On the dynamics of standards

Transcription

1 RAND Journal of Economics Vol. 32, No. 4, Winter 2001 pp On the dynamics of standards Joel Sobel I study a two-parameter family of rules governing the process of entry into a club. There are three types of agent: candidates desire to enter the club; judges vote on who should enter; elite are already in the club. Promotion requires that at least n of N judges agree that a candidate is at least as good as r of the R members of the elite. Candidates minimize the cost of acquiring a D- dimensional characteristic subject to satisfying the promotion criteria. The article characterizes how minimum entry standards change over time under different assumptions. 1. Introduction It is common to judge individuals by comparing their work to work done by others. Test scores of one generation of students may be compared to the previous generation s scores to determine passing grades. Candidates for promotion are compared to people who have been promoted in the past. Journal reviewers use the quality of recent issues of the journal to determine which articles should be published. In these examples, success is judged relative to the past performance of others; successful agents become the standard by which future agents are judged. This article studies the dynamic performance of promotion rules based on relative comparisons. It describes systematic biases that lead to increases and decreases in standards. The models contain three types of agent: members of the elite, candidates, and judges. Members of the elite have acquired a credential. They could be high school graduates, tenured faculty members, or members of a professional honors society. Candidates are people who aspire to the credential that the members of the elite already have. Judges are people who decide whether candidates should enter the elite. Standards are the level of achievement needed to enter the elite. A candidate makes a costly investment that determines his level of accomplishment (a vector called the candidate s characteristic). If sufficiently many judges believe that a candidate compares favorably to a fixed fraction of the existing elite, then the candidate is promoted. Promoted candidates become members of the elite. Judges compare future candidates to them. Three examples motivate the analysis. In faculty tenure decisions, candidates are assistant professors. The members of the elite are the senior faculty in the department or possibly individuals working in related areas at different institutions. The judges are outside referees, the senior faculty, and university administrators. The senior faculty typically selects the outside reviewers. Candidates may be compared to their senior colleagues, but they are also likely to be compared University of California, San Diego; jsobel@ucsd.edu. Various committee assignments helped to form my thoughts on this subject. Thanks to Julian Betts, Andreas Blume, Antonio Cabrales, Vincent Crawford, Mikhail Klimenko, Garey Ramey, Joel Watson, referees, an Editor of this Journal, and many seminar audiences for comments, Cynthia Bansak and Michael Davis for research assistance, and to NSF grant nos. SBR and SBR for financial support. Special thanks to John Conlisk for detailed advice and encouragement. 606 Copyright 2001, RAND.

2 SOBEL / 607 to individuals in the same area at different universities. There is usually no established rule that determines what is sufficient for promotion. However, recent legal decisions have demonstrated that relative comparisons are important, particularly when discrimination is suspected. Candidates have successfully used the argument that they are as good as recently promoted individuals to appeal negative tenure decisions. Theorems 2 and 4 identify features of the promotion process that may lead to decreasing standards. 1 The Fellows of the Econometric Society are the society s governing body. It is an honor to become a fellow. Candidates are scholars who contribute to economic and econometric theory. The elite and judges are the current active fellows of the society. A candidate must be named on 30% of the ballots cast to be elected. The analysis suggests that these rules could lead to a decreasing quality of fellows. 2 Professional baseball s Hall of Fame honors people 3 who have made great contributions to the sport. Baseball players become candidates for the elite; the elite is current members of the Hall of Fame; and the judges are veteran sportswriters. The colorful book of James (1995) presents a detailed analysis of how various rules for entry into the Hall of Fame might influence standards. He provides an informal discussion of the same kinds of forces that drive the analysis of this article. Findlay and Reid (1997) is a recent empirical study of Hall of Fame standards. The examples suggest that it could be instructive to investigate whether there are features of the standard-setting process that contain unexpected biases in favor of decreasing or increasing standards. In the model of this article, three factors influence the dynamics of standards. First, there are differences of opinion about the composition of quality. Quality is not one-dimensional, and judges put different weights on quality dimensions. Second, to be promoted a candidate must compare favorably to current members of the elite. The criteria for promotion will change as the composition of the elite changes. Third, agents gain nothing from doing strictly more than is necessary to get promoted. Since it is costly to obtain skills, no agent will acquire more than the minimum needed to meet existing standards. The results identify conditions under which standards decrease or increase. All of the promotion rules require candidates to be at least as good as at least one member of the elite. Such rules can have surprising consequences; even candidates with arbitrarily low skill levels will be promoted. Once one accepts that characteristics are multidimensional and that judges have heterogeneous preferences, the logic of the result is simple. A candidate who looks good to one subset of judges may look bad to another subset. If the rules permit this candidate to be promoted, then the candidate may create a lower standard that the unhappy subset of judges applies to future candidates. Sobel (2000) studies the dynamics of standards in an environment similar to the one of this article. Candidates invest in a costly characteristic in order to obtain a promotion. Promotion standards are relative. Sobel (2000) assumes that there is only one judge and concentrates on the case in which all candidates have identical costs. The article demonstrates that a temporary change in the preferences of the judge always leads to a reduction in standards. The result in Sobel (2000) illustrates only the possibility of declining standards; it does not illustrate the possibility of oscillation or increasing standards. Ellison (2000a) documents an increase in the time from initial submission to publication in academic economics journals over the last generation. He evaluates and rejects several explanations for this phenomenon. Ellison (2000b) presents a theoretical model that is broadly consistent with his empirical finding. In this model, journal articles have two characteristics. One characteristic represents the quality of the article (q) and the other represents the polish added through 1 Adam Smith (1776) commented on how related incentives influenced the quality of university instruction. 2 During the past 20 years the system for electing fellows has been changed several times. Some of the changes were motivated by a desire to increase the number of fellows elected. The secretary s reports that appear each year in the January issue of Econometrica contain discussion. 3 The Hall of Fame selects primarily major league baseball players, but managers, owners, and others may be honored.

