2) (3) (4) (5) (6) (7) (8)

Truth-telling 0.153 0.146 0.0565 0.0133 1.143 0.585** 0.0419 0.0127

(0.890) (0.265) (0.111) (0.0841) (1.461) (0.240) (0.117) (0.0557)

Up-report -0.327 0.341 0.119 0.390** 1.840*** 0.806*** 0.411** 0.379***

(0.201) (0.242) (0.156) (0.157) (0.330) (0.219) (0.165) (0.104)

Down-report -1.292*** -1.446*** -1.279*** -1.798*** 0.0721 0.0812 0.379** 0.112

(0.238) (0.206) (0.173) (0.194) (0.390) (0.187) (0.182) (0.129)

Rank -0.000997 0.0228** 0.00127 0.00915** 0.0338* 0.00731 0.0205*** 0.00651**

(0.0109) (0.00944) (0.00542) (0.00424) (0.0179) (0.00856) (0.00571) (0.00281)

SD-Fair

School Slots

0.0509 -0.0964 -0.00445 -0.0343 -0.124 0.0683 -0.0762** -0.0228

(0.0683) (0.0634) (0.0334) (0.0288) (0.112) (0.0574) (0.0352) (0.0191)

Female -0.302* 0.283* -0.165** 0.0826 0.00765 -0.304** -0.00983 0.0792

(0.172) (0.160) (0.0806) (0.0795) (0.283) (0.145) (0.0850) (0.0527)

Age 0.0701 0.0516 0.0202 -0.0439* -0.207** 0.00231 -0.0257 -0.0395**

(0.0614) (0.0457) (0.0332) (0.0242) (0.101) (0.0414) (0.0350) (0.0160)

Econ 0.0885 0.195 -0.00920 0.0600 -0.124 -0.165 -0.151 0.0228

(0.207) (0.319) (0.108) (0.114) (0.341) (0.289) (0.113) (0.0756)

Engineer -0.218 -0.00320 0.0315 0.194 0.648* -0.185 -0.0325 0.141

(0.217) (0.308) (0.112) (0.128) (0.356) (0.279) (0.118) (0.0850)

Science 0.719 0.100 0.0964 0.112 0.0410 -0.235 -0.0987 0.0734

(0.499) (0.350) (0.159) (0.150) (0.819) (0.317) (0.168) (0.0991)

Sigma -0.404 0.221 -0.00448 -0.00749 0.0804 0.333 -0.0644 0.00232

(0.357) (0.367) (0.160) (0.141) (0.586) (0.333) (0.168) (0.0936)

Alpha 0.453 0.737** 0.0425 -0.140 0.208 -0.459 0.0489 -0.0915

(0.387) (0.354) (0.174) (0.145) (0.634) (0.321) (0.183) (0.0963)

Lambda 0.00870 -0.0284 0.0212 0.00658 -0.0302 0.0565 0.00577 0.00529

(0.0516) (0.0548) (0.0288) (0.0216) (0.0848) (0.0497) (0.0303) (0.0143)

Constant -1.626 -1.775* -0.474 0.861 4.122* -0.218 0.684 0.762**

(1.421) (0.985) (0.739) (0.523) (2.332) (0.892) (0.780) (0.347)

Observations 108 72 108 72 108 72 108 72

R-squared 0.319 0.641 0.524 0.713 0.354 0.339 0.256 0.418

Note: Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1.λis the average value of its lower and upper bound. We also run regression with the lower bound and upper bound of λ, and the results are similar.

27

Table 7: Proportions of Top Choice Match in different mechanisms

Truth-telling Non-truth-telling

Treatment # Top Choice Match % Top Choice Match Total # Top Choice Match % Top Choice Match Total

BOS_I 1 100.00 1 87 81.31 107

BOS_C 7 87.50 8 57 89.06 64

SD_I 13 18.57 70 23 60.53 38

SD_C 8 16.00 50 15 68.18 22

Total 29 22.48 129 182 78.79 231

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Table 8: Simulation Result of Payoff per capita Panel A: Payoff per capita

Scenario BOS_I BOS_C SD_I SD_C

1 10.9556 10.97899 11.23236 11.31417

2 10.33074 10.64639 10.79264 11.04257

3 10.49718 10.77118 10.99384 11.12069

4 10.54583 10.82056 10.98083 11.16021

Panel B: Significance test of the differences in efficiency

Scenario

1 BOS_I<BOS_C < SD_I<SD_C

(p=0.0434) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)

