Transcription

American Economic Journal: Microeconomics 2 (May 2010): 34–63http://www.aeaweb.org/articles.php?doi 10.1257/mic.2.2.34Information Disclosure and Unravelingin Matching Markets†By Michael Ostrovsky and Michael Schwarz*This paper explores information disclosure in matching markets.A school may suppress some information about students in orderto improve their average job placement. We consider a setting withmany schools, students, and jobs, and show that if early contracting is impossible, the same, “balanced” amount of information isdisclosed in essentially all equilibria. When early contracting isallowed and information arrives gradually, if schools disclose thebalanced amount of information, students and employers will notfind it profitable to contract early. If they disclose more, some students and employers will prefer to sign contracts before all information is revealed. (JEL C78, D82, D83)When recruiters call me up and ask me for the three best people, I tell them, “No! I will giveyou the names of the six best.”— Robert J. Gordon, Director of Graduate Placement,Northwestern University, Department of EconomicsHarvard wants high schools to give class rank, but high schools do not want to.— Senior Harvard officialLabor market institutions often suppress some information about job candidates.For example, students at the Stanford Graduate School of Business (GSB) aregraded on a curve, resulting in transcripts that very accurately reflect students’ performance. These transcripts, however, are not revealed to potential employers:the GSB has no policy on grade disclosure; your grades belong to you andit is your right to use them as you wish. Stanford’s nondisclosure normamong MBA students, however, has existed for nearly 40 years.1Most top business schools have similar norms. High profile examples from otherareas include Yale Law School, where first semester grades are credit/no credit.Stanford Medical School conceals from residency programs a part of the student’srecord. Massachusetts Institute of Technology official undergraduate transcriptsavailable to graduate schools and potential employers also suppress available* Ostrovsky: Graduate School of Business, Stanford University, Stanford, CA 94305 (e-mail: [email protected]); Schwarz: Yahoo! Labs, 2397 Shattuck Ave., Suite 204, Berkeley, CA 94704 (e-mail: [email protected]). We are grateful to Drew Fudenberg, Kate Ho, Ed Lazear, Muriel Niederle, Al Roth, AndreiShleifer, Larry Summers, Adam Szeidl, and Robert Wilson for comments and suggestions, and to Tomasz Sadzikfor research assistance.†To comment on this article in the online discussion forum, or to view additional materials, visit the articlespage at http://www.aeaweb.org/articles.php?doi a/academics/learning methods.html, accessed March 22, 2007.34

Vol. 2 No. 2 ostrovsky and schwarz: information disclosure and unraveling35information. They contain only full letter grades, while internal transcripts distinguish between such grades as B and B . Nearly 40 percent of high schools donot disclose class rank to colleges, even though some of them maintain it internallyand report it when “absolutely necessary.”  2 In fact, more than 90 percent of privatenonparochial schools do not disclose class rank (David A. Hawkins et al. 2005).Concealing information need not require deliberate actions on the part of schools.If revealing full information is not in the interests of a school, it can add noise totranscripts by tolerating (or encouraging) grading policies that make grades lessinformative. Unless the dean clearly communicates expectations about gradingstandards, professors are likely to have different ideas regarding the appropriategrade for the average performance. Lack of consistent grading standards adds noiseto transcripts, because luck of the draw determines the grading standard adopted byan instructor. A school can reduce this sort of noise by reporting an average gradein each class alongside the grade received by a student or by mandating the use ofa forced curve in large classes. Inflated grades could also reduce informativenessof transcripts, perhaps unintentionally. For instance, after years of grade inflation,close to 50 percent of grades in undergraduate classes at Harvard College are A andA , often erasing the differences between the good and the great. Figure 1 suggeststhat the informativeness of grades at Harvard, as measured by their entropy, hasdeclined in recent years as the percentage of A and A grades has risen.3All of the practices described above are similar from the employer’s perspective.Refusal to reveal part of the student record, inconsistent grading among instructors,or coarse transcripts are all “noise” that reduces the ability of potential employersto correctly judge the ability of students. The examples above suggest that at leastsome schools are either indifferent regarding how much information to reveal orprefer to