Information acquisition and provision in school choice: a theoretical investigation

When participating in school choice, students may incur information acquisition costs to learn about school quality. This paper investigates how two popular school choice mechanisms, the (Boston) Immediate Acceptance and the Deferred Acceptance, incentivize students’ information acquisition. Specifically, we show that only the Immediate Acceptance mechanism incentivizes students to learn their own cardinal and others’ preferences. We demonstrate that information acquisition costs affect the efficiency of each mechanism and the welfare ranking between the two. In the case where everyone has the same ordinal preferences, we evaluate the welfare effects of various information provision policies by education authorities.


Introduction
When choosing a school, students often have imperfect information on their own preferences over candidate schools, partly because it is difficult to assess the potential educational outcomes for each school (Dustan et al. 2015). More importantly, acquiring this information can be costly, if a student faces too many choices, or must acquire information on a large number of factors, such as academic performance, teacher quality, school facilities, extra-curricular activities offered, and peer quality.
The literature on matching and school choice, however, typically assumes that all students have perfect knowledge about their own preferences, at least their ordinal ones. Relaxing this assumption, our study extends the literature by investigating how school choice mechanisms incentivize student information acquisition and how information provision by educational authorities affects efficiency. Specifically, we focus on two widely used mechanisms, the (Boston) Immediate-Acceptance (hereafter IA) and the Gale-Shapley Deferred-Acceptance (hereafter DA) mechanisms. By taking into account both the benefits and costs of information acquisition, this study provides a more comprehensive evaluation of the mechanisms and as a result, provides some guidance for the design of school choice or other matching markets.
Our first contribution is to show that IA and DA provide heterogeneous incentives for students to acquire information. In a setting with unknown preferences and costly information acquisition, we prove that both the strategy-proof DA and the non-strategyproof IA incentivize students to acquire information on their own ordinal preferences. However, we find that only the non-strategy-proof mechanism induces students to learn their own cardinal preferences with which IA can sometimes be more efficient than DA (Abdulkadiroglu et al. 2011;Troyan 2012). IA's lack of strategy-proofness also implies that information on others' preferences can be useful for the purpose of competing with other students. As such, the acquisition of information on others' preferences may be individually rational but socially wasteful, a disadvantage of a non-strategy-proof mechanism.
Although the above results may seem obvious, to the best of our knowledge, they have not yet been formalized in the literature. More importantly, they lead to new implications for the study of the mechanisms. For example, the welfare comparison of the two mechanisms is sensitive to costly information acquisition. Taking into account endogenous information acquisition, we provide two numerical examples showing that the cost of information acquisition affects student welfare in equilibrium. In both examples, IA achieves higher student welfare than DA when students' cardinal preferences are private information (i.e., zero information acquisition cost), a finding similar to Abdulkadiroglu et al. (2011) but in a setting with students having potentially heterogeneous ordinal preferences. As the cost of acquiring information on own preferences increases, the welfare advantage of IA diminishes to zero in the first example, while the welfare ranking between the two mechanisms flips in the second example for some cost configurations.
Extending these findings, our second contribution is to present some implications for the design of information provision policies. For example, a possible policy intervention is to provide information freely. We investigate the welfare effects of information provision by education authorities. Specifically, we consider four sets of policies with increasing information provision. The least informative policy forbids everyone from acquiring any information beyond the distribution of preferences. The second policy informs everyone about her own ordinal preferences, and the third reveals one's own cardinal preferences. The most informative policy makes everyone's cardinal preferences common knowledge. The information on a student's own preferences might be provided through presentation materials on schools (Hastings and Weinstein 2008) or by targeting disadvantaged population (Hoxby and Turner 2015). The information on others' preferences can be (indirectly) provided by publishing everyone's applications and allowing one to revise her own application upon observing others' strategies, as has been done in the school choice context in Amsterdam (De Haan et al. 2015) and North Carolina (Dur et al. 2018), as well as in the college admissions context in Inner Mongolia, China (Gong and Liang 2017).
In a setting where students have the same ordinal preferences, we analyze symmetric equilibrium under the four information provision policies. We show that the ex ante student welfare under DA is invariant to the policies while providing information on one's own cardinal preferences improves welfare under IA.
Interestingly, we find that provision of information on others' preferences has ambiguous effects under IA, implying that sometimes providing more information on others' preferences can be welfare-decreasing under IA. The reason is that, knowing there is fierce competition for a school, students who prefer that school may shy away from applying to it, and other students may be assigned that school with a positive probability. As a result, relative to the case with cardinal preferences being private information, there is a loss in (ex ante) student welfare evaluated before the realization of each student's type.
We illustrate this welfare loss in an example with three students and three schools, {s 1 , s 2 , s 3 }, with the details in "Appendix A.5.5". Each student's cardinal preferences, (v 1 , v 2 , v 3 ), are i.i.d. and equal to (1, 0.1, 0) with probability 3/4 and (1, 0.9, 0) with probability 1/4. When cardinal preferences are private information, a symmetric equilibrium strategy is to submit rank-order list (s 1 , s 2 , s 3 ) for students of type-(1, 0.1, 0) and submit (s 2 , s 1 , s 3 ) for those of type-(1, 0.9, 0). Therefore, whenever the realized game has at least one student of type-(1, 0.9, 0), s 2 will never be assigned to a student of type-(1, 0.1, 0). In contrast, when cardinal preferences are common knowledge, in a realized game that has one student of type-(1, 0.1, 0) and two of type-(1, 0.9, 0), a type-(1, 0.9, 0) student may shy away from applying to s 2 because of the competition from the other student of the same type. Specifically, a symmetric equilibrium strategy in this realized game is to submit rank-order list (s 1 , s 2 , s 3 ) for the student of type-(1, 0.1, 0) and, for those of type-(1, 0.9, 0), submit (s 2 , s 1 , s 3 ) with probability 16/19 and (s 1 , s 2 , s 3 ) with probability 3/19. As a result, in this realized game, there is a positive probability that the student of type-(1, 0.1, 0) is assigned s 2 . This lowers the ex ante welfare that is evaluated before the realization of each student's type.
The paper proceeds as follows. Section 2 reviews the related literature on information acquisition and/or school choice. Section 3 presents the theoretical results on information acquisition, and Sect. 4 discusses those on information provision. Section 5 discusses possible extensions and concludes.