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Table 9: Simulation Result of Ex-post Fairness by Number of Blocking Pairs per

capita Panel A: Number of blocking pairs per capita in different mechanisms


1 0.453704 0.561181 0.109259 0.092847

2 0.817315 0.597014 0.297407 0.184861

3 0.732454 0.514653 0.212778 0.150625

4 0.725926 0.502014 0.216852 0.151111

Panel B: Significance test of the differences in number of blocking pairs per capita

Scenario

1 BOS_C>BOS_I > SD_I>SD_C

(p=0.0000) (p=0.0000) (p=0.0000)

2 BOS_I>BOS_C > SD_I>SD_C

(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)

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Table 10:Simulation Result of Ex-post Fairness by Absolute Value of Degree of

Mismatch Panel A: Absolute value of degree of mismatch per capita in different mechanisms


1 0.58412 0.533143 0.240463 0.208056

2 0.882269 0.541736 0.502454 0.319444

3 0.803241 0.478889 0.395648 0.2775

4 0.786389 0.46125 0.403843 0.273125

Panel B: Significance test of the differences in absolute value of degree of mismatch per capita

Scenario


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)

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Table11: Simulation Result of Ex-ante Fairness by Number of Blocking Pairs per

capita Panel A: Number of blocking pairs per capita


1 0.462824 0.650972 0.340694 0.342917

2 0.818241 0.795556 0.551944 0.455625

3 0.724398 0.707431 0.461713 0.430417

4 0.709398 0.701597 0.46463 0.423194


Scenario

1 BOS_C>BOS_I > SD_I≈SD_C

(p=0.0000) (p=0.0000) (p=0.7958)


(p=0.0963) (p=0.0000) (p=0.0000)

3 BOS_I≈BOS_C > SD_I>SD_C

(p=0.2762) (p=0.0000) (p=0.0007)


(p=0.6258) (p=0.0000) (p=0.0001)

32

Table 12:Simulation Result of Ex-ante Fairness by Absolute Value of Degree of Mismatch

Panel A: Absolute value of degree of mismatch per capita


1 0.558194 0.640288 0.513657 0.489931

2 0.711898 0.815069 0.688056 0.623333

3 0.633333 0.755069 0.588935 0.588333

4 0.605741 0.743819 0.597222 0.5775


Scenario


(p=0.0000) (p=0.0000) (p=0.0030)


(p=0.0000) (p=0.0053) (p=0.0011)

3 BOS_C>BOS_I > SD_I≈SD_C

(p=0.0000) (p=0.0000) (p=0.9491)

4 BOS_C>BOS_I ≈ SD_I>SD_C

(p=0.0000) (p=0.3368) (p=0.0308)

33

Table A1: OLS Regression of Efficiency under Different Mechanisms

Independent Variable Dependent Variable: Payoff per capita

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Boston -0.345*** -0.396*** -0.350** -0.340*

(BOS_C-SD_C) (0.00302) (0.113) (0.130) (0.168) Incomplete -0.0918*** -0.250*** -0.127*** -0.179*

(SD_I-SD_C) (0.0230) (0.0395) (0.0235) (0.0921) Boston*Incomplete 0.0668 -0.0657 -0.147 -0.0953

(BOS_I-BOS_C)-(SD_I-SD_C) (0.0766) (0.120) (0.132) (0.183) Rank -0.119*** -0.118*** -0.120*** -0.116***

(0.00848) (0.00606) (0.00580) (0.00516) SD-Fair School Slots -0.219*** -0.196*** -0.183*** -0.194***

(0.0537) (0.0406) (0.0408) (0.0361) Constant 14.76*** 14.34*** 14.39*** 14.41***

(0.205) (0.137) (0.144) (0.137) Observations 72,000 72,000 72,000 72,000

R-squared 0.359 0.243 0.286 0.273 BOS_I-BOS_C 0.0250 -0.3156*** -0.2740* -0.2747

(0.0730) (0.1128) (0.1302) (0.1582) BOS_C-SD_I -0.2529*** -0.1462 -0.2227 -0.1603

(0.0232) (0.1186) (0.1285) (0.173) BOS_I-SD_I -0.2779*** -0.4619*** -0.4967*** -0.435***

(0.0765) (0.0382) (0.0219) (0.0723) Note: *** p<0.01, ** p<0.05, * p<0.1. Standard errors in parentheses, clustered at the session level.