conceal some information about the ability of their students; otherwise,they would try to implement policies that minimize the amount of noise in theirtranscripts.There is an alternative channel through which information can be suppressed.Each semester before graduation a student’s transcript becomes longer and moreinformative. Even if schools make transcripts as informative as possible, studentsand employers may choose to contract significantly before graduation, thus leading to incomplete information disclosure. We say that unraveling occurs when thetiming of contracting reduces the amount of available information in a dynamicsetting. Early action and early decision admission programs at many selective colleges (Christopher Avery, Andrew Fairbanks, and Richard Zeckhauser 2003) areexamples of unraveling. These programs allow high school students to submit theirapplications in the fall of their senior year, and admission decisions are made beforefall semester grades are available (in contrast to the regular admission process,which takes fall semester grades into account). The market for law clerks (Averyet al. 2001) is another dramatic example of unraveling. Avery et al. (2001) report2“Schools Avoid Class Ranking, Vexing Colleges,” New York Times, March 5, 2006.Some of Harvard’s policies actually encourage grade inflation. An instructor who gives an F or a D is askedto write a note explaining the reasons for the poor performance of a student. In contrast, instructors who givemany A grades are not asked to explains their reasons.3

871.78825861.75The percentage of A and A gradesAmerican Economic Journal: Microeconomics may 2010The entropy of the distribution of grades36Figure 1. The Informativeness of Grades at HarvardNotes: The dashed line shows the percentage of A- grades and A grades among all grades atHarvard. The solid line shows the entropy of the distribution of grades. We use entropy as aproxy for the informativeness of grades. It is equal to ( i      si ln(si)), where i ranges from theworst grade to the best, and si is the share of grade i among all grades. If all students receivethe same grade, no information is revealed and entropy is minimized. If a transcript structureis modified in a way that reduces the amount of information (e.g., students who had a C or a Dgrade can no longer be distinguished), entropy goes down.that interviews for clerkship positions are held at the beginning of the second yearof three-year law school programs, when only one-third of the students’ gradesare available. Clearly, a lot of information is withheld. Alvin E. Roth and XiaolingXing (1994) describe several other markets in which the timing of transactions hasunraveled.This paper shows that there is a remarkably close connection between the equilibrium (or “balanced”) amount of information revealed by schools in the static environment and the incentive for students and employers to unravel in the dynamicenvironment. If schools disclose the balanced amount of information, students andemployers will not find it profitable to contract early. If they disclose more, unraveling will occur.I. Information Disclosure in a Static EnvironmentWe begin our analysis in the static model. Schools evaluate students and givethem transcripts. Subsequently, these transcripts are used by outsiders (e.g., employers, professional schools, clerkship positions) in their hiring decisions. We assumethat the ability of each student and the distribution of students among schools are

Vol. 2 No. 2 ostrovsky and schwarz: information disclosure and unraveling37given exogenously.4 We also assume that wages offered by employers are inflexible,and so the supply of placement slots of a given desirability is exogenously fixed.5The ability of students is perfectly observed by schools, but not by outsiders.Each school decides how much information to reveal in its transcripts in order tomaximize the average desirability of placement of its alumni. Outsiders use transcripts to infer the expected ability of students and rank them solely according totheir expected ability. The desirability of each position is common knowledge, andstudents rank positions based on desirability. Thus, all students have the same preferences and so do all recruiters.The key feature of our model is that by introducing noise in students’ transcripts,a school can change the distribution of desirabilities of positions to which its students are matched in the job market. Consider, for instance, the competition foradmission to medical schools. Introducing noise into transcripts may enable a college to increase placement into moderately desirable medical schools at the cost ofreducing the number of students placed at top medical schools. The aggregate distribution of positions in the job market does not depend on the transcripts given out byschools, and so the total desirability of placements is constant. However, as we willsee in the next