Literature review
This study contributes to the matching literature. Typically, this literature assumes that agents know their preferences (Gale and Shapley 1962;Roth and Sotomayor 1990;Abdulkadiroglu and Sönmez 2003). One exception is Chade et al. (2014), who consider the case where colleges observe signals of students' ability but do not have the possibility to acquire information. Allowing the possibility of information acquisition, Lee and Schwarz (2012) and Rastegari et al. (2013) study settings where firm preferences over workers are not completely known and are revealed only through interviews.
The theoretical papers that address endogenous information acquisition in matching include Bade (2015) and Harless and Manjunath (2015). In the context of house allocations, Bade finds that the unique ex ante Pareto optimal, strategy-proof, and non-bossy allocation mechanism is that of serial dictatorship. However, in their study, Harless and Manjunath (2015) prove that the top-trading-cycles mechanism dominates the serial dictatorship mechanism under progressive measures of social welfare. Both papers focus on ordinal mechanisms. 1 As we show below, in any strategy-proof ordinal mechanism, students have no incentives to learn their cardinal preferences beyond the ordinal ones, while information on cardinal preferences can be welfare-improving, especially when students have similar ordinal preferences (Abdulkadiroglu et al. 2011). Lastly, in an ongoing study, Artemov (2016) considers an environment similar to our experimental setting to compare the performance of IA and DA.
Another unique feature of our study is the acquisition of information on others' preferences, which is in contrast with other studies that focus on the acquisition of information on one's own preferences. One exception in this body of literature is Kim (2008), who considers a common-value first-price auction with two bidders, one of whom learns her opponent's signal.
Our setting does not have pre-determined priorities. When students do have such priorities, e.g., priorities determined by siblings' school attendance or test score, they may have incentives to acquire information on others' preferences and priorities under DA as shown in Grenet et al. (2019) and Immorlica et al. (2020). The intuition is that others' preferences and priorities help a student assess the probability of being accepted by each school and that a student does not need to learn about schools that will never accept her.
In practice, students or their parents in school choice might acquire information through their social networks. For example, families in Boston, e.g., the West Zone Parents Group, used to share their knowledge about schools and discuss strategies to rank schools with each other (Pathak and Sönmez 2008). Recently, several experimental studies of school choice have examined peer information sharing within networks (Ding and Schotter 2017) or through intergenerational advice (Ding and Schotter 2019). When information is shared within a network, some isolated students under DA receive lower payoffs compared to their well-connected counterparts (Ding and Schotter 2017). When the school choice game is played as an intergenerational game, in which each student can play only once and can pass advice on to the next generation, student play does not converge to the dominant strategy equilibrium under DA (Ding and Schotter 2019). In sum, these studies indicate that acquiring information through social networks may lead to uneven outcomes, indicating that information provision by a trusted authority may help level the playing field.

Information acquisition
In this section, we outline a theoretical model of endogenous information acquisition for one's own and others' preferences under two common school choice mechanisms, the Immediate and Deferred Acceptance mechanisms.

The setup
Our model begins with a finite set of students, I , to be assigned to a finite set of schools, S, through a centralized school choice mechanism. S is supplemented by a "null school" or outside option, s 0 , and S ≡ S ∪ s 0 . For each s ∈ S, there is a finite supply of seats, q s ∈ N, and the total capacity is no more than the total number of students, s∈S q s ≤ |I |, while q s > 0 for all s. By assumption, q s 0 ≥ |I |. Furthermore, we assume that schools rank students by a post-application uniform lottery without pre-defined priorities (single tie-breaking). Therefore, students do not know its realization when they enter the mechanism. An example of this setting is the middle school choice in Beijing (He 2017). This assumption rules out pre-determined priorities in school choice, such as sibling and neighborhood priorities, and therefore there is no information acquisition about priorities. 2 We leave this generalization for future work.
Student i's valuations of schools are an i.i.d. draw from a distribution, F, denoted by a vector Morgenstern utility of school s. For notational convenience, we assume that v i,s 0 = 0 for all i, which implies that every school in S is acceptable to everyone. Therefore, this is an independent-private-value model, and we discuss how our results generalize to common-and interdependent-value models in Sect. 5.
Further, student preferences are strict: For any pair of distinct schools s and t in S, v i,s = v i,t for all i with probability one. We therefore define strict ordinal preferences P on S such that s P i t if and only if v i,s > v i,t . We also augment the set of all possible strict ordinal preferences P with a "null preference" P φ ≡ ∅ denoting that one has no information on her ordinal preference, expressed asP = P ∪ ∅ . The distribution of V conditional on P is denoted by F (V |P), while the probability mass function of P implied by F is G (P|F) (P is finite). We impose a full-support assumption on G (P|F), i.e., G (P|F) > 0, ∀P ∈ P, indicating that every strict ordinal preference ranking is possible given the distribution of cardinal preferences. Necessarily, G(P φ |F) = 0.
In our model, the value of the outside option and the distribution of preferences, F(V ) and thus G(P|F), are always common knowledge. However, in contrast to previous models of school choice, we introduce an information-acquisition stage for each i to learn her own preferences (P i and/or V i ) or others' preferences (V −i ) before entering the mechanism. Because of the independent-private-value nature, learning about others' preferences is only for the purpose of gaming or competing with other students.