34

Table A2: Regression of Ex-post Fairness by Different Measures

Independent

Variable

Dependent Variable

Absolute value of degree of mismatch

(Ordered Probit)

Number of blocking pairs

(OLS)

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 1 Scenario 2 Scenario 3 Scenario 4

Boston 0.741*** 0.421*** 0.415*** 0.429** 0.463*** 0.412*** 0.364*** 0.351**

(BOS_C-SD_C) (0.213) (0.122) (0.142) (0.180) (0.0899) (0.102) (0.105) (0.121)

Incomplete 0.182 0.358*** 0.277*** 0.327*** 0.0178 0.113*** 0.0622*** 0.0657*

(SD_I-SD_C) (0.166) (0.0303) (0.0578) (0.0932) (0.0235) (0.0166) (0.0165) (0.0331)

Boston*Incomplete -0.123 0.0406 0.139 0.0898 -0.122 0.108 0.156 0.158

(0.215) (0.121) (0.141) (0.182) (0.0944) (0.104) (0.107) (0.126)

Rank 0.0252*** 0.0111*** 0.0123*** 0.0114*** -0.00145 -0.0119*** -0.0109*** -0.0114***

(0.00754) (0.00289) (0.00304) (0.00272) (0.00329) (0.00267) (0.00251) (0.00247)

SD-Fair School

Slots

-0.0776*** -0.0342*** -0.0377*** -0.0408*** 0.00782 0.0391** 0.0397** 0.0377***

(0.0201) (0.00931) (0.00701) (0.00577) (0.0162) (0.0130) (0.0123) (0.0115)

Observations 72,000 72,000 72,000 72,000 72,000 72,000 72,000 72,000

(Pseudo)R-squared 0.0511 0.0285 0.0338 0.0325 0.094 0.086 0.101 0.094

BOS_I-BOS_C 0.0598 0.3988*** 0.4163*** 0.4167*** -0.1043 0.2203** 0.2178* 0.2239*

(0.1372) (0.1166) (0.1287) (0.1569) (0.0914) (0.1024) (0.1054) (0.1218)

BOS_C-SD_I 0.5584*** 0.0632 0.1385 0.1017 0.4454*** 0.2996** 0.3019** 0.2852**

(0.1298) (0.1189) (0.1302) (0.1626) (0.0870) (0.1032) (0.1042) (0.1217)

BOS_I-SD_I 0.6182*** 0.4620*** 0.5548*** 0.5185*** 0.3411*** 0.5199*** 0.5197*** 0.5091***

(0.0394) (0.0165) (0.0105) (0.0419) (0.0287) (0.0169) (0.0187) (0.0367)

Note: *** p<0.01, ** p<0.05, * p<0.1. Standard errors in parentheses, clustered at the session level.

35

Table A3: Regression of Ex-ante Fairness by Different Measures

Independent

Variable

Dependent Variable

Absolute value of degree of mismatch

(Ordered Probit)

Number of blocking pairs

(OLS)

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 1 Scenario 2 Scenario 3 Scenario 4

Boston 0.221*** 0.232*** 0.218*** 0.226** 0.303*** 0.340*** 0.277** 0.278**

(BOS_C-SD_C) (0.0101) (0.0712) (0.0745) (0.0915) (0.0115) (0.0931) (0.0935) (0.109)

Incomplete 0.0212 0.0807*** 0.00913 0.0402 -0.00134 0.0963*** 0.0313 0.0414

(SD_I-SD_C) (0.0339) (0.0296) (0.0244) (0.0600) (0.0198) (0.0201) (0.0176) (0.0437)

Boston*Incomplete -0.158*** -0.227*** -0.178** -0.233** -0.194*** -0.0736 -0.0143 -0.0336

(0.0402) (0.0791) (0.0775) (0.105) (0.0331) (0.0956) (0.0941) (0.115)

Rank 0.0164*** 0.00662*** 0.00759*** 0.00493*** -0.00683** -0.0151*** -0.0134*** -0.0143***

(0.00367) (0.00112) (0.00139) (0.00119) (0.00274) (0.00426) (0.00390) (0.00384)

SD-Fair School

Slots

-0.116*** -0.0910*** -0.105*** -0.0948*** -0.0483* -0.0284 -0.0355 -0.0349

(0.0156) (0.0110) (0.00984) (0.0106) (0.0230) (0.0268) (0.0261) (0.0252)