3 608 / THE RAND JOURNAL OF ECONOMICS revision (r). In Ellison s model a small systematic bias on the part of journal reviewers leads to a substitution of characteristic (r) for characteristic (q). Although the objectives of Ellison s research are different from mine, we both identify possibly surprising asymptotic properties of standards when there are multiple characteristics contributing to quality. Barberà, Maschler, and Shalev (1998), Granot, Maschler, and Shalev (2000), and Roberts (2001) analyze dynamic models of group formation in which voting behavior is strategic. Candidates characteristics are fixed and judges are the current members of the elite. Barberà, Maschler, and Shalev require candidates to obtain the support of only one judge in order to join the elite; Granot, Maschler, and Shalev require unanimous agreement to change group membership; and Roberts studies the implications of majority voting. These articles focus on the implications of a changing electorate. My article holds the electorate constant and concentrates on how voting rules that depend on the composition of the group influence the composition of the group. Leland (1979) identifies a situation in which imposing minimum quality standards enables a market for professional services to survive when sellers have more information than buyers do. Leland does not discuss how setting standards may influence the abilities of professionals. Sjöström and Weitzman (1996) study an environment in which strategic considerations lead to agents exerting low effort, but competitive pressures cause effort to rise to efficient levels. The reason for lower standards is quite different from mine. The literature on tournaments 4 identifies circumstances under which incentive schemes based on relative performance work better than individualistic schemes. Tournament models provide a reason why reward schemes based on relative performance are used: A manager who lacks information about the effort choice of his employees benefits from using a compensation scheme based on relative performance if the output of the agents is correlated. Section 2 explains the model in more detail. I present the main results in Section 3. If no restrictions are made on the composition of the population of elites, standards typically fluctuate over time. If a large supermajority of the judges must support a candidate for promotion, then standards necessarily increase over time. If a candidate can gain promotion with the support of a small fraction of the judges, then standards necessarily decrease over time. For a middle range, it is possible that standards will be unstable, rising, and falling stochastically. Sharper results are available if the number of members specializing in each skill must remain constant over time. Standards increase if the strength of the plurality needed for promotion exceeds the relative ranking a candidate needs to gain promotion. For example, if a simple majority of judges must view a candidate as better than all of the elite in order to secure promotion, then standards rise. If a simple majority of judges need only view the candidate as better than one member of the elite, then standards fall. Sections 4 and 5 study the robustness of the results in Section 3. Section 4 relaxes the assumption that candidates specialize. It assumes instead a two-dimensional characteristic space. More restrictive assumptions on the promotion rule are necessary to obtain conclusions comparable to those in Section 3. Section 5 extends the analysis of Section 4 to higher-dimensional characteristic spaces. Section 6 returns to the examples described in the Introduction. Section 7 discusses some of the limitations of the modelling assumptions, then concludes the article with a discussion of possible applications. 2. A model of promotions The following assumptions describe the basic model. Context. There are three types of agent in the model: candidates, judges, and members of the elite. Candidates apply to judges to be promoted into the elite. There is a fixed number 4 See, for example, Green and Stokey (1983), Holmström (1982), Lazear and Rosen (1981), and Nalebuff and Stigliz (1983a and 1983b). A related article by Shleifer (1985) studies the benefits of regulation schemes based on relative performance.