Remark 1
Other than the full-support assumption on G, the distribution functions of preferences (F and G) are rather flexible. For example, G can put a close-to-one probability that a particular school, s, is the most-preferred school for a student; this corresponds to the case in which students have a good idea about which school is their most preferred. Moreover, since G is common knowledge, it also implies that school s is also the most popular school among all students. This becomes known to students as they know G.

School choice mechanisms
We focus on two mechanisms popular in both research literature and practice: the Boston Immediate Acceptance and the Gale-Shapley Deferred Acceptance mechanism.
The Immediate Acceptance mechanism (IA) asks students to submit rank-ordered lists (ROL) of schools. Together with the pre-announced capacity of each school, IA uses pre-defined rules to determine the school priority ranking over students and consists of the following rounds: Round 1. Each school considers all students who rank it as their first choice and assigns its seats temporarily in order of their priority at that school until either there is no seat left at that school or no such student left.
Generally, in: Round (k > 1). The kth choice of the students who have not yet been assigned is considered. Each school that still has available seats assigns the remaining seats to students who rank it as kth choice in order of their priority at that school until either there is no seat left at that school or no such student left.
The process terminates after any round k when either every student is assigned a seat at some school, or the only students who remain unassigned have listed no more than k choices. The Gale-Shapley Deferred Acceptance mechanism (DA) can be either studentproposing or school-proposing. We focus on the student-proposing DA mechanism in this study. Specifically, the mechanism collects school capacities and students' ROLs for schools. With strict rankings of schools over students that are determined by pre-specified rules, it proceeds as follows: Round 1. Every student applies to her first choice. Each school rejects the least ranked students in excess of its capacity and temporarily holds the others.
Generally, in: Round (k > 1). Each student who is rejected in Round (k −1) applies to the highestranked school in her ROL that has not rejected her. Each school pools together new applicants and those on hold from Round (k − 1). It then rejects the least ranked students in excess of its capacity. Those who are not rejected are temporarily held.
The process terminates after any Round k when no rejections are issued. Each school is then matched with those students it is currently holding.

Acquiring information on own preferences
We first investigate the incentives to acquire information on one's own value. The timing of the game and the corresponding information structure are described as follows and also in Fig. 1: for each i, but i knows only the value distribution F(V ); (ii) Each individual i decides whether to acquire a signal on her ordinal preferences; If yes, she decides how much to invest in information acquisition, denoted by α ∈ [0,ᾱ]. (iii) If ordinal preferences are learned, she then chooses the investment, β ∈ [0,β], to acquire a signal on her cardinal preferences. (iv) Regardless of the information acquisition decision or outcome, every student plays the school choice game under either IA or DA.
We differentiate between the learning of ordinal and cardinal preferences, as the former represents acquiring coarse information about the schools, whereas the latter represents obtaining more detailed information and therefore is more costly. In a similar vein, the literature on one-sided and two-sided matching usually assumes that agents know their own ordinal preferences (Roth and Sotomayor 1990), while cardinal preferences being possibly unknown due to "limited rationality" (Bogomolnaia and Moulin 2001).

Technology of information acquisition
Information acquisition in our model is covert. That is, i knows that others are engaging in information acquisition, but does not know what information they have acquired.
After the two-stage information acquisition, the information i has is summarized by signals ω i = ω 1,i , ω 2,i ∈P ×V. If i pays (α, β), the distribution of signals is H (ω i |α, β), as outlined below: (learning both ordinal and cardinal).
Together, they imply that In other words, an agent cannot receive anything other than the three types of signals.
Upon observing signal ω i , the posterior distributions of cardinal and ordinal preferences are: where 1 V i (or 1 P i ) is the probability distribution placing probability 1 on point V i (or P i ).

Game of school choice with information acquisition
In our model, after observing the signal ω i , students enter the school choice game under either DA or IA. Each student i submits an ROL denoted by L i ∈ P such that s L i t if and only if s is ranked above t. 5 When i submits L i and others submit L −i , the payoff is represented by: is the vector of the probabilities determined by the mechanism. We further distinguish between two types of mechanisms: strategy-proof and non-strategy-proof. A mechanism is strategy-proof if: that is, reporting true ordinal preferences is a dominant strategy. It is well-known that the student-proposing DA is strategy-proof (Dubins and Freedman 1981;Roth 1982), while IA is not (Abdulkadiroglu and Sönmez 2003).
Under either mechanism, a symmetric Bayesian Nash equilibrium is defined by a tuple (α * , β * (P, α * ) , σ * (ω)) such that, for all i: With her own signal ω, everyone plays a best response, recognizing that others have paid α * −i , β * −i to acquire information. This leads to a value function given .
Here, β * (P, α * ) is the optimal decision given that one has learned her ordinal preference (P) after paying α * to acquire P. (iii) Acquisition of information on ordinal preferences α * ∈ [0,ᾱ], The above expression has already taken into account that the optimal β equals zero if one obtains a signal ω 1 = P φ in the first stage: β * (P φ , α) = 0 for all α.
Given the above, we can now state our existence result in Lemma 1.