section, in a broad range of situations, noise is a necessary feature oftranscripts given out in equilibrium.6Consider a population of students. The ability of each student is a real number ain the interval [aL  , aH  ]. Each student attends one of I schools. The distribution λi(·)of ability levels at each school i is continuous, exogenous, and commonly known.Without loss of generality, we assume that schools observe the true abilities of theirstudents.7 Each school decides how much of this information to reveal, i.e., how precise to make its transcripts. A school can make transcripts completely informative,revealing the ability level of each student, or it can make them completely uninformative, or anything in between.Formally, a school chooses a transcript structure, which is a mapping from theabilities of students into expected abilities   a    ˆ [aL  , aH  ]. This mapping may be stochastic, i.e., for each ability a there can be a probability distribution over the setˆ that a student of ability a can get. However, this mappingof expected abilities    a has to be statistically correct, in the following sense: the average ability of studentsˆˆ in school i has to equal  a .“labeled” with expected ability    a 4The effects of allowing agents in a matching market to invest in their “quality” are analyzed in Harold L.Cole, George J. Mailath, and Andrew Postlewaite (1992, 2001); Michael Peters and Aloysius Siow (2002); Peters(2007); and Ed Hopkins (2010).5One can show that our results remain valid when the wages of some (or all) firms are flexible. Moreover,if the wages of all firms are flexible, then under full information revelation the wage schedule will be convex inability (see Michael Sattinger 1993, for a survey of the literature on assignment models with flexible wages), andtherefore, as we explain in the discussion following Theorem 1, full information revelation by all schools will bean equilibrium outcome.6Several recent papers study strategic disclosure of information in a variety of environments (Steven Matthewsand Postlewaite 1985; Masahiro Okuno-Fujiwara, Postlewaite, and Kotaro Suzumura 1990; Alessandro Lizzeri1999; Archishman Chakraborty and Rick Harbaugh 2007). The distinguishing features of our setup are the general equilibrium approach and the competitive nature of the market.7Suppose nobody observes the true ability, but each school observes a signal regarding the true ability ofeach of its students. Based on this signal, a school can form an expectation about a student’s ability. All resultsin the paper continue to hold if instead of “true ability” we use “expected ability based on information availableto schools.”

38American Economic Journal: Microeconomics may 2010ˆ a) is aDefinition 1: A transcript structure is a function F(· ·), where F(    a probability distribution with which a student of ability a [aL  , aH ] is mapped toexpected ability   a  ˆ , such that the average ability of students labeled with expectedˆ  ˆ .8ability  a is equal to   a Essentially, the definition says that schools give out grades and transcripts to students using some commonly known grading scheme, and then employers can backout each student’s expected ability based on his or her transcript, the grading scheme,and the distribution of student abilities in the school. We assume that schools cancommit to their transcript structure. This is not a critical assumption.9 What is critical is that employers know the distribution of transcripts given out by a school, aswell as the distribution of student abilities there. Employers know the distribution oftranscripts if they receive applications from many candidates from a given school.Likewise, the distribution of student abilities in large schools is known to recruiters fairly well, at least if it does not change drastically year-to-year. We rule out thepossibility that a school can “fool” employers into thinking that it has better studentsthan it actually does by giving out too many good grades (as in William Chan, HaoLi, and Wing Suen 2007), and focus solely on information compression. This restriction is conceptually similar to the one made by Matthew O. Jackson and Hugo F.Sonnenschein (2007), who show that by linking independent decisions and requiringeach agent to report a vector of his realized types in these decisions that mirrors theunderlying distribution of types, a mechanism designer can essentially relax incentiveconstraints as the number of independent decisions becomes large. The key difference between our restriction and theirs is that in our case, while the schools cannot lie“on average,” they do have the ability to compress the distribution of reported studentabilities, whereas in the case of Jackson and Sonnenschein (2007), the mechanismdesigner requires the reported distribution of types to coincide with the expected one.After schools announce transcript structures and announce expected abilities