4 SOBEL / 609 N of judges, a continuing supply of candidates capable of promotion, and a fixed number R of members of the elite. Retirements and promotions. Periodically, a member retires from the elite. Every current member of the elite has a positive probability of being the one to retire. For simplicity, assume that the retirement probability is time invariant. At the time of the retirement, the judges evaluate applications and locate a replacement for the retiree. The retirement/replacement events will be indexed t =1, 2,... This index will be called time, though it need not reflect calendar time. The replacement candidate will sometimes be referred to as candidate t. Candidate characteristics. Each candidate is characterized by a D 1vector x of nonnegative characteristics; D > 1. The characteristics are favorable qualities. Larger amounts make a candidate more impressive to judges. Linear preferences of judges. Judge i s preferences over characteristics x can be represented by a weighted sum of the elements of x. The weights are specific to the judge and fixed over time. The preferences of the N judges can thus be arrayed as an N D matrix W =[w ij ] with all elements positive. The ith row of W is the weight vector of the ith judge. The N 1 vector W x gives the evaluations of the N judges for a candidate with characteristics x. Distinct preferences. Judges differ. For judge i, the ratio w ij /w ik is the marginal rate at which a candidate s characteristics j and k could be traded off while holding the judge s valuation fixed. That is, w ij /w ik is the slope of the judge s linear indifference curves over candidates characteristics j and k, other characteristics fixed. Judges preferences differ in the sense that, for each pair of characteristics j and k, w ij /w ik is different for each judge i. That is, different judges have differently sloped indifference curves. Promotion standards. I consider a simple class of promotion rules, characterized by the number of judges required to support promotion and the criteria they use. Under an (n, r) rule, promotion requires at least n judges to rank the candidate at least as highly as the rth-best member of the current elite. The most stringent such rule is an (N, 1) rule; it requires that all N judges rank the candidate at least as highly as the best of the current elite. The least stringent such rule is a (1, R) rule; it requires only that one judge rank the candidate as highly as the worst of the current elite. Quality cutoffs. For each judge at time t, there is a cutoff quality for the candidate that will lead the judge to vote yes, depending on the characteristics of the current elite and depending on the promotion rule. Let the N 1vector q t =[q it ] denote the quality cutoffs for the N judges. This vector defines the standards at time t. For a mathematical definition of q t, let X t 1 be a matrix whose columns are the x-vectors of the elite existing at the end of time t 1. The matrix WX t 1 arrays the judges evaluations of those members of the elite, where the ith row gives the ratings of the ith judge. Then q it is the rth-largest element in row i under an (n, r) rule. The article studies the asymptotic behavior of the sequence q t. Promotion requirements. The N judges would unanimously elect candidate t, with characteristics vector x t,ifthen scalar inequalities of W x t q t were all satisfied. Let M(W x t q t ) denote the number of inequalities in W x t q t that are satisfied. The requirement on x t for promotion is then M(W x t q t ) n. Candidate training. For each t, each successful candidate qualifies for promotion through costly training. The spirit of the assumption is that there is a continuing supply of sufficiently capable people who are able to qualify. There may be many people behind the scenes who never try for promotion, or who try and fail; their existence is not recorded in the model. To acquire

5 610 / THE RAND JOURNAL OF ECONOMICS characteristics x, candidate t must pay x c t + δ(x) K t. Here c t is a D 1 marginal-cost vector specific to candidate t, K t is a D 1fixed-cost vector specific to candidate t, and δ(x)isad 1 vector of zeros and ones indicating which elements of x the candidate has acquired in positive amounts (δ j (x) =1ifand only if x j > 0). Candidate variability. Candidates differ in their comparative advantage at acquiring individual characteristics. The marginal-cost vectors c 1, c 2,...,c t,...of the candidates are assumed to be independent random drawings from a fixed distribution. Without loss of generality, the support of the individual marginal costs c jt is assumed to be equal to the unit interval. Similarly, the fixed-cost vectors K 1, K 2,...,K t,...of the candidates are drawn from a fixed distribution, independent of each other and of the marginal-cost vectors. Candidate efficiency. Candidate t does not overqualify. That is, candidate t selects characteristic x t to solve the problem min x x c t + δ(x) K t subject to M(W x t q t ) n. (1) Consistent with this assumption, one could suppose that there is only one candidate considered at time t or that there are many, among which one is selected at random. The problem differs from a linear programming problem in two ways. First, the objective function is nonlinear due to the fixed costs K t. Second, since the candidate can qualify by satisfying any subset of n judges, the constraint set is the union of sets determined by linear inequalities. The constraint set would be linear if n = N.Tojustify problem (1), assume that a candidate earns nothing if he is not promoted and V otherwise. Provided that the value of the solution to (1) is less than V, (A) determines the best way to maximize net earnings. Candidate specialization. In Section 3, fixed costs for all but one dimension are assumed to be sufficiently large that all candidates specialize. That is, for all possible costs, the solution x to (1) involves x j > 0 for at most one component j. For example, specialization would occur if K t were drawn in two steps as follows. (i) Select an element of K t at random and set it equal to zero. (ii) Set all other elements of K t equal to a large number. 3. Results with specialized candidates Throughout this section, specialization is assumed. Theorem 1 states that for an (n, r) rule, standards will rise if candidates have to please enough judges (if n is large enough), and standards will fall if candidates have to please few enough judges (if n is small enough). The notation lim t q t = means that lim t q it = for each i. Theorem 1. Assume that judges use an (n, r) promotion rule and that all candidates specialize. (i) If n/(n +1)< 1/D, then standards decline to zero (lim t q t = 0). (ii) If n/(n +1)> (D 1)/D, then standards rise without bound (lim t q t = ). Proof. See the Appendix. A graphical argument provides intuition. A member of the elite is characterized by a value of x, which is a point in D-dimensional space. Consider the special case of D = 2. Since specialization is assumed, a member of the elite will be a point on one of the positive axes. (In Section 4, characteristics are permitted to lie in the interior of the first quadrant.) A family of indifference curves represents the preferences of each judge. These indifference curves will be parallel straight lines (due to the linear-preferences assumption) and will have different slopes for different judges (due to the distinct-preferences assumption). The preferences remain fixed over time. Consider the simple case when there are R =3members of the elite and N =3judges. Depending on the (n, r) rule, candidates will need to please n =1,n =2,orn =3judges.