Lemma 1 Under DA or IA, a symmetric Bayesian Nash equilibrium exists.
This also leads to our first proposition: Proposition 1 (Information acquisition incentives: own preferences) In any symmetric Bayesian Nash equilibrium (α * , β * (P, α * ) , σ * (ω)) under DA or IA, the following is true:
Remark 2 Similar to the results for DA, students have no incentive to learn their own cardinal preferences under a strategy-proof mechanism that elicits only ordinal preferences, for example, top-trading cycles. Moreover, these results also apply to the school-proposing DA in our setting because it is equivalent to the student-proposing DA when students are only prioritized by a single post-applicant lottery. 6

Acquiring information on others' preferences
We now consider a student's incentive to acquire information on others' preferences.
Here, we assume that everyone knows exactly her own cardinal preferences (V i ) but not others' preferences (V −i ), and that the distribution of V i , F (V i ), is common knowledge with the same properties as before. The purpose of such a setting is to highlight the incentive to collect information for strategic purposes, above and beyond the incentive to learn one's own preferences. The process and technology for information acquisition are depicted in Fig. 2.
To acquire information, student i may pay δ to acquire a signal of , she learns nothing. The distribution of signals and the posterior distribution of preferences are: The technology has the following properties: The cost for information acquisition is e (δ) such that e (0) = 0, e , e > 0 and e (0) < ∞. Similarly, we restrict our attention to δ ∈ 0,δ , where e δ = v.
Information acquisition is again covert. We focus on a symmetric Bayesian Nash equilibrium, (δ * (V ) ,σ * (ω 3 , V )), where: That is, given one's own signal ω 3,i , everyone plays a best response, recognizing that everyone has paid δ * to acquire information (denoted as δ * −i ). We further define the value function given ω 3,i , δ * −i and V i as:
The existence of such an equilibrium can be proven by similar arguments in the proof of Lemma 1, and the properties of information acquisition in equilibrium is summarized as follows: Proposition 2 (Information acquisition incentives: others' preferences) Suppose (δ * (V ) , σ * (ω 3 , V )) is an arbitrary symmetric Bayesian Nash equilibrium under a given mechanism. We have: (ii) There always exists a preference distribution F such that δ * (V ) > 0 under IA for V in some positive-measure set. Remark 3 Similar to the results for DA, students have no incentive to learn others' preferences under a strategy-proof mechanism that elicits either ordinal or cardinal information from students, for example, top-trading cycles and the school-proposing DA in our setting.
In short, this result provides another perspective on strategy-proofness as a desideratum in market design: a strategy-proof mechanism makes the school choice game easier to play by reducing the incentive to acquire information on others' preferences to zero.

Welfare effects of information acquisition: two examples
By considering the cost of information acquisition, our setting shows that the welfare comparison between the two mechanisms is sensitive to information acquisition. Below, we provide two numerical examples. As information acquisition becomes more costly, the welfare advantage of IA diminishes in the first example, while the welfare order between IA and DA flips in the second example.
Neither example allows students to acquire information on others' preferences due to computational difficulties in solving for an equilibrium. 7 We allow students to acquire their own ordinal preferences and then possibly their own cardinal preferences. The following three information structures are useful for our discussion: (i) uninformed (UI) in which every student only knows the distribution of preferences, (ii) ordinally informed (OI) in which everyone's own ordinal preferences are private information, and (iii) cardinally informed (CI) in which everyone's own cardinal preferences are private information. In all three information structures, the preference distribution is common knowledge.
Example 1: The advantage of IA over DA diminishes with information acquisition costs. Let us start with an example in which IA dominates DA in welfare given the information structure being either CI or OI. There are two schools. Each school has one seat, and student preference distribution is described in Table 1. There are three students, and each student's preferences are an i.i.d. draw from the distribution. 8 With the information structure UI, students submit (s 1 , s 2 ) in equilibrium under either IA or DA and receive 0.44 in terms of expected utility. However, with either OI or CI, IA dominates DA in a symmetric Bayesian Nash equilibrium. 9 The technology of information acquisition is the same in Sect. 3.3 (in particular, Fig. 1), and there is no possibility of acquiring information on others' preferences. We further specify that a(α) = √ α k (for ordinal information) and b(β) = √ β 10k 2 (for cardinal information). 10 The total cost function is c(α, β) = α + β + 10kαβ. To see how welfare changes with information acquisition, we let k be one of the 17 values {0, 0.05, 0.09, 0.15, . . . , 100, ∞}. Between 0.05 and 100, k increases on a logarithmic scale. When k = 0, there is no cost to acquire information on either ordinal or cardinal preferences; when k = ∞, it is impossible to acquire any information.
For a given k under a mechanism, we solve for a symmetric Bayesian Nash equilibrium as defined in Sect. 3.3 and calculate ex ante equilibrium payoffs (net of information acquisition costs). Figure 3 depicts how the efficiency of each mechanism is affected by information acquisition costs.
When k = 0 (free information), the information structure is CI (i.e., cardinal preferences are private information), and IA delivers higher welfare than DA. However, when k increases, the welfare advantage of IA decreases and essentially disappears when k ≥ 1.30. This is because students invest less in information acquisition and thus more frequently fail to acquire information. The welfare performance of DA also decreases when the cost becomes higher. The two mechanisms converge to the same equilibrium outcome as k → ∞ (i.e., impossible to acquire information and thus the information structure is UI).
Example 2: Reversal of the welfare order between IA and DA. Let us consider another example in which IA dominates DA when the information structure is CI, but DA dominates IA when the information structure is OI.
There are two schools. Each school has one seat, and student preference distribution is described in Table 2. There are three students, and each student's preferences are an i.i.d. draw from the distribution. With the information structure UI, students receive 0.5917 in terms of expected utility in equilibrium under either mechanism. 11 With CI, IA dominates DA in a symmetric Bayesian Nash equilibrium. 12 However, with OI, DA dominates IA in a symmetric Bayesian Nash equilibrium. 13 The reason for this reversal is that, with OI, a type-(1, 0.45) student submits (s 2 , s 1 ) too often because she cannot distinguish between the two preference types, (1, 0.45) and (1, 0.9); as a result, a type-(1, 0.45) student is sometimes assigned school 2 even when the other student is of type-(0.1, 1).
For the technology of information acquisition is the same as in the first example. Moreover, a(α) = √ α k (for ordinal information) and b(β) = √ β