Observations 72,000 72,000 72,000 72,000 72,000 72,000 72,000 72,000

(Pseudo)R-squared 0.0124 0.00668 0.00907 0.00854 0.056 0.054 0.059 0.059

BOS_I-BOS_C -0.137*** -0.1460** -0.1686** -0.1929** -0.195*** 0.02268 0.0169 0.0078

(0.0206) (0.0735) (0.0739) (0.0868) (0.0265) (0.0935) (0.0924) (0.1060)

BOS_C-SD_I 0.1994*** 0.151** 0.209*** 0.189* 0.3045*** 0.244** 0.247** 0.237*

(0.0325) (0.0749) (0.0743) (0.1014) (0.0203) (0.0946) (0.0929) (0.1109)

BOS_I-SD_I 0.0628 0.0051 0.0402* -0.0072 0.109*** 0.266*** 0.263*** 0.245***

(0.0398) (0.0356) (0.0237) (0.0529) (0.0310) (0.0219) (0.0105) (0.0367)

Note: *** p<0.01, ** p<0.05, * p<0.1. Standard errors in parentheses, clustered at the session level.

36

Table A4: Simulation Result of Payoff per capita with Score Fluctuation Range

-20 to +20 Panel A: Payoff per capita


1 10.92069 10.96375 11.2075 11.27708

2 10.34148 10.62299 10.74042 11.02271

3 10.49005 10.72687 10.93519 11.10313

4 10.5081 10.77194 10.91375 11.09889

Panel B: Significance test of the differences in efficiency

Scenario


(p=0.0007) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)

37

Table A5: Simulation Result of Ex-post Fairness by Number of Blocking Pairs

per capita with Score Fluctuation Range -20 to +20 Panel A: Number of blocking pairs per capita in different mechanisms


1 0.621759 0.684583 0.137778 0.107292

2 0.921713 0.571806 0.332407 0.189028

3 0.876759 0.533611 0.238426 0.154792

4 0.863333 0.522917 0.254028 0.157153


Scenario


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)

38

Table A6: Simulation Result of Ex-post Fairness by Absolute Value of Degree of

Mismatch with Score Fluctuation Range -20 to +20 Panel A: Absolute value of degree of mismatch per capita in different mechanisms


1 0.672454 0.600875 0.294306 0.220208

2 0.945324 0.54 0.551111 0.325694

3 0.889213 0.492569 0.436157 0.283958

4 0.871111 0.4675 0.466389 0.274236


Scenario


(p=0.0000) (p=0.0000) (p=0.0000)

2 BOS_I>BOS_C ≈ SD_I>SD_C

(p=0.0000) (p=0.2901) (p=0.0000)


(p=0.0000) (p=0.0000) (p=0.0000)

4 BOS_I>BOS_C ≈ SD_I>SD_C

(p=0.0000) (p=0.9154) (p=0.0000)

39

Table A7: Simulation Result of Ex-ante Fairness by Number of Blocking Pairs per

capita with Score Fluctuation Range -20 to +20 Panel A: Number of blocking pairs per capita


1 0.578889 0.859306 0.598287 0.630694

2 1.018287 0.975764 0.828241 0.711944

3 0.93787 0.934583 0.75375 0.675139

4 0.939074 0.937778 0.761991 0.695833


Scenario

1 BOS_C>SD_C > SD_I>BOS_I

(p=0.0000) (p=0.0117) (p=0.0396)


(p=0.0155) (p=0.0000) (p=0.0000)


(p=0.8663) (p=0.0000) (p=0.0000)


(p=0.9463) (p=0.0000) (p=0.0000)

40

Table A8: Simulation Result of Ex-ante Fairness by Absolute Value of Degree of Mismatch with Score Fluctuation Range -20 to +20 Panel A: Absolute value of degree of mismatch per capita


1 0.617824 0.753258 0.624769 0.639583

2 0.757546 0.920069 0.814722 0.750417

3 0.680926 0.872153 0.717546 0.706875

4 0.674352 0.850486 0.727315 0.716736


Scenario

1 BOS_C>SD_C > SD_I≈BOS_I

(p=0.0000) (p=0.0508) (p=0.2976)

2 BOS_C>SD_I > BOS_I≈SD_C

(p=0.0000) (p=0.0000) (p=0.4833)

3 BOS_C>SD_I ≈ SD_C>BOS_I

(p=0.0000) (p=0.3066) (p=0.0019)

4 BOS_C>SD_I ≈ SD_C>BOS_I

(p=0.0000) (p=0.3219) (p=0.000)

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