oftheir students, students and positions are matched. On one side of the market thereis a population of students. On the other side of the market there is a set of positions.The desirability of each position, q [ qL  , qH ], is common knowledge. The distribution μ(·) of position desirabilities is continuous, exogenous, commonly known,and has positive density on [ qL  , qH ]. The mass of positions is equal to the massof students.10 Students rank positions by desirability, and employers rank studentsby expected ability.11 The resulting rankings induce a unique (up to permutations8This definition is very similar to the definition of “information structure” in Dirk Bergemann and MartinPesendorfer (2007). That paper, however, considers information disclosure in a very different environment (asingle-seller, single-object auction), whereas we consider a matching market.9If schools could not commit to their transcript structures, equilibrium information disclosure that we explorein the following section would still remain an equilibrium outcome of the resulting cheap-talk game (see VincentP. Crawford and Joel Sobel 1982, for a formal analysis of cheap-talk games). Of course, the cheap-talk game hasmany other equilibria.10This is not a restrictive assumption, because unemployment can be viewed as a position of the lowest desirability, and because if the mass of positions is greater than the total mass of students, the same subset of positionsgets assigned a student under any information disclosure.11As long as the output of a worker is a function of his ability, we can find a rescaling of ability, such that aparticular firm is indifferent between having a worker of ability a 0 for sure and a worker of uncertain ability withexpectation a 0. However, we do have to assume that this rescaling is the same for all firms.

Vol. 2 No. 2 ostrovsky and schwarz: information disclosure and unraveling39of equally desirable positions) assortative stable matching between students andpositions.Each school selects a transcript structure to maximize the total desirability ofpositions obtained by its students. Each school is small relative to the labor marketand is a “price taker”—its actions have no effect on the placement of students of agiven expected ability.12The following series of examples illustrates the model. In these examples, wediscuss equilibrium information disclosure—the concept we formally define in thefollowing section.Example 1: Student abilities at each school are distributed uniformly on [ 0, 100 ],and position desirabilities are also distributed uniformly on [ 0, 100 ]. If all schoolsˆ a), the resulting mapping Q from expectedfully reveal student abilities (i.e., set    a   ˆ ), and no school can benefit byˆ )      a abilities to position desirabilities is linear (Q(    a deviating. Thus, fully informative transcripts form an equilibrium.Example 2: Now, suppose that at one-half of all schools, student abilities ared istributed uniformly on [ 0, 100 ], while the other half has a more able population—student abilities are distributed uniformly on [ 50, 100 ]. There is a mass 0.5 ofstudents at each type of school. There is also a mass 1 of positions, distributed uniformly on [ 0, 100 ], as before. If all schools fully reveal student abilities, the resultingmapping from expected abilities to desirabilities has two linear pieces:ˆ   a  ,for   a    ˆ 50ˆ2Q(  a ) e     ˆ        3a   2   50, fora    ˆ 50.For instance, a student with expected ability 50 is in the twenty-fifth percentile of thestudent population, and gets a job of the twenty-fifth desirability percentile. Figure 2illustrates this desirability mapping Q. Note that again, no school can benefit bydeviating and suppressing some information. If a “better than average” schoolmixes some students of different abilities together, it gets exactly the same payoff aswithout mixing, while if an “average” school mixes students with abilities above 50and below 50, it gets a strictly lower payoff than without mixing.Example 3: Finally, suppose that there is an “oversupply” of less able students: atone-half of all schools student abilities are distributed uniformly on [0, 100], whilethe other half has a less able population—student abilities are distributed uniformlyon [0, 50]. As before, there is a mass 0.5 of students at each type of school and a mass12This can be reconciled with a finite number of schools by using the standard general equilibrium approach;assume that there are I school types and an infinite number of schools of each type. Technically, different schoolsof the same type could select different transcript structures. However, in any equilibrium in which that could happen, the average transcript structure of schools of a given type would also be an optimal transcript structure for aschool of that type to use, and so there exists another equilibrium with the same aggregate properties in which allschools of the sa