2500
√ k (for cardinal information). The total cost function is c(α, β) = α + β + 100αβ. To see how welfare changes with information acquisition, we let k be one of the 9 values For a given k under a mechanism, we again solve for a symmetric Bayesian Nash equilibrium and calculate ex ante equilibrium payoffs (net of information acquisition costs). Figure 4 shows that whenever information acquisition is costly (k > 0), DA dominates IA, in contrast to IA dominating DA given CI. This reversal is due to two factors. First, as discussed above, with the information structure OI, DA dominates IA in equilibrium. Second, the cost of acquiring information on cardinal preferences is high in this example. Given that ordinal preferences have been acquired, to have a probability p c ∈ [0, 1) of learning one's own cardinal preferences, the direct cost is (2500 p c ) 2 k. Even for the smallest positive value of k considered here, the probability of acquiring information on cardinal preferences in equilibrium is 1.92 × 10 −4 , while the equilibrium probability of acquiring ordinal preferences is 1. Essentially, for any positive value of k in the example, we are in a case very similar to the one with the information structure being OI.
due to the costs. In this section, we examine the impact of information provision by education authorities.
In our model, we assume that the provision of information decreases the cost of information acquisition to zero, while the lack of it increases such cost to infinity. For simplicity, we focus on a special setting where everyone has the same ordinal (but different cardinal) preferences, similar to the setting in Abdulkadiroglu et al. (2011) and Troyan (2012). This setting is unfortunately not a special case of the model in sections 3.3 and 3.4 , because student preferences are correlated. However, it can be shown that the main results, Propositions 1 and 2 , still hold true in the setting of this section.
We start with a prior F and thus G (P|F) such that after a P is drawn, it becomes everyone's ordinal preference. Again, every school to be acceptable: v i,s > 0 for all i and s. We use F v s to denote the marginal distribution of the cardinal preference for school s.
We next represent the education authority's decision regarding how much information to release by sending a vector of signals to every i:ω i = ω 1,i ,ω 2,i ,ω 3,i ∈ P ×V ×V (|I |−1) , whereω 1,i andω 2,i are the signals of i's ordinal and cardinal preferences respectively, andω 3,i is the signal of others' cardinal preferences. All signals e., they are either perfectly informative or completely uninformative.
We study the ex ante welfare in equilibrium under each of the following information structures: Note that UI, OI, and CI are the same as in Sect. 3.5. The identical ordinal preference is common knowledge under OI, CI, or PI. However, under UI, no one knows the realization of ordinal preference, but everyone knows that the ordinal preference will be the same across students. These four information structures reflect possible outcomes of different school choice policies. When the education authority makes it difficult for students to acquire information on schools, we are likely to be in the UI scenario. When it makes some information easy to access or directly sends signals to students about their ordinal preferences, students may find it costless to learn their ordinal preferences, and thus we are likely in the OI scenario. If all information on own preferences is readily available, we are likely to be in the CI scenario.
We are also interested in the PI scenario, which relates to the gaming part of school choice under a non-strategy-proof mechanism. From Proposition 2, individual students have incentives to acquire information on others' preferences under IA. The literature has shown that this additional strategic behavior may create additional inequalities in access to public education. More precisely, if one does not understand the game and does not invest enough to acquire information on others' preferences, she may have a disadvantage when playing the school choice game. As a policy intervention, education authority can choose to make this information easier to obtain by publishing students' strategies and allowing students to revise their applications upon observing others' strategies as in Amsterdam (De Haan et al. 2015) and Wake County, NC (Dur et al. 2018).
Note that a symmetric Bayesian Nash equilibrium, possibly in mixed strategies, always exists under any of the four information structures by the standard fixed point arguments. We summarize the results on ex ante welfare under DA and IA in the following two propositions. This implies that there is no gain in ex ante student welfare when students receive more information under DA.
Finally, we state our last proposition.
Proposition 4 (Ex ante welfare under IA) Under IA, we obtain the following ex ante student welfare comparisons in terms of Pareto dominance in a symmetric equilibrium: (i) When uninformed or ordinally informed, the ex ante student welfare is s∈S Ex ante welfare for cardinally informed students weakly dominates that for uninformed or ordinally informed students: CI ≥ OI = UI; (iii) Ex ante welfare for perfectly informed students weakly dominates that for uninformed or ordinally informed students: PI ≥ OI = UI; (iv) The ranking between the ex ante welfare for perfectly informed students and that for cardinally informed students is ambiguous.
The above proposition suggests that it is always beneficial to provide more information on one's own cardinal preferences, but the effect of providing information on others' preferences is ambiguous.
To prove part (iv), we use two examples in "Appendix A" (Sects. A.5.4, A.5.5). The intuition for more information on others' preferences being welfare-decreasing is as follows. Suppose that there are two types of students who are categorized by their preferences for school s 1 , high-and low-type. It is optimal for a high-type student to top-rank s 1 if there are not more than n other students top-ranking s 1 . Because the cardinal preferences are i.i.d. draws, some realized preference profiles of this game can have more than n + 1 high-type students, which is observed by every student when they are perfectly informed. In a symmetric equilibrium in this case, always top-ranking s 1 becomes sub-optimal for high-type students. Instead, they may play a mixed strategy in equilibrium by top-ranking some other school with a positive probability. Consequently, school s 1 will be assigned to a low-type student with a positive probability, lowering the ex ante student welfare that is evaluated before the realization of student preferences. In contrast, when cardinally informed, students do not observe others' preferences and thus play against a distribution of student types.
When other students are high-type with a low probability, a high-type student will choose to always top rank s 1 in a symmetric equilibrium. The concrete example in "Appendix A.5.5" details this intuition.
On the other hand, observing others' preferences can sometimes facilitate coordination among students and thus become welfare improving. Consider again the above example with high-and low-type students in terms of preferences for school s 1 . Suppose that when cardinally informed, high-type students choose to never top rank s 1 in a symmetric equilibrium, because their preferences for s 1 are not too high while other students are very likely to be high-type. Now consider PI. Some realized preference profiles of this game can have only one high-type student. In these realizations, it can be optimal for the high-type student to top-rank s 1 , increasing the chance that a hightype student is assigned s 2 . As a result, the ex ante student welfare, evaluated before the realization of student preferences, can be improved. "Appendix A.5.4" provides a numerical example showing this intuition.