6 SOBEL / 611 According to Theorem 1, there will be falling standards if n =1and rising standards if n =3. The theorem gives no information for n =2.Consider the three cases. Case n =3. Given n =3,the least stringent (n, r) rule occurs when r =3.When r =3, a candidate need only be as good as one existing member of the elite in order to gain a judge s support. If standards rise when r =3,they will certainly rise when r < 3. Hence assume that r =3in the discussion of this case that follows. Sooner or later, all three members of the elite will have the same specialization (their characteristics will lie on the same axis). This is because retirement is random and any member of the current elite might retire in any period, and because the costs that determine a candidate s dimension of specialization (x 1 or x 2 ) are random. Further, since replacement candidates will go out on an axis only as far as necessary to qualify, it will sooner or later occur that all three members of the current elite fall on the same point on the same axis. Examine what happens after such a tie point has been reached. Consider Figure 1. Assume that all three members of the elite are at point A, the initial tie point. Indifference curves for the three judges passing through A are also shown. They fan out under the distinct-preferences assumption. When a member of the elite retires, the replacement candidate must specialize, must be as good as point A from the viewpoint of all three judges, and must not overqualify. The only points on the graph meeting these conditions are the point A itself, which is on all three indifference curves, and point B, which is on one curve and above the other two. As long as some member of the elite is at point A, the points A and B will remain as the two points that are as good as A or B in the minds of all three judges. Sooner or later, however, all three members of the elite will happen to fall at point B. This event is critical. Once the elite lies entirely at point B, acandidate can no longer qualify by locating at point A. There is a new tie point at B. The relevant figure becomes Figure 2, with the three indifference curves fanning out from point B. Now the two points at which a candidate can efficiently qualify are B and C. At random, point C will sooner or later become the new tie point. Then we will have completed a standards-rising cycle. The latest tie point C must be to the right of the original tie point A. A new cycle begins. The sequence of tie points will ratchet out forever, thus explaining part (ii) of the theorem. FIGURE 1

7 612 / THE RAND JOURNAL OF ECONOMICS FIGURE 2 The argument gives insight about why the r in an (n, r) rule does not matter to the theorem. At a tie point, being as good as one member of the elite is the same as being as good as all three members of the elite. Of course, r may influence the details of the dynamic, such as the rate at which standards increase. The geometric argument illustrates the roles of various assumptions. For example, specialization keeps the points on the axes; randomness bumps the tie points from one axis to the other; and distinct preferences push successive tie points outward. The assumption that D > 1 is critical. With only one dimension in the characteristic vector, once all members of the elite supplied the same characteristic, there would be no further change. On the other hand, the qualitative properties of the geometric example work for higher dimensions. If there are more than D =2elements in a characteristic vector x, the tie points will merely jump among more axes. If there are more than N =3judges, then the indifference curves will have more spokes. If there are more than R =3members of the elite, randomness will still produce tie points, although it may take longer. The argument also suggests ways to relax assumptions. There is nothing critical about the assumption that the judges have linear indifference curves, provided that indifference curves fan out from a tie point. Cost efficiency of candidate training is not needed for a rising-standards result. Overqualification would only raise standards faster. Efficiency does play a critical role in the declining-standards part of the theorem. Case n =1. When only n =1judge needs to be pleased, analogous intuition applies to a randomly timed succession of tie points. The difference is that the tie points ratchet inward instead of outward, and standards shrink toward zero. Case n = 2. In this case, the judge with the intermediately sloped indifference curve is decisive. The elite gets stuck at the two ends of one of his indifference curves. Standards converge to a positive, finite level. When D = 2, anything stronger than a simple majority implies increasing standards. Anything less implies decreasing standards. In general, the greater the number of dimensions, the stronger the assumption needed to guarantee that standards either decrease or increase. When n/(n +1) [1/D, (D 1)/D], it is possible to construct examples in which standards fluctuate widely. I present such an example in the Appendix. One reason that the conclusions of Theorem 1 do not depend on r is because the retirement process has little structure. It is always possible for all the members of the elite to supply the same characteristic. r is not relevant when the population is homogeneous. Theorem 2 considers

8 SOBEL / 613 what happens when the composition of specialties in the elite must remain constant over time. The ranking r used in the promotion rule plays a role in the theorem. Assume that the composition of the elite does not change over time. One can interpret the assumption as requiring the judges to find a candidate who has the same specialty as the member of the elite who just retired. For simplicity assume that there is exactly one member of the elite for each specialty. Hence R = D. Under an (n, r) rule, promotion requires that at least n judges must view the candidate to be no worse than the best r members of the elite. I assume that the candidate is compared to the member of the elite that he is to replace. 5 Theorem 2. Assume that a promoted candidate must have the same specialty as the person he replaces, that all candidates specialize, and that judges use an (n, r) promotion rule. (i) If n/n (r 1)/R, then standards decline to zero (lim t q t = 0). (ii) If (n 1)/N > r/r, then standards rise without bound (lim t q t = ). Proof. See the Appendix. Theorem 2 states that if n/n is small, then standards decrease, while if n/n is large, then standards increase. These results are similar to Theorem 1. The principal difference is that in Theorem 2, whether standards decrease or increase depends on the r that appears in the promotion rule. While Theorem 1 had nothing to say about the dynamics of standards when n/(n +1) [1/D, (D 1)/D], Theorem 2 predicts that standards move to an extreme for a large set of (n, r) rules. It is straightforward to relax the assumption that there are exactly the same number of dimensions in the space of characteristics as there are members of the elite. 6 A graphical intuition for the result is again available for a two-dimensional characteristic space. Consider Figure 3. In the example there are four judges (N =4)and two members of the elite (R = 2). Initially, one member of the elite has characteristic A and the other has characteristic B. The specialization assumption requires that there is always one member of the elite with a characteristic along each axis. Case r =1. When r =1,standards increase when n =3.The argument is similar to the argument given for Theorem 1. As the figure is drawn, judge I thinks that characteristic A is better than characteristic B; judge II is indifferent between A and B; and judges III and IV strictly prefer characteristic B to characteristic A. When the member of the elite with characteristic B retires, a candidate supplying the same characteristic will be promoted with the support of judges II, III, and IV. When the member with characteristic A retires, however, a candidate must supply C, which is more than A,togain the support of three judges (I, II, and III). Once characteristic C is represented in the elite, a new cycle begins forcing standards in the other dimension above B. When (n, r) =(2, 1), standards do not increase. For example, the initial configuration located at A and B in Figure 3 is stable in the sense that subsequent candidates will always supply either A or B. It is apparent that when r = 1, standards cannot decrease. Since candidates are compared to the member of the elite that they replace, they must be at least as good as their predecessors. This conclusion is in contrast to Theorem 1, where decreasing standards are always possible if n/(n +1)issufficiently small. Case r =2. When r = 2,standards decrease if n = 2.The argument works like the argument for increasing standards in the r = 1case. Again consult Figure 3. Upon retirement, a candidate must convince two judges that his characteristic is at least as good as the worst characteristic in 5 Theorem 2 continues to hold assuming that a candidate is not compared to the person he replaces. Because the candidate need not be better than the retiring member of the elite, conditions for decreasing standards are weaker [n/n r/r] and conditions for increasing standards slightly stronger [(n +1)/N (r +1)/R]. 6 If there are precisely a members of the elite specializing in each component, then, setting d = r/a, the conclusions of Theorem 2 become (1) n/n (d 1)/D, then lim t q t = 0; and (2) if (n 1)/N d/d, then lim t q it = for each i.

9 614 / THE RAND JOURNAL OF ECONOMICS FIGURE 3 the elite. The replacement for the member of the elite with characteristic B will be able to supply less than B and still gain the support of judges III and IV. When r =2,acandidate who supplies the same characteristic as an existing member of the elite will always be promoted. Hence standards cannot increase. With a two-dimensional characteristic space, Theorem 1 states that standards would be either decreasing or increasing (except in the knife-edge case of n =(N + 1)/2). The example demonstrates that stable standards arise for a wider range of parameters under the assumptions of Theorem 2. The added power of Theorem 2 arises when N and R are large, so it is not apparent from the analysis of the example. In this case, standards decrease when n/n < r/r and increase when n/n > r/r. Theorem 2 describes the dynamics of essentially all (n, r) rules, 7 while Theorem 1 makes no prediction unless n/(n +1)isclose to zero or one. Theorem 2 quantifies an intuitive tradeoff. Increasing the number of judges needed to support a candidate tends to create higher standards. Reducing the ranking that a candidate must achieve to be promoted tends to create lower standards. Under the assumptions of the theorem, standards cannot be maintained merely by requiring a candidate to receive a great deal of support to get promoted. The level of support needed for promotion must be adjusted to how good the candidate must be in order to gain that support. The graphical analysis provides an understanding of how Theorems 1 and 2 work when there are two possible specialties. I conclude this section with an intuition for the general case. Imagine that one member of the elite specializes in each of the D specialties. Consider a situation in which, for each j, one member of the elite specializes in j, and this member supplies x j of his specialty. Summarize this configuration by the vector x =[x j ]. It is possible to find balanced configurations x in which, for each j, atleast rn/d judges rank the candidate supplying x j as one of the r best. Why should this be possible? Each judge will view (at least) r of the members of the elite as one of the best r. Since there are N judges and D members of the elite, on average (loosely speaking) each member of the elite is one of the r best to rn/d judges. The Appendix contains a fixed-point argument that makes this intuition precise. Note that it follows from linearity of preferences that if x is a balanced configuration, then so is λx for all λ>0. 7 The proof of Theorem 2 shows that standards are bounded above if n/n r/r and bounded below if n/n r/r.

10 SOBEL / 615 If the population of elites is at a balanced configuration, at least r N/D judges support any candidate who duplicates the characteristic of a member of the elite. Hence (assuming that no judge is indifferent between a pair of members of the elite), under an (n, r) rule, a candidate with the jth specialty could be promoted by supplying strictly less than x j when n < rn/d (leading to decreasing standards), while a candidate would need to supply strictly more than x j to be promoted when n > rn/d (leading to increasing standards). When the population is not at a balanced configuration, it can be bounded by balanced configurations. In the case of decreasing standards, the upper bound can be shown to converge to 0; in the case of increasing standards, the lower bound can be shown to increase without bound. The results in Theorem 1 come from an application of this result for r =1.The results in Theorem 2 use R = D. The precise condition for decreasing standards differs slightly from n < rn/d to account for the possibility of indifference. 4. No specialization in a two-dimensional characteristic space The previous theorems place strong restrictions on the preferences of the judges and the cost functions of the candidates. Other results are available without these assumptions. In this section and in Section 5Idrop some of the restrictive assumptions. Conclusions qualitatively similar to (but weaker than) those established in Section 3 still hold. Relax the assumption that candidates specialize. Assume instead that 0 is in the support of the distribution of fixed costs. Theorem 3. Assume that D =2and that judges use an (n, r) promotion rule. (i) If n/(n +1) 1/2 and r/(r+1) 1/2, with one strict inequality, then standards decrease to zero. (ii) If n/(n +1) 1/2 and r/(r +1) 1/2, with one strict inequality, then standards increase without bound. Proof. See the Appendix. The assumptions on (n, r) needed are stronger in Theorem 3 than they were in (twodimensional versions of) Theorems 1 and 2. Stronger assumptions on the promotion rule are needed because there are weaker assumptions on the distribution of candidate costs. When characteristics have two dimensions, the preferences of the judges can be ordered along a single dimension, the slope of their linear utility function. If the judge with the median slope prefers a candidate to a particular member of the elite, then a majority of the judges will share that preference. Assume that every member of the elite has the same characteristic. If n > (N +1)/2, then satisfying the judge with the median slope is generally not enough to get promoted. Unless the next candidate chooses a characteristic identical to the one currently represented, standards will go up in the sense that the median judge will view any candidate as strictly better than the previous elite. Figure 4 illustrates this point. If the characteristic of a representative member of the population is given by A, and a candidate is promoted if two of the three judges support him, then in order to be promoted the candidate will choose a characteristic on or above YAX. This argument suggests that if n > (N +1)/2, then standards should be increasing. It neglects the possibility that membership in the elite may be heterogeneous. If candidates have different cost functions, then they may choose to acquire different characteristics. When they do, different judges will use different characteristic vectors to evaluate candidates. Generally, a more diverse elite and a higher r create pressure for lowering standards. Consider Figure 5. There are two members of the elite supplying characteristics labelled A and B. There are three judges. Under a (2, 2) rule, to be promoted a candidate must be at least as good as the worst member of the elite according to two judges. The median judge, judge II, is indifferent between A and B. However, judge I believes that A is the worst member of the elite, while judge III thinks that B is the worst. Hence a candidate supplying X can get promoted (by satisfying judges I and III), although judge II strictly prefers both A and B to X. This example illustrates the power of the assumption that candidates specialize. A specialized candidate could not get promoted unless his characteristic was greater than or equal to either A or B.

11 616 / THE RAND JOURNAL OF ECONOMICS FIGURE 4 5. No specialization: arbitrary characteristic space When the set of characteristics has more than two dimensions, there is no longer a median judge. A much weaker version of Theorem 3 is available in this case. Standards are increasing under the (1, 1) rule that promotes anyone who, according to at least one judge, is at least as good as the best person in the population. The argument only gets stronger if promotion is possible even if the candidate ranked among the r-best members of the elite (rather than the first). This argument establishes the first part of Theorem 4 below. More precisely, under the (1, 1) rule, q t q t 1 for all t. That is, standards cannot increase. If any candidate did select a characteristic that was, according to some judge, strictly better than the current best, then he could have been promoted by doing less. The best person may get worse (if someone retires). Since the judges have distinct preferences and the candidates supply sufficiently diverse characteristics, for each i, there is positive probability that q it < q it 1. Hence the new standards q t will not exceed q t 1 and may be strictly less. Theorem 4. Under a (1, r) rule, lim t q t = 0. Under an (N, r) rule, lim t q t =. Proof. The discussion prior to the statement of the theorem establishes its first sentence. The second sentence concerns the symmetric situation in which promotion requires the support of all of the judges and follows from a symmetric argument. Q.E.D. The extreme assumption on the number of judges needed for promotion is essential to Theorem 4. If a candidate must dominate the most accomplished elite according to more than one judge, then it is likely that he would exceed at least one judge s threshold in order to be acceptable to another judge. The best member of the elite would probably get better over time. The assumption that support of only one judge is sufficient for promotion is not frivolous, however. Take the case of academic promotions. Tenure letters come from individuals who are likely to be favorably disposed to the type of work that the candidate does. The heterogeneity of tastes could come from outside reviewers rather than internal colleagues. Every candidate could get unanimous, honest letters stating that he or she is as good as any member of the department, because each candidate has a different homogeneous panel of reviewers who like that person s

12 SOBEL / 617 FIGURE 5 work. An alternative justification for the assumption is that there is submajority rule, so that the support of a minority is sufficient for advancement, and there are distinct, homogeneous groups of judges that are sufficient on their own to guarantee advancement. Nevertheless, the need to satisfy judges with different preferences will weaken the results in realistic situations. 6. Discussion The Introduction described three situations that might fit my analysis. Let us now return to these examples. Several of my results may apply to the study of rules governing academic promotions. Tenure candidates typically require strong support from the senior faculty to be promoted. With a naive interpretation of this requirement, Theorems 1 and 2 suggest that standards should rise over time, unless the relative comparisons needed for promotion are weak. Relative comparisons invoked on the grounds of equity are weak. Theorem 2 suggests that standards have a tendency to decrease when candidates must only surpass the least accomplished members of the elite. Different systems will exhibit different biases. My analysis identifies some of the characteristics important in determining the dynamics of standards. In promotion deliberations, judges (external reviewers) are often selected strategically. Theorem 4 is then relevant. A candidate may receive near-unanimous support by selecting an appropriate subset of the judges. Preferences are so diffuse in some disciplines that it is not difficult to find supporters for almost any candidate. Combined with procedures that base promotions on relative comparisons to the current faculty, Theorem 4 identifies a bias toward decreasing standards. 8 Theorems 1 and 2 are relevant results for the election of fellows of the Econometric Society. While the population of fellows grows over time, the most recently elected fellows form a plausible comparison group for the voters. Given the 30% plurality needed for election, standards should 8 There is some evidence that the percentage of assistant professors hired by the University of California who eventually receive tenure is increasing (see Scull et al., 1995). Increasing promotion rates alone, of course, are not convincing evidence that the quality of the people promoted is declining.

13 618 / THE RAND JOURNAL OF ECONOMICS fall if there are only two or three dimensions over which candidates may specialize. 9 The nature of the nominating process that generates potential fellows may provide a slate of candidates that corresponds roughly to the composition of the most recently elected group of fellows. Under these circumstances, Theorem 2 suggests that standards would fall under a 30% plurality rule unless judges withhold votes from candidates who do not rank in the top third of recently elected fellows. This analysis understates the complexity of the election process. It is possible that the typical fellow who votes uses only candidates in his or her area as a comparison group. Winning the support of 30% of the judges would be difficult in this case, especially if fellows only vote for candidates in their specialty. Findlay and Reid (1997) analyze baseball s Hall of Fame voting. Their main motivation was to determine whether there was discrimination against African-American and Latino players in the voting. They claim, however, that entry standards increased during the 1960s and 1970s. Given that candidates require the support of 75% of the voters to be elected, the finding is consistent with Theorems 1, 2, and 3. Findlay and Reid s study does not provide measures of how the quality of the Hall of Fame changes over time. James (1995) argues that at different times (the middle 1940s and early 1970s), actions of the Veterans Committee 10 led to significant declines in the quality of the membership of the Hall of Fame. These selections may have a more significant impact on the composition of the Hall of Fame than the selections of the sportswriters. James presents fans arguments that support candidates by comparing them favorably to current members of the Hall of Fame. If sportswriters use the same criterion, then results in an earlier version of this article suggest that entry into the Hall of Fame should become easier over time. Resolving the apparent conflict between Findlay and Reid and James is an empirical question that is beyond the scope of this article. 7. Conclusion This article presents a model of promotions that identifies conditions for rising and falling standards. The analysis ignores some realistic factors that would tend to reverse the strongest conclusions. Several aspects deserve special mention. The analysis ignores competition. Anyone who meets the standard is promoted. If there are more candidates than positions, then promotions will be based in part on how candidates compare to each other rather than how they compare to the elite. This force acts strongly in favor of rising standards when there is an abundance of candidates. More generally, holding the value of promotion constant, the supply of suitable candidates should rise when standards are low and fall when standards are high. These pressures would moderate fluctuations in standards. Variability of candidates and standards means that some draws from the distribution of candidates would prefer not to pay the cost necessary to get promoted. The time between the entry of viable candidates will vary. The model assumes that a member of the elite will retire precisely when a qualified candidate appears. If retirements were independent of the arrival of new candidates, then standards would need to adjust to maintain a constant membership in the elite. In the article, candidates supply characteristics at efficient levels. That is, candidates never exceed standards. If membership in the elite signals quality to otherwise ignorant consumers, then the assumption is appropriate. It is an extreme assumption, which is likely to be approximately true only when there are problems associated with learning an individual s productivity. Studies of secondary education express concern that imposed standards may become ceilings rather than floors, as my model assumes. For example, the National Commission on Excellence in Education 9 Gordon (1996) reports that the ratio of the Econometric Society s fellows to its membership has risen from 7.7% in 1974 to 13.6% in 1994, and the percentage increased 14 of the last 20 years. The ratio of fellows to active members is an arbitrary and incomplete statistic. It is unlikely to be an accurate measure of whether the quality of fellows is declining. Other measures, for example, the ratio of fellows elected to members eligible, have shown no trend over the past 20 years. 10 The Veterans Committee selects players who are no longer eligible to be elected by the sportswriters. James (1995) suggests that the standards applied by the Veterans Committee vary dramatically over time.

14 SOBEL / 619 (1983) proposes ways to improve skills of high school graduates, but argues against minimum competency examinations, which fall short of what is needed, as minimum tends to become maximum, thus lowering educational standards for all (p. 20). The assumption that candidates never exceed standards is most suspect in the case of baseball s Hall of Fame. Election to the Hall of Fame comes after a player has completed his career. During a player s career there are strong incentives to excel that are largely independent of whether or not he eventually is able to get elected to the Hall of Fame. Results in an earlier version of this article demonstrate that pressure for declining standards would exist even if candidates did not choose efficient effort levels. The article does not justify the particular choice of promotion rules. It does present a simplified description of the origin and incentives of judges. Judges must come from somewhere. In many potential applications, the judges are especially distinguished members of the elite. To the extent that candidates and members of the elite aspire to be judges, their optimization problems are incompletely specified. To the extent that judges care about the aggregate composition of the elite, rather than characteristics of the marginal member, their preferences are incorrectly specified. In many situations judges will modify their voting behavior in response to changes in the promotion rule. I hope to investigate the dynamics of standards in a setting where preferences and voting behavior are determined endogenously. This article is a first step in the study of the dynamics of standards. Ihave assumed without justification that candidates are judged relative to the existing elite. Judges compare the skills of candidates to those of the elite in order to determine who should be promoted. Considerations of equity motivate the use of comparable cases to determine fairness in hiring and promotions. If standards were absolute (for example, promotion requires x x, where x is fixed without reference to the characteristics of the elite), then my arguments would have no force. If judges were forward looking, taking into account how changing the composition of the elite might influence future decisions to change standards, 11 my results would change. The last problem is related to the lack of welfare analysis. The analysis does not allow one to identify an optimal level of standards. Under conditions that give rise to decreasing standards, candidates need to invest less and less over time to get promoted. There is, however, no explicit consideration of the value of their output as a function of the level of their skills (and the level of standards). A more elaborate model that completely specifies judges preferences is needed to determine whether decreasing standards are harmful either to society or to an individual. 12 There is another criticism of the article s approach. Relative standards require that candidates be judged against agents who have succeeded in meeting standards in the past. Alternatively, candidates could be compared to candidates who failed to meet standards in the past. Instead of asking whether someone is as good as people who have been promoted, one could ask if they are as bad as people who have not been promoted. If promotion decisions are based on this standard (so that, for example, one does not promote a candidate who is worse in someone s opinion than someone who was not promoted in the past), then decreasing standards would mean that it becomes easier to not get promoted. Hence it becomes harder to get promoted. In practice both the current members of the elite and the set of failed candidates play a role in determining standards. With these limitations in mind, it is useful to identify general situations in which the approach of this article may provide useful insights. The model is most relevant to situations in which the number of promotions is unlimited; where there are disagreements about how to evaluate quality; and where relative comparisons are important. I close the article by mentioning three situations that may fit the model. Start with the debate on the quality of secondary schools in the United States. There are differences of opinion about how to weight different factors in academic performance. There is no reason why all qualified students cannot receive high school degrees. Relative performance 11 Barberà, Maschler, and Shalev (2001) make this assumption. 12 Betts (1998) and Costrell (1994) study the welfare implications of models of educational standards in a static model.