Concluding remarks
This paper provides insights for designing better school choice policies by studying endogenous information acquisition and the effects of information provision.
We distinguish between two types of information acquisition. One is to learn one's own preferences over schools, and the other is to discover others' preferences. Acquiring information on own preferences is necessary in school choice, given the complex nature of education production and the usual lack of information on schools.
In contrast, learning about others' preferences is more related to competing with other students.
The two popular mechanisms, DA and IA, provide heterogeneous degrees of incentives for students to acquire information on preferences. Only IA incentivizes students to learn their own cardinal and others' preferences, while students under DA have no incentive to acquire information beyond their own ordinal preferences. We demonstrate that information acquisition costs affect the efficiency of each mechanism and the welfare ranking between the two. This implies that it is important to endogenize information acquisition in welfare analyses of school choice.
In the case where everyone has the same ordinal preferences, we show the welfare effects of various policies of information provision. The results reveal that information provision is irrelevant in DA, while providing more information on own cardinal preferences is always welfare-improving in IA. However, more information about others' preferences can sometimes be welfare-decreasing in IA.
Our model can be potentially extended in several dimensions. The results can be generalized to the setting in which students have interdependent values over schools. In this case, acquiring information on own values can be achieved by learning more about the schools as well as learning from others' preferences. With interdependent values, students decipher signals on others' preferences in two ways, useful information on one's own values and that on others' values. Our results then describe under each mechanism which deciphering is necessary.
Our model considers the sequential acquisition of information, but, in reality, students may acquire information on one's own and others' preferences simultaneously. Given the lack of strategy-proofness and the role of cardinal utility under IA, we expect our results to hold.
Our model does not allow pre-determined admission priorities. In practice, such priorities are common and can be correlated with student preferences. For example, public school quality is often cited as one of the most important reasons for families to decide where to live (Hoxby 2003, page 10), while schools often give priorities to students in its neighborhood. 14 Therefore, it would be interesting to examine an extended information acquisition game: first, students acquire information about schools to decide where to live; second, the residential location decision determines student priority; third, students then enter another information acquisition game for in-depth information about each school given their priority. Such a game can create a correlation between student preferences and priorities. We leave this extension for future research. Finally, our model assumes every student is rational; however, this assumption is not born out in laboratory or field studies (Chen and Sönmez 2006;Abdulkadiroglu et al. 2006;He 2017). Further studies might explore a theoretical model with students of heterogenous sophistication levels, as in Pathak and Sönmez (2008). Given these considerations, the laboratory experiment in our companion paper (Chen and He 2018) may help us better understand how the theoretical predictions correspond to actual participant decisions in a school choice context.

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Appendix A: Proofs
Before proving the propositions, let us summarize the properties of the two mechanisms. As the results can be easily verified by going through the mechanisms, we omit the formal proof. 15 Lemma 2 DA and IA (with single tie breaking) have the following properties: (i) Monotonicity: If the only difference between L i and L i is that the positions of s and t are swapped such that t L i s, s L i t, and # s ∈ S|s L i s = # s ∈ S|s L i s for all s ∈ S\ {s, t}, then: the inequality is strict when L j = L i , ∀ j = i.
(ii) Guaranteed share in first choice: If school s is top ranked in L i by i, a s (L i , L −i ) ≥ q s / |I |, for all L −i .
(iii) Guaranteed assignment: s∈S a s (L i , L −i ) = 1 for all L −i .

A.1 Proof of Lemma 1
The proof applies to either DA or IA. Note that given any (α −i , β −i ) of other students, σ * (ω) exists. This can be proven by the usual fixed point argument. Note that σ * (ω) does not depend on one's own investments in information acquisition, although it does depend on the signal that one has received (ω).
Given ω, i's payoff function can be written as: which is continuous in σ . Therefore, the value function Π (ω, For student i, the optimal information acquisition is solved by the first-order conditions (second-order conditions are satisfied by the assumptions on the functions a () , b (), and c ()): Given the non-negative value of information and the properties of a () , b (), and c (), one can verify that there must exist α * and β * (P) for all P ∈ P such that the first-order conditions are satisfied.

A.2.1 Proof of˛ * > 0
Given the existence of a symmetric equilibrium, let us suppose instead that α * = 0. It implies that β * (P) = 0 for all P ∈ P and that the value function can be simplified as: Since α * = 0 and β * = 0 (a |P|-dimensional vector of zeros) is a best response for i, ∀α > 0, which can be satisfied if and only if Π P, V φ , 0, 0 = Π P φ , V φ , 0, 0 for all P ∈ P, given that Π P, V φ , 0, 0 dG (P|F) ≥ Π P φ , V φ , 0, 0 and c α (0, 0) < a (0) = ∞. In a given symmetric equilibrium σ * , the finiteness of the strategy space implies that a finite set of lists L (1) , . . . , L (N ) are played with positive probabilities p (1) , . . . , p (N ) (N ∈ N). Suppose that s 1 is bottom ranked in L (1) and s 2 is the second to the bottom. Moreover, there exists an ordinal preference P * such that s 1 P * s P * s 2 for all s = s 1 , s 2 . We also define L (1) which only switches the ranking of the bottom two choices in L (1) , s 1 and s 2 . Since Π P * , V φ , 0, 0 = Π P, V φ , 0, 0 , it implies that L (1) is also a best response to σ * even if i has learned P i = P * . We then compare i's payoffs from submitting L (1) and L (1) .
By the monotonicity of the mechanism (Lemma 2), a s 1 L (1) , Besides, under either of the two mechanisms, given a list, lower-ranked choices do not affect the admission probabilities at higher-ranked choices. Together with the guaranteed assignment (Lemma 2), it implies that a s 1 L (1) , σ * leads to a probability distribution over a finite number of possible profiles of others' actions (L −i ). With a positive probability, everyone else plays L (1) . In this event, therefore, by submitting L (1) , i strictly increases the probability of being accepted by s 1 and decrease the probability of the least preferred school s 2 , comparing with that of submitting L (1) . Furthermore, in any other possible profile of L −i , the probability of being assigned to s * is also always weakly higher when submitting L (1) . Hence, L (1) is not a best response to σ * when P i = P * , and thus This contradiction proves that α * = 0 is not an equilibrium. Since an equilibrium always exists, it must be that α * > 0.

A.2.3 Proof ofˇ * (P) > 0 for some P under IA
We construct an example where β * (P) > 0 for some P given the distribution F under IA. For notational convenience and in this proof only, we assume the upper bound of utility v = 1 and the lower bound v = 0, although we bear in mind that all schools are more preferable than outside option. Suppose that F implies a distribution of ordinal preferences G (P|F) such that for s 1 and s 2 : The distribution of cardinal preferences is: (ε, η, ξ ) are all small positive numbers in (0, 1). Otherwise, there is no additional restriction on F (V |P) for P =P nor on v s , ∀s ∈ S\ {s 1 , s 2 }. Suppose that β * (P) = 0 for all P ∈ P. Section A.2.1 implies that α * > 0. If ω i = P , V φ (i.e., ordinal preferences are known but not cardinal ones), the expected payoff of being assigned to s 2 is: And (η, ξ ) are small enough such that E v i,s 2 |P < q s 1 / |I |. Therefore, obtaining s 2 with certainty is less preferable than obtaining q s 1 / |I | of s 1 . In equilibrium, with a small enough (ε, η, ξ ), it must be that: Therefore, from i's perspective, any other player, j, playsP with probability: It then suffices to show that student i has incentive to deviate from such equilibrium strategies. Suppose that i has learned her ordinal preferences and P i =P. If furthermore she succeeds in acquiring information on V i , there is a positive probability that v s 1 , v s 2 = (1, 1 − ξ ). In this case, if she plays L i s.t., s 2 L i s 1 L i s 3 . . . L i s |S| (or other payoff-equivalent strategies), her expected payoff is at least: While playing P i (=P) leads to an expected payoff less than: This upper bound is obtained under the assumption that one is always assigned to s 1 when not everyone submitsP. When (ε, ξ ) are close to zero, it is strictly profitable to submit L i instead ofP: because in other realizations of V , i cannot do worse than submittingP. The marginal payoff of increasing β P from zero by Δ is then: which is strictly positive given c β (α * , 0) < b (0) = +∞. This proves that under IA β * (P) > 0 for some P ∈ P given F.

A.3 Proof of Proposition 2
For the first part, by the definition of strategy-proofness, information on others' types does not change one's best response. Therefore, δ * (V ) = 0 for all V under any strategy-proof mechanism.
To prove the second part, we construct an example of F (V ) to show δ * (V ) > 0 for some V under IA. For notational convenience and in this proof only, we assume the upper bound of utility v = 1 and the lower bound v = 0, although we bear in mind that all schools are more preferable than outside option. The distribution of cardinal preferences is: where (ε, ξ, η) are small positive values. Besides, Suppose that for student i, V i = V (3) . If δ * (V ) = 0 for all V , the best response for i in equilibrium is to top rank either s 1 or s 2 .
Given F (V ), there is a positive probability, 1 2 − ε |I |−1 , that every other student has V (1) and top ranks s 1 . In this case, the payoff for i top-ranking s 1 is less than q s 1 / |I | + ξ , while top-ranking s 2 leads to (1 − η).
There is also a positive probability, 1 2 − ε |I |−1 , that every other student has V (2) and top ranks s 2 . In this case, the payoff for i top-ranking s 1 is 1, while the one when top-ranking s 2 is at most (1 − η) q s 2 / |I | +ξ .
−i and the above shows they are different for some realization of (V i , V −i ), thus, The marginal payoff of acquiring information (increasing δ (V i ) from zero to Δ) is: which is positive for a small (ε, ξ, η) because e (0) < d (0) = ∞. This proves that δ * (V ) > 0 for some V with a positive measure given F.

A.4 Proof of Proposition 3
Under UI, the only information i has is that her preferences follow the distribution F (V ). Denote W E i as the expected (possibly weak) ordinal preferences of ∈ P are all the strict ordinal preferences that can be generated by randomly breaking ties in W E i if there is any. Therefore, M ≥ 1. When others play L −i , the expected payoff of i playing L i is: Since DA with single tie breaking is essentially the random serial dictatorship, it is therefore a dominant strategy that i submits any P E,m i m ∈ {1, . . . , M}. Moreover, a strategy that is not in P E,1 i , . . . , P E,M i can never be played in any equilibrium, because there is a positive-measure set of realizations of the lottery that such a strategy leads to a strictly positive loss.
We claim that in equilibrium for any L * −i such that L * j ∈ P E,1 Note that for any L * −i , s∈S a s P E,m Since everyone has the same expected utility for being assigned to every school, the maximum utilitarian sum of expected utility is: If Equation (2) is not satisfied and there exists i such that for someL * −i : The maximum utilitarian social welfare in (3) where which contradicts the inequalities (4) and (5). This proves (2) is always satisfied. Under OI, CI, or PI, the unique equilibrium is for everyone to report her true ordinal preferences, and thus the expected payoff (ex ante) is: the ex ante assignment probability, i.e., the probability before the realization of P and V i , is which must be the same across students by symmetry. Therefore, we must have: as in equilibrium all seats at all s ∈ S must be assigned. Suppose i plays an alternative strategy σ i such that That is, i plays the "average" strategy of the equilibrium strategy regardless of her preferences. Her payoff given any realization of P is: The last equation is due to (6). Since σ i may not be optimal for i upon observing her preferences V i , we thus have for ex ante welfare: which proves CI ≥ OI = UI in terms of Pareto dominance of ex ante student welfare.

A.5.3 Proof of PI ≥ OI = UI under IA
Under PI, everyone's cardinal preferences V i are common knowledge. Given a symmetric equilibrium, by the same argument as above, we must have PI Pareto dominates OI and UI. Suppose that σ N E (V i , V −i ) : [0, 1] |S|×|I | → Δ (P) is a symmetric Nash equilibrium. We show that: Note that a s σ N E (V i , V −i ) , σ N E V j , V − j j∈I \{i} is i's probability of being assigned to s in equilibrium when the realization of cardinal preferences is (V i , V −i ). Furthermore, the ex ante assignment probability, i.e., the probability before the realization of P and which must be the same across students by symmetry. Therefore, we must have, ∀s ∈ S: as in equilibrium all seats at all s ∈ S must be assigned. Suppose i plays an alternative strategy σ i such that That is, i plays the "average" strategy of the equilibrium strategy regardless of her and others' preferences. Her payoff given a realization of (V i , V −i ) is: The last equation is due to (7). Therefore, her payoff given a realization of P is: Since σ i may not be optimal for i upon observing her and others' preferences (V i , V −i ), we thus have: which thus proves that P I > O I = U I in terms of Pareto dominance. We use two examples to show part (iii) in Proposition 4: Section A.5.4 shows that PI can dominate CI in symmetric equilibrium while the example in Section A.5.5 shows the opposite.
That is, everyone submits her true preference ranking. This leads to an ex ante welfare of: which is lower than the above symmetric equilibrium under PI. Also note that always playing (s 1 , s 2 , s 3 ) is also a symmetric Nash equilibrium under PI in all realizations of preference profile, which leads to the same ex ante student welfare as σ B N .
For a type-(1, 0.1, 0) student, it is a dominant strategy to play (s 2 , s 1 , s 3 ). Conditional on her type, her equilibrium payoff is: