aa r X i v : . [ ec on . T H ] D ec HIRING FROM A POOL OF WORKERS
AZAR ABIZADA AND INÁCIO BÓ
Abstract.
In many countries and institutions around the world, the hiring of work-ers is made through open competitions. In them, candidates take tests and are rankedbased on scores in exams and other predetermined criteria. Those who satisfy someeligibility criteria are made available for hiring from a “pool of workers.” In each ofan ex-ante unknown number of rounds, vacancies are announced, and workers arethen hired from that pool. When the scores are the only criterion for selection, theprocedure satisfies desired fairness and independence properties. We show that whenaffirmative action policies are introduced, the established methods of reserves and pro-cedures used in Brazil, France, and Australia, fail to satisfy those properties. We thenpresent a new rule, which we show to be the unique rule that extends static notions offairness to problems with multiple rounds while satisfying aggregation independence,a consistency requirement. Finally, we show that if multiple institutions hire workersfrom a single pool, even minor consistency requirements are incompatible with varia-tions in the institutions’ rules.
JEL Classification: C78, J45, L38, D73Keywords: public organizations, hiring, affirmative action.
Abizada: School of Business, ADA University, 11 Ahmadbay Aghaoglu St., BakuAZ1008, Azerbaijan. Email:[email protected]ó (Corresponding author): University of York, Department of Economics and Re-lated Studies, York, United Kingdom; website: ; e-mail: [email protected].
Date : December 2020.The authors thank Oguz Afacan, Samson Alva, Julien Combe, Janaína Gonçalves, Rosalia Greco, Rus-tamdjan Hakimov, Philipp Heller, Sinan Karadayi, Morimitso Kurino, Alexander Nesterov, ThibaudPierrot, Bertan Turhan, Chiu Yu Ko, and two anonymous referees for helpful comments. Research as-sistance by Florian Wiek is much appreciated. Financial support from the Leibniz Association throughthe SAW project MADEP (Bó) is gratefully acknowledged. Introduction
While most companies are free to use almost any criteria to decide which workersto hire and when, that is not the case in many governments and institutions aroundthe world. To reduce the agency problems of government institutions and increase thetransparency of the hiring process, those institutions have to follow clear and strictcriteria for selecting workers. In particular, when the number of workers hired is large(such as police officers, tax agents, etc.), the selection procedure may consist of severalsteps, such as written exams, physical and psychological tests, interviews, and so on,which may also be time consuming. Due to the high costs of executing such selectionprocedures, these hirings often occur in two phases: the evaluation phase, in whichworkers apply for the job and take part in the above-mentioned tests and exams, andthe second phase, in which the institutions select, over time and on a need basis, workersfrom the “pool” of workers who took part in the first phase. After a certain period, thepool of workers is renewed, with new ones coming through a new evaluation phase. Asdescribed by the Public Service Commission of the New South Wales government:“A talent pool is a group of suitable candidates (whether or not existingPublic Service employees) who have been assessed against capabilities atcertain levels. (...) Using a talent pool enables you to source a candi-date without advertising every time a vacancy occurs. You can eitherdirectly appoint from the pool without further assessment (for example,to fill a shorter-term vacancy), or conduct a capability-based behavioralinterview with one or more candidates from the pool to ensure a fit withorganizational, team and role requirements (or additional assessment foragency, role specific or specialist requirements – this is recommended forlonger term or ongoing vacancy). This considerably reduces the time andcosts associated with advertising.” The main characteristics of these procedures, which will be essential to our analysis, arethat (i) the selection of workers to hire at any time, follows a well-defined rule, which isa systematic way of selecting workers to fill a specified number of positions, (ii) workersare hired in rounds , on a need-basis, and must be selected from the pool of workers whotook part in the evaluation phase, and (iii) the institutions do not necessarily knowex-ante the number of workers they will hire during the pool’s validity. Therefore, ingeneral, not all workers in the pool will be hired. This aspect is emphasized in thedescription of the selection process used for all personnel hiring in the European Unioninstitutions: IRING FROM A POOL OF WORKERS 3 “The list is then sent to the EU institutions, which are responsible forrecruiting successful candidates from the list.
Being included on areserve list does not mean you have any right or guaranteeof recruitment. ” [emphasis from the original article]European Union(2015)Notice that the quote above refers to a “reserve list”. A reserve list is a group ofcandidates who are not hired initially but may (or may not) be hired later. Proceduresthat mention reserve lists are equivalent to those with a pool of workers: the poolconsists of the first set of workers hired together with the reserve list.A vast number of hirings occur around the world following this type of procedure.Most developed countries use them when hiring public sector workers to the best of ourknowledge. Below we provide three examples, which are informative about the numberof jobs involved.All hirings for the U.K. Civil Service occur using open competitions which result ina “order of merit list”, with a reserve list valid for 12 months. In 2019, there were445,480 civil servants in the U.K., with 44,570 of them being hired in 2018. In Brazil, the federal constitution mandates that the hiring of public sector workersin all government levels (Federal, State, and Municipal), and state-owned companies,are made through open competition. It moreover states that their results are “valid fortwo years”, and that the workers with a non-expired competition result have priorityover those with later results. In 2017, there were more than 11.37 million public sectorworkers in Brazil. In France, most public sector workers’ hiring is also made through annual opencompetitions ( concours ). These result in an order of merit list and must also include a“complementary list”, with a number of candidates that is at most 200% of the numberof positions hired in the first round. In 2018, there were 5.48 million public sectorworkers in France. In 2016, 40,209 workers at the federal level were hired using theseprocedures. Very often, the rules used for hiring workers involve scores in the selection process.This is not uncommon: the criteria that are used mostly consist of a weighted averageof performance points in multiple dimensions, such as written exam results, education Source: Civil Service Commission. 2018. “Recruitment Principles.” Source: Civil Service Statistics 2019, Cabinet Office National Statistics Source: Brazilian Federal Constitution (1988), Article 37. In practice, this article implies that aftera public competition, workers who “pass” the competition are put on hold and might be hired for twoyears. Those who are not hired have to reapply to a new open competition to be considered. Source: Atlas do Estado Brasileiro, IPEA Décret n°2003-532 du 18 juin 2003 relatif à l’établissement et à l’utilisation des listes complémentairesd’admission aux concours d’accès aux corps de la fonction publique de l’Etat Source: Ministère de l’action et des comptes publics
IRING FROM A POOL OF WORKERS 4 level, etc. When the workers’ scores constitutes the sole element for determining whichworkers to hire, a very natural rule, namely sequential priority , is commonly used: if q workers are to be hired, hire the q workers with the highest scores among those whoremain in the pool. This rule is simple but has many desirable characteristics. First,it is fair in the sense that every worker who is not (yet) hired has a lower score thanthose who were hired. This adds a vital element of transparency to the process: if theworker can see, as is often the case, the scores of those who were hired (or at least thelowest score among those who were hired), then she has a clear understanding of whyshe was not hired. Secondly, it responds to the agency problem: an institution cannotarbitrarily select low-scoring workers before selecting all those who have a score higherthan that worker. Finally, the selected workers’ quality and identity do not depend onthe number of rounds and vacancies in each round. That is, selecting 20 workers in fourrounds with five workers in each results in selecting the same workers as if 20 workerswere selected at once. We denote this last property by aggregation independence . Oneimplication of this requirement is that the set of selected workers is independent of thenumber of rounds and vacancies in each round: selecting 10 workers in two rounds offive workers in each results in the same selection as selecting two workers in each of fiverounds.While sequential priority satisfies those desirable properties, the criteria used forhiring workers often combine scores with other compositional objectives, in the formof a desired proportion of workers belonging to some subset of the population, such asethnic minorities, people with disabilities, or women. In section 4 we formalize theseobjectives, noting that these cannot be achieved by the sequential priority rule evenif scores are designed to incorporate them, and show that minority reserves, which isarguably the best method for implementing these objectives in static problems, doesnot satisfy desirable properties when used multiple times over a single pool of workers.In section 5 we present our main contribution, which is a new rule for hiring workers,that is the unique rule that satisfies natural concepts of fairness for this family ofproblems. We also show that it is essentially the only rule that extends static notionsof fairness with compositional objectives to a problem with sequential hirings whilebeing aggregation independent.In section 6, we evaluate rules used in real-life applications in different parts ofthe world, combining scores with compositional objectives. These include “quotas” forindividuals with physical or mental disabilities in public sector jobs in France, for blackworkers in public sector workers in Brazil, and the gender-balanced hiring of firefightersin the Australian province of New South Wales. We show that these fail most of thetime to satisfy natural concepts of fairness and aggregation independence. Real-life examples of selection rules based on the ranking of workers are the selection of policemen inBerlin and public sector workers in Brazil and France.
IRING FROM A POOL OF WORKERS 5
Finally, in section 8 we consider the cases where there are multiple institutions (orlocations, departments, etc.) hiring from a single pool of workers. While this scenariois widespread, our main result shows that a mild requirement, saying that the orderin which firms hire workers should not change whether some of the institutions hire aworker, essentially leaves us with a single rule, which says that all institutions musthire workers following a single common priority over them.Other than the sections described above, the rest of the paper is organized as follows.In section 2 we introduce the basic model of hiring by rules and justify the desirabilityof aggregation independence. In section 3 we restrict our focus to rules that are basedon scores associated with each worker, in section 7 we evaluate the properties of therules evaluated when they are used for a single round of hiring, and in section 9 weconclude. Proofs and formal descriptions of the rules absent from the main text arefound in the appendix.1.1.
Related literature.
The structure and functioning of the hiring process for publicsector workers have many elements that makes it a clear target for market design:salaries and terms of employment are often not negotiable, the criteria for decidingwho should be hired are exogenously given (or “designed”) and there is a clear concernwith issues of fairness and transparency. This paper is, to the extent of our knowledge,the first to evaluate from a theoretical perspective this type of hiring that occurs inthe public sector, in which workers are sequentially hired following a predeterminedcriterion.There are a few branches of the literature, however, that are related to our analysis.First, the description and analysis of methods for hiring public sector workers aroundthe world and the incentives involved. Sundell (2014) evaluates to what extent the useof examinations constitutes a meritocratic method for recruiting in the public sector.The author observes that exams may not be the most adequate way to identify fitnessfor each function, but that the patronage risk involved when using more subjectivecriteria such as interviews and CV screening often overcomes those losses. In fact,in an empirical analysis in different ministries in the Brazilian federal government,Bugarin and Meneguin (2016) found a positive relationship between corruption casesand the proportion of employees hired by using subjective criteria.The property of aggregation independence, which we propose is important for thisproblem, is related to consistency (Thomson, 1990; Tadenuma and Thomson, 1991;Thomson, 1994) notions. Loosely speaking, an allocation rule is consistent if wheneveragents leave the problem with their own allocations, the residual problem’s solutionmakes the same allocation among the remaining agents. On the other hand, aggregationindependence says that the order (or timing) in which the allocation of a given numberof jobs occurs does not change the identity of those who will get the jobs. Different
IRING FROM A POOL OF WORKERS 6 notions of consistency have been used in other matching and allocation problems basedon priorities as well (Ergin, 2002; Klaus and Klijn, 2013).Finally, a big part of our analysis concerns what we denote compositional objectives:objectives regarding characteristics that some portions of the workers hired should have,such as a minimum proportion of workers with disabilities, ethnic minorities, or certaingenders. Sönmez et al. (2019) evaluated the constitutionally mandated affirmative ac-tion policy used in the hiring of public sector workers in India. Similar to the cases westudy, workers are also selected based on open competitions that result in an order ofmerit of the candidates. They also identify some shortcomings that result from howthe rules are used to implement affirmative action objectives and propose solutionsfor them. However, they do not consider the cases in which hirings occur in multiplerounds, which, as we show in this paper, might have significant consequences.Most of the positive and normative literature on the market design consequences ofaffirmative action policies focus on college admissions and school choice. Kojima (2012)and Hafalir et al. (2013) evaluate the use of maximum quotas (which limit the numberof non-minority students who can be admitted in a school) with minority reserves inthe context of a centralized school choice procedure. They show that majority quotasmay paradoxically hurt minority students, while minority reserves improve upon thisproblem. However, these welfare results have no parallel in our analysis, in whichworkers are either hired or not. Several other papers also evaluate affirmative actionprocedures currently used to select students into schools or universities, identifyingshortcomings and proposing alternative procedures. Aygun and Bó (Forthcoming) showhow the affirmative action procedure used in university admissions in Brazil results infairness and incentive problems. Dur et al. (2020) studied the allocation of students toChicago’s elite public high schools and compared various reservation policies. All ofthem, however, treat the problem from a static perspective: either only one choice ismade, or a complete allocation is produced once and for all.2.
Hiring by rules and aggregation independence A rule determines which workers an institution should hire, given a number of work-ers to hire, a pool of workers, and, potentially, the workers that the institution hiredbefore. Each time an institution attempts to hire workers from the pool is denoted asa round .Let A be the set of workers hired in previous rounds, and W be the set of workersavailable. For each ( W, A, q ) , a rule ϕ determines which q workers from W should be The Indian civil service, like the Brazilian one, uses reserve lists that are valid for two years. Source:Indian Union Public Service Commission Other papers, such as Abdulkadiroğlu and Sönmez (2003); Echenique and Yenmez (2015); Bo (2016);Abdulkadiroğlu (2005) evaluated affirmative action policies in school and college matching.
IRING FROM A POOL OF WORKERS 7 hired. That is, for each ( W, A, q ) , ϕ ( W, A, q ) ⊆ W and | ϕ ( W, A, q ) | = min { q, | W |} .For simplicity of notation, we will sometimes use the following shorthand: ϕ ( W, A, h q , . . . , q t i ) ≡ ϕ (cid:0) W , A , q (cid:1) ∪ ϕ (cid:0) W , A , q (cid:1) ∪ · · · ∪ ϕ (cid:0) W t − , A t − , q t (cid:1) Where A = ∅ , W = W and for i > , A i = A i − ∪ ϕ ( W i − , A i − , q i ) and W i = W \ A i . Furthermore, for simplicity, we will use the shorter notation ϕ ( W, q ) when A = ∅ . Unless stated explicitly, none of our results rely on situations in which there arenot enough workers, either in general or with some characteristics, to be hired. Thatis, in all of our results, we will assume that the number of workers in W is at least aslarge as P q i , and the same holds for the cases that we will evaluate in which someworkers belong to minority groups.One crucial property of the process of hiring by rules is that the sequence of hires λ = h q , . . . , q t i is ex-ante unknown. That is, every round may or may not be the lastone. The total number of workers who will be hired is also unknown. Therefore, theproperties that we will deem as desirable should hold at any point in time. In thiscontext, a critical property that a rule should satisfy is aggregation independence .A rule is aggregation independent if the total set of workers hired after a certain numberof rounds does not depend on how they are distributed among rounds. Definition 1.
A rule ϕ is aggregation independent if for any q ≥ q ≥ and setsof workers W and A , ϕ ( W, A, q ) = ϕ ( W, A, h q , q − q i ) .Therefore, when the rule being used is aggregation independent, an institution thathires q workers in the first round and q in the second will select the same workersthat it would by hiring q + q in a single round. One can easily check that if a rule isaggregation independent, this extends to any combination of rounds: if P i q i = P j q ′ j ,a sequence of hires q , . . . , q n will select the same workers as q ′ , . . . , q ′ m .We now provide three reasons to justify aggregation independence as a stronglydesired property for rules for hiring by rules: transparency, non-manipulability, androbustness. Transparency
One of the main reasons driving governments and institutions to use hiring by rulesis that, for those who are not hired, the reason that happens is made clear and straight-forward. For example, take the rule that consists of always hiring the workers with thehighest exam scores. By knowing the rule and observing the hired workers (and theirscores), any worker who was not hired knows that there was no obscure reason whyshe was not yet hired: it is merely because her score was lower than all those who werehired.Suppose, however, that the rule that is used is not aggregation independent. Then,a worker who was not hired, by just looking at the set of workers who were hired, may
IRING FROM A POOL OF WORKERS 8 not be able to easily understand why she was not hired, even understanding the rulethat was used, because it would also be necessary for her to know the precise sequenceof the number of workers that were hired in each round.
Non-manipulability
While many times the rules which govern the hiring process are chosen in a way thatreduces the ability of managers to make arbitrary choices of whom to hire, they mayhave freedom in choosing the sequence of hires. For example, instead of hiring fourworkers in one month, she may choose to hire two workers first and then two additionalworkers.If the rule is aggregation independent, different choices of sequences of workers hiredwill not lead to different sets of workers hired. However, if the rule is not aggregationindependent, that may not be the case, and a manager may choose a specific sequence ofhiring decisions, which will allow a particular worker to be hired, whereas she would notbe, absent the specific sequence chosen. An aggregation independent rule, by definition,is not manipulable by the choice of the sequence of hires.
Robustness
The third reason why aggregation independence is a desirable property is that thedegree to which the set of workers hired satisfies the objectives represented in the ruleis robust to uncertainty or bad planning on the part of the manager in terms of thenumber of workers that is needed. In other words, assuming that the criterion forchoosing workers which is set by the rule represents the desirability of the workersit chooses (for example, it chooses the most qualified set of workers subject to someconstraint), an aggregation independent rule will always choose the best set of workers,whether the manager makes hiring decisions all at once or continually re-evaluatesthe number of workers to be hired. Aggregation independent rules do not have thatproblem: managers may hire workers based on demand, and that will not result in aless desirable set of workers hired.In Section 4.1 we show specific examples of how aggregation independence relates tonon-manipulability and robustness.3.
Score-based rules
A common way workers are selected when hiring by rules is through a scoring ofall workers. Using criteria such as written exams, evaluation of diplomas, certificates,and experience, workers receive a score (or a number of points). These scores becomethe deciding factor of who to hire: when hiring q workers, hire the q workers with thehighest scores from the pool. For a set of workers W , let s W = ( s w ) w ∈ W be the scoreprofile of workers in W , where for all w = w ′ , s w = s w ′ . Denote by top q ( W ) the q Our assumption that no two workers have the same score is based on how the procedures that weconsider work in real life. Even when discretized scores are used for evaluating the candidates, theprocess always results in a strict ordering of these workers. This can be seen in how the legislation
IRING FROM A POOL OF WORKERS 9 workers with the highest scores in W . A natural property for a score-based rule is forit to be fair . That is, after any number of rounds, if a worker w was hired and w ′ wasnot, then s w > s w ′ . Definition 2.
A rule ϕ is fair if for any W , A and λ = h q , . . . , q t i , w ∈ ϕ ( W, A, λ ) and w ′ ϕ ( W, A, λ ) implies that s w > s w ′ .A natural rule for these kinds of problems is what we denote by sequential priority .When hiring q workers, it consists of selecting the q workers with the highest score fromthe pool of workers. If the pool contains less than q workers, then hire all of them. Thefollowing remark comes immediately from the definition of the rule. Remark . The sequential priority rule is aggregation independent and fair.When the selection of workers is based on scores, which is a very typical setup, thesequential priority rule gives us all we need: it is fair and aggregation independent.4.
Compositional objectives
It is common for hiring processes based on rules to combine the use of scores with com-positional objectives, such as affirmative action. Typically, the objective is to reservesome of the jobs for workers with a certain characteristic, sometimes those belongingto an ethnic minority or those who possess some type of disability. Denote by M theset of workers who belong to the minority group (that is, M ⊆ W ) and ω ( W ) be thenumber of minority workers in W . The affirmative action policy also has a minorityratio m , where ≤ m ≤ , which represents the proportion of hires that should bebased on affirmative action.As argued in Section 2, the desirable properties associated with affirmative actionshould also hold after any number of rounds. Our first requirement is that, whenpossible, the proportion of selected minorities should be at least m after each round. Definition 3.
A rule ϕ respects minority rights if, for any W and sequence of hires λ = h q , . . . , q t i , ( i ) when | M | ≥ m × P ti =1 q i we have ω ( ϕ ( W, λ )) / | ϕ ( W, λ ) | ≥ m , or ( ii ) when | M | < m × P ti =1 q i we have M ⊂ ϕ ( W, λ ) . Remark . The sequential priority rule does not respect minority rights. In general,fairness is incompatible with respecting minority rights . refers to an “order of merit” ’, or in the details of the hiring posts, which often describe multiple(deterministic) methods for breaking ties. Although W is a set, for simplicity of notation we will consider s W following the order in which theelements of W are written. For example, if we denote W = { w , w , w } , s W = (10 , , impliesthat worker w has a score of . Assume that there are three workers: one minority (call him K) and two non-minority (L and V),where the scores are as follows s L > s V > s K . If the rule needs to select two workers and m is 0.3,then in order to respect minority rights , the rule should select K and L, which is not fair , as fairness requires L and V to be selected. IRING FROM A POOL OF WORKERS 10
Therefore, we define a weaker notion of fairness, which takes into account the minorityrestriction. A rule is minority fair if, conditional on respecting minority rights, the hiringdecision is based on scores.
Definition 4.
A rule ϕ is minority fair if, for any W , M ⊆ W and λ = h q , . . . , q t i ,where H = ϕ ( W, λ ) : ( i ) for each w, w ′ ∈ W \ M or w, w ′ ∈ M , if w ∈ H and w ′ / ∈ H , then s w > s w ′ , ( ii ) for each w ∈ W \ M and w ′ ∈ M , if s w < s w ′ and w ∈ H , then w ′ ∈ H , ( iii ) if there is w ∈ W \ M and w ′ ∈ M with s w > s w ′ , w / ∈ H and w ′ ∈ H , then ω ( H ) / | H | ≤ m .In words, a rule is minority fair if it (i) chooses between workers from the same group(minorities or non-minorities) based on their scores, (ii) does not hire low-scoring non-minorities while higher-scoring minorities are available, and (iii) only hires low-scoringminorities over higher-scoring non-minorities when that is necessary to bring the ratioof minorities closer to m from below.One natural question one might have is whether a carefully designed “standardized”scoring system could turn the sequential priority rule into one that respects minorityrights and/or is minority fair. In other words, is there a transformation s ′ W of thescores s W , for which the sequential priority rule under s ′ W respects minority rights andis minority fair under s W ? The answer for the first part is yes, and for the second, no.To see this, let s be the highest score any worker might have, and s ′ W be a scoring systemthat replicates s W for non-minorities, and that adds s to the scores of minorities. Thatis, s ′ W makes all minorities have higher scores than non-minorities but keep all relativescores the same otherwise. It is clear that the sequential priority rule under this scoringsystem respects minority rights but might not be minority fair since it will always hireminority candidates whenever some are still available, regardless of the value of m .To see that no transformation of a scoring system can respect minority rights andbe minority fair, consider the following problem: W = { w , w , w , w } , where M = { w , w } , s W = (100 , , , and m = 0 . . Let s ′ W be the scoring of the workersin W derived from this transformation. Since the sequential priority rule under s ′ W respects minority rights and is minority fair, it is easy to check that it must satisfy s w > s w > s w > s w . Imagine, however, that the set of workers is W = { w , w , w } .In this case, the sequential priority under s ′ W would imply that if q = 1 , worker w would be hired, which is a violation of minority rights since it would require w to behired instead. An alternative question that one might ask is whether there is a scoring function, defined for eachgiven set of workers and original scores, which respects minority rights and is minority fair. The answerfor this one will be yes, since one can produce a ranking of workers from any aggregation independentrule by choosing one worker at a time, and as we will show later in this paper, such a rule exists. Interms of interpretation, we believe that a scoring rule that is endogenous to the set of workers being
IRING FROM A POOL OF WORKERS 11
Sequential use of minority reserves.
Perhaps the most natural candidate fora hiring rule for this family of problems is the use of reserves . With this method, ineach period in which there are vacancies to be filled, the institution uses a choice pro-cedure generated by reserves (Hafalir et al. , 2013; Echenique and Yenmez, 2015), witha proportion m of vacancies reserved for minority workers.Given a set of workers W , of minorities M ⊆ W , a number of reserved positions q m and of hires q , a choice generated by reserves consists of hiring the top min { q m , | M |} workers from M and then filling the remaining q − min { q m , | M |} positions with thetop workers in W still available. In a static setting, this procedure is shown to havedesirable fairness and efficiency properties while satisfying the compositional objectives(Hafalir et al. , 2013). We denote the sequential use of minority reserves rule by ϕ SM .In our setting, therefore, the sequential use of minority reserves rule consists of, in around r , hiring q r workers, reserving m × q r of them for minorities. Proposition 1.
The sequential use of minority reserves respects minority rights. How-ever, it is neither minority fair nor aggregation independent.
The next example shows the problems associated with this rule.
Example 1.
Let W = { w , w , w , w , w } , M = { w , w , w } , and s W = (100 , , , , .Let r = 2 , q = q = 2 and m = 0 . . In the first round, the top worker from M andthe top from W \ { w } are hired, that is, { w , w } . In the second round, the pools ofremaining workers are W = { w , w , w } and M = { w } . The top worker from M ,that is, { w } , and the top from W \ { w } are hired. Therefore, { w , w } are hired inthe second round and ϕ SM ( W, h q , q i ) = { w , w , w , w } .Now consider the case where q = q + q = 4 . Then in the first and unique round, thetwo top workers from M , { w , w } , and the top two workers from W \ { w , w } are hired,that is, { w , w } . Therefore, ϕ SM ( W, q + q ) = { w , w , w , w } . Therefore, the ϕ SM rule is not aggregation independent . Moreover, note that w ∈ ϕ SM ( W, h q , q i ) , w ∈ ϕ SM ( W, h q , q i ) and s w = 50 >
20 = s w while ω (cid:0) ϕ SM ( W, h q , q i ) (cid:1) / (cid:12)(cid:12) ϕ SM ( W, h q , q i ) (cid:12)(cid:12) =0 . > . m , implying that the ϕ SM rule is not minority fair .The sequential use of minority reserves rule is a good rule for providing examples ofproblems associated with rules that are not aggregation independent. First, considerthe issue of manipulability . Take the example 1 above and suppose that the managerprefers to hire the worker w . If she hires the four workers that she needs all at once, w would not be hired. If, instead, she chooses to hire first two workers, and then later twomore workers, w will be hired. That is, by choosing a sequence of hires strategically,the manager can hire the person she wanted. considered is more of a technical property that results from aggregation independence than a criterionthat can be described as a scoring method for hiring before knowing who will apply for the jobs. IRING FROM A POOL OF WORKERS 12
Next, we show that the lack of aggregation independence may lead to the hiring ofa group of workers who are not in line with some common objectives of desirability,(the issue of robustness , as described in section 2). To see how this can be a problem,consider the example below:
Example 2.
Let W = { w , w , w , w , w , w , w , w } , M = { w , w , w , w } , s W =(100 , , , , , , , and m = 0 . .If workers are hired in four rounds, where q = q = q = q = 1 , the set of workershired will be { w , w , w , w } . If, on the other hand, workers are hired all at once, with q = 4 , the set of workers hired will be { w , w , w , w } Assuming that the scores are a good representation of the degree of desirability of aworker for a task, the example above shows that a lack of planning could lead to hiringa set of workers that are substantially less qualified.5.
Sequential adjusted minority reserves
We now present a new rule, sequential adjusted minority reserves , denoted by ϕ SA .It consists of the sequential minority reserves rule in which the number of vacanciesreserved for minorities is adjusted in response to hires made in previous rounds. Morespecifically, the rule works as follows: Round 1
Let m = m , M = M and W = W . The top m × q workers from M arehired. Denoted those workers by A ∗ . Additionally, the top (1 − m ) × q workers from W \ A ∗ are hired. Let M be the workers in M who were notyet hired, and W be the workers in W who were not yet hired. Round r ≥ Let A r = ϕ SA ( W, h q , . . . , q r − i ) and m r = max n m − ω ( A r ) P ri =1 q i , o . The m r × q r top scoring minority workers in M r − are hired. Denote thoseworkers by A ∗ . Additionally, the top (1 − m r ) × q r workers from W r \ A ∗ arehired. Let M r +1 be the workers in M r who were not yet hired, and W r +1 be the workers in W r who were not yet hired.Therefore, the sequential adjusted minority reserves adapts the set of workers hiredaccording to those hired in previous rounds. This makes sense: if we do not take intoaccount, for example, that after the last round, the number of minority workers greatlyexceeded the minimum required, some high-scoring non-minority workers may not behired, leading to a violation of minority fairness. The theorem below shows that this isessentially the only way of achieving these objectives. Theorem 1. If ϕ is a rule that is minority fair and respects minority rights, then forevery set of workers W and sequence of hires λ , ϕ ( W, λ ) = ϕ SA ( W, λ ) . For simplicity, the description below assumes that the number of workers in M and W is large enoughso that in every round there is a sufficient number of them to be hired. A more general descriptioncan be found in the appendix. IRING FROM A POOL OF WORKERS 13
The sequential adjusted minority reserves is also the only rule that extends “static”notions of fairness to this dynamic setting while being aggregation independent. To seethat, we first define the static counterparts of definitions 3 and 4.
Definition 5.
A rule ϕ respects static minority rights if, for any W and q > , ( i ) when | M | ≥ m × q we have ω ( ϕ ( W, h q i )) / | ϕ ( W, h q i ) | ≥ m , or ( ii ) when | M | < m × q we have M ⊂ ϕ ( W, h q i ) . Definition 6.
A rule ϕ is static minority fair if, for any W , M ⊆ W and q > ,where H = ϕ ( W, h q i ) : ( i ) for each w, w ′ ∈ W \ M or w, w ′ ∈ M , if w ∈ H and w ′ / ∈ H , then s w > s w ′ , ( ii ) for each w ∈ W \ M and w ′ ∈ M , if s w < s w ′ and w ∈ H , then w ′ ∈ H , ( iii ) if there is w ∈ W \ M and w ′ ∈ M with s w > s w ′ , w / ∈ H and w ′ ∈ H , then ω ( H ) / | H | ≤ m .In other words, a rule satisfies the static versions of these two notions if they holdwhen there is only one round of hiring. As a result, a rule that respects minority rightsalso respects static minority rights, and a rule that is minority fair is also static minorityfair. However, the converse does not apply: definitions 5 and 6 restrict only the firstset of workers hired in a sequence of hires.The following result shows that the sequential adjusted minority reserves is essen-tially the only rule that extends these static notions to multiple hires while beingaggregation independent. Theorem 2. If ϕ is a rule that respects static minority rights, is static minority fair,and aggregation independent, then for every set of workers W and sequence of hires λ , ϕ ( W, λ ) = ϕ SA ( W, λ ) . Moreover, notice that the first round of hiring in the sequential adjusted minorityreserves is equivalent to the one in the sequential use of minority reserves. Therefore,the sequential adjusted minority reserves is also the aggregation independent extensionof the static minority reserves rule (Hafalir et al. , 2013).
Corollary 1. If ϕ is a rule for which ϕ ( W, h q i ) = ϕ SM ( W, h q i ) and ϕ is aggregationindependent, then for every set of workers W and sequence of hires λ , ϕ ( W, λ ) = ϕ SA ( W, λ ) . Hiring rules around the world
In the following sections, we present the rules currently being used in France, Brazil,and Australia, and show that they suffer from different issues. Since the definition of a rule includes an arbitrary set of previous hires, there can be more rules thatcan satisfy those properties by defining them differently, for example, for sets of previous hires thatcould not result from using the rule from the beginning. If we restrict ourselves to the case in which A = ∅ (the rule is used for every hire ever made), this is a uniqueness result. IRING FROM A POOL OF WORKERS 14
Public sector workers in France.
By law, every vacancy in the French publicsector must be filled through an open competition . When vacancies are announced, adocument explaining deadlines, job specifications, and the criteria that will be usedto rank the applicants is published. Workers who satisfy some stated requirementsthen take written, oral, and/or physical exams. In some cases, diplomas or othercertifications can also be used for evaluating the workers. At the end of this process,all workers’ results in these tests are combined in a predetermined way, to produce aranking over all workers. If the number of vacancies announced was q , then the top q workers are hired. An additional number of workers are put on a “waiting list.” Theseworkers may be hired if some of the top q workers reject the job offer or if additionalvacancies need to be filled before a new open competition is set.The French law also establishes that at least 6% of the vacancies should be filled bypeople with physical or mental disabilities. Instead of incorporating the selection ofthose workers into the hiring procedure in a unified framework, the institutions insteadopen, with unclear regularity, vacancies exclusive for workers who have those disabili-ties. The hiring of workers over time continues following the same procedure as theopen positions described above. However, nothing prevents workers with disabilitiesfrom applying for open positions. In fact, the authorities provide some accommodationfor these workers during the selection, such as, for example, allowing for extra time towrite down the exams. These are meant as an attempt to make up for some disadvan-tages that those workers have with respect to those without disabilities, and not to giveany advantage.Let W ∗ be the set of workers who applied for the open competitions, and M ∗ thosewho applied for the competitions reserved for candidates with disabilities. Workersin M ∗ must have a disability, but workers with disabilities might also apply for opencompetitions. Therefore, M ∗ ⊆ M , and in general M ∗ ∩ W ∗ = ∅ . Since these constitutedifferent competitions, there are scores for the workers in each competition, and morespecifically, a worker who applies to both competitions might obtain different scoresin each. Therefore, we denote by s Ow the score obtained by worker w in the opencompetition and by s Dw the score that worker w obtained in the competition for workerswith disabilities. Since W ∗ and M ∗ are usually different sets of workers, some workersmight have a score in only one competition, but some workers with disabilities mighthave in both.The number of vacancies that are open for workers with disabilities, and when theyare opened, is not determined by any law and is mostly done in an ad hoc manner. Toevaluate the consequences of the method used in France in a formal way, however, we In some cases, different procedures are also used, such as making candidates with disabilities competefor the same vacancies as those without disabilities, but giving them a “bonus” in their scores. Similarmethods are used in affirmative action policies elsewhere, such as in local universities in Brazil. In thispaper, however, we focus on examples involving quotas and reserved vacancies.
IRING FROM A POOL OF WORKERS 15 will consider two alternative policies. In both cases, we will assume that there are twopools of workers, W ∗ and M ∗ , and given a number of positions to be hired q , a totalof q workers from these pools must be hired. In what follows, we consider an arbitrarysequence of hires h q , q , . . . , i . Policy 1 : This policy consists of first hiring the top q workers from W ∗ and thenadjusting the number of workers in M ∗ hired in later rounds. For example, say that q = 100 , but only four workers among the top workers in W ∗ (with respect to s OW )hired have disabilities. Then, considering the objective of hiring at least 6% workerswith disabilities, if q = 50 , then open five vacancies exclusive for workers in M ∗ (selected with respect to s DW ) and for those in W ∗ (selected with respect to s OW ).As a result, by the end of the second round, at least workers with disabilities, or m × ( q + q ) , will be hired. Policy 2 : This policy consists of first hiring m × q from M ∗ (selected with respect to s DW ), (1 − m ) × q workers from W ∗ (selected with respect to s OW ) and then adjusting thenumber of workers in M ∗ hired in later rounds. For example, say that q = 100 . Thenthe policy will result in hiring six workers from M ∗ and from W ∗ in the first round.At least of the workers hired would be among those with a disability, therefore, butpotentially more. Suppose that eight workers with disabilities were hired in the firstround and that q = 50 . Then in the second round, two vacancies exclusive for workersin M ∗ would be open, and the remaining would be open for all workers in W ∗ .While the two policies above represent what we believe are the best efforts to satisfythe objectives stated in the law under the current existing procedures, Policy 1 differs inthat when only one round of hiring is done, the proportion of workers with a disabilityhired might be below the minimal proportion stated in the law. This fact will moreevident in Proposition 2.Whenever necessary, we will refer to the rules defined by policies 1 and 2 by ϕ F and ϕ F . Since under the French assignment rule each worker may have one or twoscores, what constitutes minority fairness is less clear in this context. However, theexample below shows that policy 2 may lead to outcomes that clearly violate the spiritof minority fairness. Example 3.
Let W ∗ = { w , w , w } and M ∗ = { w , w , w } , with scores s OW =(50 , , and s DW = (50 , , and m = 0 . . If q = 2 , then ϕ F ( { W ∗ , M ∗ } , q ) will select { w , w } . Worker w , however, has a disability and a better score than w in the competition in which both participated. We have no evidence that any of these policies constitute actual practice by French institutions,but we believe that they represent the two most natural attempts at satisfying the legal requirementsunder the current rules. For simplicity, here and in the rest of the main text we will assume that every expression involvingnumbers of workers or vacancies are integers. In the appendix, we relax that restriction and show thatnone of the results presented depend on that.
IRING FROM A POOL OF WORKERS 16
Notice that if worker w also applied for the open vacancies and in that competitionobtained a score that is also better than the one obtained by w , she would have beenhired instead of w . If the relative rankings of the workers in both competitions aredifferent, more intricate violations of the spirit of minority fairness can also occur.If we make the (strong) assumptions that all workers with disabilities apply to bothcompetitions and that the relative rankings between those workers in both competitionsare the same, we can obtain a clear distinction between both policies, as shown below. Proposition 2.
Suppose that M ∗ ⊆ W ∗ and that for every w, w ′ ∈ M ∗ , s Ow > s Ow ′ ⇐⇒ s Dw > s Dw ′ . Policy 1 of the French assignment rule does not respect minority rightsand is not aggregation independent. Policy 2 respects minority rights, is aggregationindependent, and minority fair. It is crucial to notice, however, that the result in proposition 2 depends on the relative rankings of the workers with disabilities being the same in bothcompetitions , but perhaps most importantly, on workers with disabilities partic-ipating in both competitions . This is not a minor issue, since these competitionsoften involve a significant amount of time and effort.6.2.
Quotas for black public sector job workers in Brazil.
The rules for thehiring of public sector workers in Brazil work essentially in the same way as in France:vacancies are filled with open competitions that result in scores associated with theworkers, and workers are hired in each period by following their scores in descendingorder. Differently from France, however, there is no quota for workers with disabilities,but instead, since 2014, there are quotas for black workers.The use of racial and income-based quotas has increased significantly in many areasof the Brazilian public sector and higher education. At least 50% of the seats in federaluniversities, for example, are reserved for students who are black, low-income, or stud-ied in a public high-school (Aygün and Bo, 2013). Many municipalities also employquotas for black workers in the jobs that they offer. One of the most significant recentdevelopments, however, is a law which establishes that 20% of the vacancies offered ineach job opening should give priority to black workers. Differently from France, the quotas for black workers are explicitly incorporated intothe hiring process. More specifically, the rule currently used in Brazil (denoted the ϕ B rule) works as follows. Let k be a number that is higher than any expected number ofhires to be made. Initial step
Workers are partitioned into two groups: (i) Top Minority (
T M ) and (ii)Others ( O ). The T M group consists of the highest scoring ⌈ m × k ⌉ workers Lei N. 12.990, de 9 de junho de 2014.
IRING FROM A POOL OF WORKERS 17 from M , and O be the top k −⌈ m × k ⌉ workers in W \ T M . Let
T M = T M and O = O . Round r ≥ The ⌈ m × q r ⌉ top scoring minority workers from T M r , and the top q − ⌈ m × q ⌉ workers from O r are hired. By removing these workers hired,we obtain T M r +1 and O r +1 .For the Brazilian law specifically, m = 0 . . The example below shows that the Brazilianrule is not minority fair. Example 4.
Let W = { w , w , w , w } , M = { w , w } , and s W = (100 , , , .Let q = 2 , m = 0 . and k = 4 . Then T M = { w , w } and O = { w , w } . TheBrazilian rule states that, when hiring two workers, the top worker from T M and thetop from O should be hired. Therefore, ϕ B ( W, q ) = { w , w } . Since w / ∈ ϕ B ( W, q ) and s w > s w , the Brazilian rule is not minority fair . Notice that in this example, worker w , who is part of the minority, has a higherscore than w , who is not a minority. Worker w is hired, while w is not. Given thatthe affirmative action rules were designed with the intent of increasing the access thatminorities have to these jobs, this type of lack of fairness is especially undesirable, sinceif the hiring process was purely merit-based, worker w would have been hired.Aygün and Bo (2013) describe the implementation of affirmative action in the admis-sion to Brazilian public universities. There, as here, the problems arise from the factthat positions (in that case seats) and workers are partitioned between those reservedfor minorities and the open positions. Differently from there, however, unfair outcomesmay not be prevented by workers even if they strategically manipulate their minoritystatus. In the example above, even if w applied as a non-minority he would not behired. Proposition 3.
The Brazilian rule is aggregation independent and respects minorityrights. However, it is not minority fair.
Gender balance in the hiring of firefighters in New South Wales.
Thehiring of firefighters in the Australian province of New South Wales attempts to achievea gender-balanced workforce by following a simple rule: Notice that the set W \ T M , in general, contains both minority and non-minority workers. As a result,if there are not enough minority workers, the remaining positions are filled with the top non-minorityworkers. One may conjecture that the scenario above is very unexpected, since the affirmative action lawmust have been enacted in response to minority workers not being hired based solely on scores. Asshown in Aygün and Bo (2013), however, this conjecture may be misleading. For example, even if theaverage score obtained by minority workers is lower, one can have situations in which the preferencesof the higher achieving minority workers are correlated, leading to the top minority workers in theentire population applying to a specific job.
IRING FROM A POOL OF WORKERS 18 “Candidates who have successfully progressed through the recruitmentstages may then be offered a place in the Firefighter Recruitment train-ing program. Written offers of employment will be made to an equalnumber of the most meritorious male and female candidates based onperformance at interview and the other components of the recruitmentprocess combines.” We denote this rule by
NSW rule , or ϕ NSW . Although not stated explicitly in theinstitution’s website, we will assume that if there are not enough individuals of somegender, the remaining hirings will be made among those candidates available, based ontheir scores. Moreover, to avoid results that rely simply on whether q is odd or even,we assume it is always even. The example below shows the problems involved in thatrule: Example 5.
Let W = { w , w , w , w } , and W F = { w , w } and W M = { w , w } bethe set of female and male workers, respectively. Suppose that the scores are s W =(100 , , , . Let q = 2 . Then ϕ NSW ( W, q ) = { w , w } . Since w is not hired but s w > s w , the NSW rule is not fair. Moreover, it is easy to see that if either gender isconsidered a minority, the rule is also not minority fair.The result below summarizes the properties of the NSW rule. Proposition 4.
The NSW rule is aggregation independent but not fair. If one of thegenders is deemed as a minority, then it respects minority rights but is not minorityfair. Single-round hiring
Until now, we evaluated rules from the perspective of whether they satisfy the de-sirable properties we introduced in the previous sections: aggregation independence,respecting minority rights, and minority fairness. While our analysis focuses on hir-ings potentially involving multiple rounds, one might wonder whether some of theseproblems would be present when the hiring is made in a single round.The property of aggregation independence does not imply anything regarding a sin-gle round of hiring. As we mentioned in section 5, since the properties of respectingminority rights and minority fairness are only satisfied when they are satisfied for anynumber of rounds, the rules for which they are satisfied will also satisfy them for asingle round of hiring. For the French policies, the results do not change.
Remark . Suppose, as in Proposition 2, that M ∗ ⊆ W ∗ , and that for every w, w ′ ∈ M ∗ , s Ow > s Ow ′ ⇐⇒ s Dw > s Dw ′ . Policy 1 of the French assignment rule does not respect staticminority rights, and Policy 2 respects static minority rights and is static minority fair. IRING FROM A POOL OF WORKERS 19
Next, consider the (sequential) use of minority reserves. When only a single roundof hiring occurs, the rule satisfies all the desirable characteristics. Moreover, as wementioned in section 5, when there is a single round of hiring, it is equivalent to thesequential adjusted minority reserves.
Remark . The (sequential) use of minority reserves respects static minority rights andis static minority fair.The problems with minority reserves are not present under single hiring. As shownin Example 1, the issues reside on the fact that minority fairness requires that anasymmetric priority is given to minority candidates only when their proportion amongthose hired is below m . The sequential use of minority rights “lacks memory”, in thesense that it always gives this asymmetric priority to minority workers, regardless ofhow much it is still needed given past hires. When only one round of hiring occurs,that is not a problem.Regarding the Brazilian rule, the variable that determines its characteristics is k ,that is, what is the number of workers from W what will be put in the sets T M and O .To see this, let q be the number of hires in the single-round hiring, and let moreover k = q . One can easily verify that the hiring that will be made is the same as the onedone by the (sequential) use of minority reserves. If k > q , on the other hand, we canhave the situation shown in Example 4. Therefore: Remark . Let q be the number of hires that occur in a single round using the Brazilianrule. If k = q , then the rule respects static minority rights and is static minority fair.If k > q , then the rule respects static minority rights, but is not static minority fair.Finally, since the negative results for the NSW rule are based on a single round ofhiring, we have the following remark: Remark . When only one round of hiring occurs, the NSW rule is not fair. If one ofthe genders is deemed as a minority, then it respects static minority rights but is notstatic minority fair. 8.
Multiple institutions
While often the hiring processes that we describe involve one or more positions in asingle job specification – and therefore the pool of applicants, or reserve list, being usedonly for that position – in many cases, a pool of workers is shared between multiple As we noted in section 1, most of the hiring in the public sector in France and Brazil, for example,requires the use of order of merit lists and reserve lists. While for some positions, such as police officers,it is natural to expect that the workers might be matched to different locations, many other positionsare more specific and do not result in a pool of candidates shared by more than one job. Examplesinclude the hiring of doctors with a specific specialty for a municipality with a single hospital, the roleof economist in state companies, who only work at the headquarters, the entry-level diplomatic career,etc.
IRING FROM A POOL OF WORKERS 20 institutions or locations. For example, in the hiring process for the Brazilian Federalpolice, workers may be allocated to different locations. In the selection process for theNew Zealand police, the candidate’s preference is also taken into account when decidingwhich district a worker who will be hired from the pool will go to:“The candidate pool is not a waiting list. The strongest candidates arealways chosen according to the needs and priorities of the districts. Thetime it takes to get called up to college depends on your individualstrengths and the constabulary recruitment requirements in your pre-ferred districts. (...) We will look to place you into your preferred districtbut you may also be given the option to work in another district whererecruits are needed most.” In this section, we evaluate how the fact that workers may be hired by more than oneinstitution affects the attainability of basic desirable properties. Now, in addition tothe set of workers W , there is also a set of institutions I = { i , . . . , i ℓ } , where | I | ≥ .Institutions make sequences of hires, and there is no simultaneity in their hires: in eachround only one institution may hire workers. Therefore, when we describe a round, wenow must determine not only how many workers are hired, but also which institutionthose workers will be assigned to. Some additional notation will be, therefore, necessary.A matching µ is a function from I ∪ W to subsets of I ∪ W such that: • µ ( w ) ∈ I ∪ {∅} and | µ ( w ) | = 1 for every worker i , • µ ( i ) ⊆ W for every institution i , • µ ( w ) = i if and only if w ∈ µ ( i ) .At the end of each round r ≥ , we define the matching of workers to institutions asa function µ r .A plural sequence of hires Λ is a list of pairs ( i, q ) , where i ∈ I , and q is the numberof workers hired. A plural sequence of hires Λ = h ( i , , ( i , , ( i , i , for example,represents the case in which in the first round institution i hires three workers, in thesecond round institution i hires two workers, and then in the third round institution i hires two workers.When considering hiring with multiple institutions, a rule, therefore, can be general-ized to produce matchings instead of allocations. Given a pool of workers W , an initialmatching µ , and a plural sequence of hires Λ = h ( i , q ) , ( i , q ) , . . . , ( i k , q k ) i , a hiringrule Φ is derived from a set of institutional rules (Φ i ) i ∈ I by returning the matchingcombining all institutional rules. That is, if Φ (
W, µ , Λ) = µ , then µ ( i ) = Φ i ( W, µ , Λ) . Source: Brazilian Department of Federal Police. We abuse notation and consider µ ( w ) to be an element of I , instead of a set with an element of I . IRING FROM A POOL OF WORKERS 21
Given W and some matching µ t − , Φ i ( W, µ t − , h ( i, q ) i ) ⊆ W \ S i ∈ I µ t − ( i ) . That is, itselects workers from W who are not yet matched to some institution in µ t − . Moreover: Φ i (cid:0) W, µ , Λ (cid:1) = k [ t =1 Φ i (cid:0) W, µ t − , h ( i t , q t ) i (cid:1) For any t > , the matching µ t is such that for all i = i t , µ t ( i ) = µ t − ( i ) , but µ t ( i t ) = µ t − ( i t ) ∪ Φ i t ( W, µ t − , h ( i t , q t ) i ) . We assume that Φ i ( W, µ, h ( i ′ , q ) i ) = ∅ whenever i = i ′ .That is, one institution cannot “hire for another institution”. We also assume that if µ and µ ′ are such that ∪ i ∈ I µ ( i ) = ∪ i ∈ I µ ′ ( i ) and µ ( i ∗ ) = µ ′ ( i ∗ ) , then Φ i ∗ ( W, µ, h ( i ∗ , q ) i ) =Φ i ∗ ( W, µ ′ , h ( i ∗ , q ) i ) . That is, an institution i ∗ ’s hiring decision can depend on the setof workers who were not yet hired and the workers who were already hired by i ∗ , butnot on the identity of the workers who were hired by each other institution.Let µ ∅ be a matching in which for all i ∈ I , µ ∅ ( i ) = ∅ . We denote by Φ i ( W, Λ) and Φ ( W, Λ) the values of Φ i (cid:0) W, µ ∅ , Λ (cid:1) and Φ (cid:0) W, µ ∅ , Λ (cid:1) . Finally, we abuse notation and if Λ = h ( i , q ) , ( i , q ) , . . . , ( i k , q k ) i , we can append the plural sequence of hires with thenotation h Λ , ( i ∗ , q ∗ ) i ≡ h ( i , q ) , ( i , q ) , . . . , ( i k , q k ) , ( i ∗ , q ∗ ) i .The example below clarifies these points. Example 6.
Consider a set of workers W = { w , w , w , w , w } with scores s W =(100 , , , , , a set of institutions I = { i , i , i } and let Φ be a rule that, in anyround, matches the highest scoring workers to the institution in that round. Then if Λ = h ( i , , ( i , , ( i , i , the matchings µ , µ and µ produced at the end of eachround are: µ = i i i w ∅ ∅ ! µ = i i i w ∅ { w , w } ! µ = i i i { w , w } ∅ { w , w } ! We will consider two properties for rules when there are multiple institutions. Thefirst is related to the desirability of workers.
Definition 7.
A rule Φ satisfies common top if there exists a worker w ∗ ∈ W suchthat, for every institution i ∈ I , w ∗ ∈ Φ i ( W, h ( i, i ) .In words, common top requires that there is at least one worker that, wheneveravailable, every institution would hire.Next, we consider a weak notion of consistency across the hirings made by the insti-tutions. Definition 8.
A rule Φ satisfies permutation independence if for any plural se-quence of hires Λ and any permutation of its elements σ (Λ) , S i ∈ I Φ i ( W, Λ) = S i ∈ I Φ i ( W, σ (Λ)) .Permutation independence, therefore, simply requires that the set of workers hired,regardless of where, should not change if we adjust the order of hiring.
IRING FROM A POOL OF WORKERS 22
We also adapt the notion of aggregation independence to multiple institutions byrequiring that each institution’s rules are aggregation independent.
Definition 9.
A rule Φ is aggregation independent if for any q ≥ q ≥ , sets ofworkers W , institution i ∈ I , and matching µ , Φ i ( W, µ, h ( i, q ) i ) = Φ i ( W, µ, h ( i, q ) , ( i, q − q ) i ) .The family of rules that we will use in our next result is elementary but also veryrestrictive.A rule Φ is single priority if there exists a strict ranking ≻ of the workers in W suchthat when Λ is any plural sequence of hires in which at least two different institutionsmake hires, for every i ∈ I : Φ i ( W, h Λ , ( i, q ) i ) = Φ i ( W, Λ) ∪ q max ≻ W \ Φ i ( W, Λ) Where, given a set X , max q ≻ X is the set with the top q elements of X with respect tothe ordering ≻ . In words, a rule is single priority if, whenever more than one institutionmake hires, all hirings from all institutions consist of hiring the top workers, among theremaining ones, when all of these institutions share a common ranking.The result below shows that, for a wide range of applications, having multiple insti-tutions is incompatible with most objectives a policymaker may have. Theorem 3.
A rule satisfies common top, aggregation independence and permutationindependence if and only if it is a single priority rule.
Theorem 3 is a fundamentally negative result. It shows that aggregation and permu-tation independence, both arguably simple desirable characteristics, are incompatiblewith institutions following different criteria when evaluating these candidates. This istrue even when all institutions share the same scores for workers, but institutions mighthave different values of m (the proportion of minorities that must be hired). Notice, however, that if the compositional objectives are interpreted as being appliedto the entire set of workers hired by these institutions, as a whole, then we can simplyuse the sequential adjusted minority reserves, as defined in section 5, every time aninstitution wants to hire a given number of workers. This procedure satisfies aggregationindependence and is also permutation independent. Moreover, it satisfies a naturaladaptation of what it means to respect minority rights and minority fairness. Insteadof applying to whether workers are hired by a specific institution, it applies to beinghired at some institution. Take, for example, I = { i , i } , W = { w , w , w , w , w } , M = { w , w , w } , s w > s w > s w >s w > s w , and the value of m for the two institutions being m = 0 . and m = 0 . If both institutionsuse the sequential adjusted minority reserves and i hires two workers before i also hires two, worker w is hired and w is not. If the order is that i hires first, then w is hired and w is not. A violationof permutation independence. IRING FROM A POOL OF WORKERS 23 Conclusion
In this paper, we evaluate a hiring method that is widely used around the world,especially for public sector jobs, where institutions select their workers over time froma pool of eligible workers. While the simple and natural rule of sequential prioritysatisfies all desirable characteristics, the addition of compositional objectives such asaffirmative action policies increases the complexity of the procedures. We show thatthe rules being used in practical hiring processes, as well as the direct application ofminority reserves, fail fairness or aggregation independence. When the compositionalobjectives can be modeled as affirmative action for minorities, the sequential adjustedminority reserves, which we introduced, is therefore the unique solution that satisfiesthose desirable properties.If multiple institutions hire from the same pool of applicants, however, we show thatthe space for different hiring criteria between institutions, is highly restricted when aminimal requirement of independence is imposed.
ReferencesAbdulkadiroğlu, A. (2005). College admissions with affirmative action.
Interna-tional Journal of Game Theory , (4), 535–549. 10 Abdulkadiroğlu, A. and
Sönmez, T. (2003). School choice: A mechanism designapproach.
American economic review , (3), 729–747. 10 Aygün, O. and
Bo, I. (2013).
College Admission with Multidimensional Reserves: TheBrazilian Affirmative Action Case . Tech. rep., Mimeo. 1. 6.2, 6.2, 22
Aygun, O. and
Bó, I. (Forthcoming). College admission with multidimensional priv-ileges: The brazilian affirmative action case.
American economic journal : Microeco-nomics . 1.1
Bo, I. (2016). Fair implementation of diversity in school choice.
Games and EconomicBehavior , , 54–63. 10 Bugarin, M. and
Meneguin, F. B. (2016). Incentivos a corrupcao e a inacao noservico publico: Uma analise de desenho de mecanismos. (Incentives to Corruptionand Inaction in the Civil Service: A Mechanism Design Approach. With Englishsummary.).
Estudos Economicos , (1), 43–89. 1.1 Dur, U. , Pathak, P. A. and
Sönmez, T. (2020). Explicit vs. statistical targetingin affirmative action: Theory and evidence from chicago’s exam schools.
Journal ofEconomic Theory , , 104996. 1.1 Echenique, F. and
Yenmez, M. B. (2015). How to control controlled school choice.
The American Economic Review , (8), 2679–2694. 10, 4.1 Ergin, H. I. (2002). Efficient resource allocation on the basis of priorities.
Economet-rica , (6), 2489–2497. 1.1 IRING FROM A POOL OF WORKERS 24
European Union (2015). General rules governing open competitions.
Official Journalof the European Union , , C70 A/01. 1 Hafalir, I. E. , Yenmez, M. B. and
Yildirim, M. A. (2013). Effective affirmativeaction in school choice.
Theoretical Economics , (2), 325–363. 1.1, 4.1, 5 Klaus, B. and
Klijn, F. (2013). Local and global consistency properties for studentplacement.
Journal of Mathematical Economics , (3), 222–229. 1.1 Kojima, F. (2012). School choice: Impossibilities for affirmative action.
Games andEconomic Behavior , (2), 685–693. 1.1 Sönmez, T. , Yenmez, M. B. and
Others (2019). Affirmative action in india viavertical and horizontal reservations.
Unpublished mimeo . 1.1
Sundell, A. (2014). Are Formal Civil Service Examinations the Most MeritocraticWay to Recruit Civil Servants? Not in All Countries.
Public Administration , (2),440–457. 1.1 Tadenuma, K. and
Thomson, W. (1991). No-envy and consistency in economies withindivisible goods.
Econometrica: Journal of the Econometric Society , pp. 1755–1767.1.1
Thomson, W. (1990). The consistency principle.
Game theory and applications , ,215. 1.1 — (1994). Consistent solutions to the problem of fair division when preferences aresingle-peaked. Journal of Economic Theory , (2), 219–245. 1.1 Appendix
Formal description of the rules.
For the descriptions in this section, consider asgiven a set W of workers, a set M ⊆ W of minority workers, a set A of workerspreviously hired, a sequence of hires q r = h q , q , . . . , q k i , and a score profile s W . Sequential Priority (SP rule) . Round 1:
Let W = W . The highest scoring q workers in W areselected. Let A , be the set of selected workers, where for each w ∈ A and each w ′ ∈ W \ A we have s w > s w ′ , and | A | = q . Round k > : Let W k = W k − \ A k − . The highest scoring q k workersin W k are selected. Let A k be the set of selected workers, where for each w ∈ A k and each w ′ ∈ W k \ A k we have s w > s w ′ , and | A k | = q k .The assignment selected by SP rule is ϕ SP ( W, h q , . . . , q r i ) = [ a ≤ r A a . Sequential Adjusted Minority Reserves (SA rule) . Round 1:
IRING FROM A POOL OF WORKERS 25
Step 1.1:
Let W , = W , M , = M ∩ W , and q , = ⌈ m × q ⌉ . Thehighest scoring min { q , , | M , |} workers in M , are selected. Let A , bethe set of selected workers, where A , ⊆ M , . Step 1.2:
Let W , = W , \ A , , M , = M ∩ W , and q , = q −| A , | .The highest scoring q , workers in W , are selected. Let A , be the setof selected workers. Round k > : Step k.1:
Let W k, = W k − , \ A k − , , M k, = M ∩ ( W k − , \ A k − , ) and q k, = ⌈ min { max { m − ω ( A , )+ ... + ω ( A k − , ) q k , } × q k , | M k, |}⌉ . The highestscoring q k, workers in M k, are selected. Let A k, be the set of selectedworkers. Step k.2:
Let W k, = W k, \ A k, , M k, = M ∩ W k, and q k, = q k −| A k, | .The highest scoring q k, workers are selected from W k, . Let A k, be theset of selected workers.The assignment selected by the SA rule is ϕ SA ( W, A, h q , . . . , q r i ) = [ a ≤ ri ∈{ , } A a,i . Sequential use of minority reserves (SM rule) . Round 1:
Step 1.1:
Let W , = W , M , = M ∩ W , and q , = ⌈ m × q ⌉ . Thehighest scoring min { q , , | M , |} workers are selected from M , . Let A , be the set of selected workers. Step 1.2:
Let W , = W \ A , , M , = M ∩ W , and q , = q − | A , | .The highest scoring q , workers are selected from W , . Let A , be theset of selected workers. Round k > : Step k.1:
Let W k, = W k − , \ A k − , , M k, = M ∩ ( W k − , \ A k − , ) and q k, = ⌈ m × q k ⌉ . The highest scoring min { q k, , | M k, |} workers from M k, are selected. Let A k, be the set of selected workers. Step k.2:
Let W k, = W \ A k, , M k, = M ∩ W k, and q k, = q k − | A k, | .The highest scoring q k, workers from W k, are selected. Let A k, be theset of selected workers.The assignment produced by the SM rule is ϕ SM ( W, q r ) = [ a ≤ ri ∈{ , } A a,i . Brazilian assignment rule (B rule) . The rule first identifies a large number k (which is larger than the totalnumber of vacancies to be filled but no larger than | W | ). Then two groupsare identified: ( i ) T M , which is the set with the top k × m minority IRING FROM A POOL OF WORKERS 26 workers:
T M ⊆ M with | T M | = ⌈ k × m ⌉ such that for each w ∈ T M and each w ′ ∈ M \ T M , we have s w > s w ′ and ( ii ) O , which is the setwith the top k (1 − m ) workers among those who were not chosen in ( i ) ,that is: O ⊆ W \ T M such that | O | = ⌊ k (1 − m ) ⌋ and for each w ∈ O and w ′ ∈ W \ ( O ∪ T M ) , we have s w > s w ′ . Within each round a ≤ r ,we have two steps. Round 1:
Step 1.1:
Let O , = O , T M , = T M and q , = ⌈ m × q ⌉ . The high-est scoring min { q , , | T M , |} minority workers are selected from T M , .Let A , be the set of selected workers. Step 1.2:
Let O , = O , , T M , = T M \ A , and q , = q − | A , | .The highest scoring q , workers are selected from O , . Let A , be theset of selected workers. Round k > : Step k.1:
Let O k, = O k − , \ A k − , , T M k, = T M k − , and q k, = ⌈ m × q k ⌉ . The highest scoring min { q k, , | T M k, |} minority workers areselected from T M k, . Let A k, be the set of selected workers. Step k.2:
Let O k, = O k, , T M k, = T M \ A k, and q k, = q k − | A k, | .The highest scoring q k, workers are selected from O k, . Let A k, be theset of selected workers.The assignment produced by the B rule is ϕ B ( W, q r ) = [ a ≤ ri ∈{ , } A a,i . French assignment rule (F rule) . Let m be the target ratio of people with disabilities, s OW be a scoringprofile for workers in the open competition and s DW be a scoring profilefor workers in the competition for workers with disabilities. Round 1:
Policy 1:
Let W , = W , M , = M ∩ W , . The highest scoring min { q , , | W , |} workers, with respect to s OW , are selected from W , . Let A be the set of selected workers. Policy 2:
Let W , = W , M , = M . The highest scoring min {⌊ (1 − m ) × q , ⌋ , | M , |} workers, with respect to s DW , are selected from M , .Let A , be the set of workers selected in this step. Then, the highestscoring min {⌈ m × q , ⌉ , | W , \ A , |} workers, with respect to s OW , areselected from W , \ A , . Let A , be the set of selected workers in thisstep, and let A = A , ∪ A , . Round k > : IRING FROM A POOL OF WORKERS 27
Step k.1:
Let W k, = W k − , \ A k − , , T A k, = S k − i =1 A i . Let q k, = min n max n m × (cid:16)P ki =1 q i (cid:17) − ω ( T A k, ) , o , | M k, | o . The highest scor-ing q k, workers, with respect to s DW , are selected from M k, . Let A k, bethe set of workers selected in this step. Step k.2:
Let W k, = W k, \ A k, , and q k, = q k − | A k, | . The highestscoring q k, workers, with respect to s OW , are selected from W k, . Let A k, be the set of selected workers, and A k = A k, ∪ A k, .The assignment produced by the F rule is ϕ F ( W, q r ) = [ a ≤ r A a . Proofs.
Proof of Proposition 1.
Example 1 shows that the sequential use of minority reserves isneither aggregation independent nor fair. To see that it respects minority rights, noticethat every time q workers are hired, at least m × q minority workers are among them.As a result, a proportion of at least m of the workers hired, at any point, is in M andtherefore the rule respects minority rights. (cid:3) Proof of Theorem 1.
First, we show that the SA rule respects minority rights and isminority fair.By definition, the SA rule respects minority rights , at the step k. of each round k ,selects minority workers to satisfy the minimum requirement up to that round. Notethat when there are not enough minority workers, SA selects all the available minorityworkers.Now, we show that the rule is minority fair.Let A ≡ ϕ SA ( W, h q , . . . , q r i ) be the selection made for the problem. We want toshow that ( i ) for each w, w ′ ∈ W \ M , if w ∈ A and w ′ / ∈ A , then s w > s w ′ , ( ii ) foreach w, w ′ ∈ M , if w ∈ A and w ′ / ∈ A , then s w > s w ′ . ( iii ) for each w ∈ W \ M and w ′ ∈ M , if s w < s w ′ and w ∈ A , then w ′ ∈ A , ( iv ) if there is w ∈ W \ M and w ′ ∈ M with s w > s w ′ , w / ∈ A and w ′ ∈ A , then ω ( A ) / | A | ≤ m .First note that cases ( i ) , ( ii ) and ( iii ) hold trivially as at step k. of each round k ,the rule selects the highest scoring workers in M , and in step k. it selects the highestscoring workers.Suppose, for contradiction, that there is w ∈ W \ M and w ′ ∈ M with s w > s w ′ , w / ∈ A and w ′ ∈ A , but ω ( A ) / | A | > m . Note that w ′ cannot be selected at step k. of anyround k , as w would have been selected as well. The only case in which the candidate w ′ is selected is during step ℓ. of some round ℓ . Since s w > s w ′ , w / ∈ A and w ′ ∈ A ,then | top q ( W ) ∩ M | < m × q , where q = P a ≤ r q a . Thus, at step r. of the last round r , That is, the only way to hire a minority worker with a lower score and not the non-minority with ahigher score, is to satisfy the minority requirements. As we mentioned earlier, the worker w ′ is hiredduring step ℓ. of some round ℓ , where selection occurs among minorities only. IRING FROM A POOL OF WORKERS 28 a selection is made so that | ( [ a For any given set of workers W , minority workers M ⊆ W , score profile s W , q ≥ and m ≥ , there exists only one set W ∗ ⊆ W that respects minority rights,is minority fair and such that | W ∗ | = q .Proof. First, note that property (i) of minority fairness implies that if a set W ∗ isminority fair, it contains the set top ω ( W ∗ ) ( M ) , that is, the top ω ( W ∗ ) highest scoringworkers in M , and the set top q − ω ( W ∗ ) ( W \ M ) , the q − ω ( W ∗ ) highest scoring workersin W \ M , both with respect to s W .Suppose, for contradiction, that there are W ⊆ W and W ⊆ W , where both W and W respect minority rights and are minority fair, | W | = | W | = q , and W = W .Note first that if | M | < m × q , respecting minority rights implies that M ⊂ W and M ⊂ W . Minority fairness implies, moreover, that top q −| M | ( W \ M ) ⊆ W and top q −| M | ( W \ M ) ⊆ W . But then W = W , a contradiction. It must be, therefore,that | M | ≥ m × q .Next, note that minority fairness implies that ω ( W ) = ω ( W ) . To see that, noticethat if ω ( W ) = ω ( W ) = m ∗ , top m ∗ ( M ) ⊆ W , top q − m ∗ ( W \ M ) ⊆ W , top m ∗ ( M ) ⊆ W , and top q − m ∗ ( W \ M ) ⊆ W . But this would imply that W = W , a contradiction.Suppose now, without loss of generality, that ω ( W ) > ω ( W ) . Since W respectsminority rights, ω ( W ) ≥ m × q , and therefore ω ( W ) > ω ( W ) ≥ m × q . Therefore,there is a worker w ∗ ∈ W \ M such that w ∗ ∈ W and w ∗ W , and a worker w ∗ ∈ M such that w ∗ ∈ W and w ∗ W .We have two cases to consider. First, suppose that s w ∗ > s w ∗ . This would violate W being minority fair, since m ( W ) > m × q and w ∗ W . Then it must be that s w ∗ > s w ∗ . But then, since W is minority fair, condition (ii) implies that w ∗ ∈ W , acontradiction.We conclude, therefore, that W = W is false, proving uniqueness. (cid:3) IRING FROM A POOL OF WORKERS 29 Since the SA rule respects minority rights and is minority fair, lemma 1 implies thatthis is the only such rule. (cid:3) Proof of Theorem 2. Let ϕ be a rule that is static minority fair, satisfies static minorityrights, and is aggregation independent. Let λ ∗ be any sequence of hires.We will follow by induction on the rounds in λ ∗ . First, the base h q i : from lemma1, there is a unique set W ⊆ W that is minority fair and respects minority rights.Both ϕ and ϕ SA are static minority fair and respect static minority rights. Therefore, ϕ ( W, h q i ) = ϕ SA ( W, h q i ) = W .For the induction step, assume that ϕ ( W, h q , q , . . . , q ℓ i ) = ϕ SA ( W, h q , q , . . . , q ℓ i ) .Since ϕ is aggregation independent, the following is true: ϕ ( W, h q , q , . . . , q ℓ i ) = ϕ ( W, h q i ) where q = P ℓi =1 q i . Let H = ϕ ( W, h q i ) . Aggregation independence of ϕ implies,moreover, that: ϕ ( W, h q i ) ∪ ϕ ( W, H, h q ℓ +1 i ) = ϕ ( W, h q + q ℓ +1 i ) ( ∗ ) Since both ϕ and ϕ SA are static minority fair and respect static minority rights, ourclaim above implies that ϕ ( W, h q i ) = ϕ SA ( W, h q i ) and ϕ ( W, h q + q ℓ +1 i ) = ϕ SA ( W, h q + q ℓ +1 i ) .Since workers cannot be hired more than once, ϕ ( W, h q i ) ∩ ϕ ( W, H, h q ℓ +1 i ) = ∅ .Therefore, there is a unique value of ϕ ( W, H, h q ℓ +1 i ) that satisfies the equality ( ∗ ) above, implying that ϕ ( W, H, h q ℓ +1 i ) = ϕ SA ( W, H, h q ℓ +1 i ) , and therefore that: ϕ ( W, h q , q , . . . , q ℓ , q ℓ +1 i ) = ϕ SA ( W, h q , q , . . . , q ℓ , q ℓ +1 i ) finishing our proof. (cid:3) Proof of Proposition 2. Let W ∗ = { w , w , w , w , w } with scores s W = (50 , , , , .For simplicity, we will use m = 0 . .Consider first the case M ∗ = { w , w } . If q = 2 , ϕ F ( { W ∗ , M ∗ } , q ) = { w , w } , whichfails to satisfy minority rights.Consider now the case M ∗ = { w , w } . Consider two possibilities: q = q = 2 and q = 4 . Then ϕ F ( { W ∗ , M ∗ } , h q , q i ) = { w , w , w , w } but ϕ F ( { W ∗ , M ∗ } , q ) = { w , w , w , w } , a violation of aggregation independence. It is easy to see that the rule that results from policy 2, under the given assumptions,is equivalent to the sequential adjusted minority reserves rule. Therefore, Policy 2 ofthe French assignment rule respects minority rights, is aggregation independent, and isminority fair. (cid:3) IRING FROM A POOL OF WORKERS 30 Proof of Proposition 3. We will show that the Brazilian rule respects minority rightsand is aggregation independent, but fails to be minority fair.By assumption, no more than k workers may be hired in total. Therefore, forany q workers to be hired in any given round there should be at least ⌈ q × m ⌉ mi-nority workers in T M and q − ⌈ q × m ⌉ workers in O . As a result, the Brazilianrule acts as two parallel sequential priority rules: one in T M and one in O . There-fore, the combination of both is evidently aggregation independent. Next, notice thatagain because of the assumption on the value of k , | M | ≥ m × P ti =1 q i . Moreover,since for any q ∈ { q , . . . , q t } there are at least ⌈ q × m ⌉ minority workers in T M , ω ( ϕ ( W, h q , . . . , q t i )) ≥ m × P q i and by assumption on k , | ϕ ( W, h q , . . . , q t i ) | = P q i therefore ω ( ϕ ( W, h q , . . . , q t i )) / | ϕ ( W, h q , . . . , q t i ) | ≥ m , implying that the Brazilianrule respects minority rights. Finally, example 4 shows that the rule is not minorityfair. (cid:3) Proof of Proposition 4. Example 5 shows that the NSW rule is neither fair nor minorityfair. Moreover, since in our results we assume that the number of men and women arealways large enough, the NSW consists of two parallel sequential priority hirings (one formales, the other for female workers), and therefore satisfies aggregation independence.Finally, it respects minority rights, since the number of male and female workers hiredis always the same. (cid:3) Proof of Theorem 3. The single priority rule satisfying common top, aggregation inde-pendence and permutation independence is straightforward to see.Denote by sequence of single hirings a plural sequence of hires of the form Λ = h ( i , , ( i , , ( i , , . . . i . That is, every hire made by any institution in any roundconsists of only one worker. Claim 1. Let W be a set of workers, and Φ be a rule that satisfies common top andpermutation independence. There exists a ranking ≻ ∗ over W such that for any sequenceof single hirings Λ , Φ( W, Λ) = Φ ≻ ∗ ( W, Λ) , where Φ ≻ ∗ is the single priority rule thatuses ≻ ∗ .Proof. We will prove by induction on the number of hires in a plural sequence of hires.That is, we will show that there exists a ranking ≻ ∗ , independent of the sequence ofhires, that is followed by Φ as a single priority.In the remaining steps of the proof, the set W and the rule Φ are given, and sofor any plural sequence of hires Λ , we will use the notation { Λ } to represent the set S i ∈ I Φ i ( W, Λ) . That is, { Λ } is the set of workers in W hired by some institution under Φ after the sequence of hires Λ . Since we will only look at single hirings, we willrepresent plural sequences of hires as sequences of institutions and use h i , i , . . . i torepresent h ( i , , ( i , , . . . i . IRING FROM A POOL OF WORKERS 31 We will use (PI) to indicate that we used the property of permutation independence of Φ , (AI) to indicate that we used aggregation independence , and (CT) for commontop .Moreover, we will use (P*) to indicate that we are using the following fact:If Λ , Λ , and i ∈ I are such that { Λ } = { Λ } and Φ i ( W, Λ ) =Φ i ( W, Λ ) , then Φ i ( W, h Λ , i i ) = Φ i ( W, h Λ , i i ) . That is, if Λ and Λ are such that institution i hires the same set of workers, and the set ofworkers remaining after all of the hires in both plural sequences of hiresis the same, then i would hire the same worker after both Λ and Λ .This comes directly from the definition of a hiring rule Φ i . Induction Base The induction base is the case where the smallest number of hiresis made while still having at least two institutions hiring. Therefore | Λ | = 2 . Supposethat the claim is not true. That is, there might be plural sequences of hires with twohires that cannot be explained by a ranking ≻ ∗ over W . That implies that there are Λ = Λ , where Λ = h i , i i , Λ = h i , i i , and { Λ } 6 = { Λ } .Since the sequences of hires involve at least two institutions, i = i and i = i .Since Λ = Λ , there are two cases to consider: (i) i = i , and (ii) i = i . Consider (i).By (PI) , {h i , i i} = {h i , i i} . By (P*) , (CT) and the fact that i = i , {h i , i i} = {h i , i i} . By (PI) , {h i , i i} = {h i , i i} . By (P*) , (CT) and the fact that i = i , {h i , i i} = {h i , i i} . But then {h i , i i} = {h i , i i} , a contradiction. For case (ii), (PI) implies that {h i , i i} = {h i , i i} and {h i , i i} = {h i , i i} , which makes this caseequivalent to (i). Induction Step We now assume that for every sequence of single hirings Λ such that | Λ | ≤ k , the rule Φ hires according to the ranking ≻ ∗ . We will use (IA) to indicate that we are usingthis induction assumption .Suppose now that the claim is not true. That is, there are sequences of single hirings Λ , Λ , such that | Λ | = | Λ | = k , and institutions i , i ∈ I , for which {h Λ , i i} 6 = {h Λ , i i} . There are two cases to consider. Case (i): i = i . Let Λ a be a sequence of single hires, such that: • | Λ a | = k • The rounds in which i hires in Λ a are exactly the same in which i hires in Λ ,if any. • For the rounds in which i does not hire: – Let i be the institution hiring at the first round in which i doesn’t hire(note that this must exist, since Λ has hirings from at least two institu-tions). – Let i be the institution hiring at every other round of Λ a , if any. IRING FROM A POOL OF WORKERS 32 In Λ a , therefore, hires made by i , if any, are the same as in Λ , there is exactly onehire by i , and the remaining hires, if any, are made by i .By (IA) and (P*) , {h Λ , i i} = {h Λ a , i i} . Next, let Λ b be exactly as Λ a , exceptthat the single place where i is is replaced by i . By (PI) , {h Λ a , i i} = (cid:8) h Λ b , i i (cid:9) .Notice that Λ b contains all the hires made by i in Λ , in addition to one extra hirefrom i . All other hires in Λ b are made by i . That is, there is no hire from i in Λ b .Next, let Λ c a sequence of hires where the hires in Λ c are exactly the same as Λ ,except that: • Every round in which i hires in Λ , the hire is made by i instead, • Denote by t ∗ the first round in which i does not hire in Λ . Note that this mustexist, since Λ has hirings from at least two institutions. Let i hire in round t ∗ in Λ c instead.Notice, therefore, that there is no hire from i in Λ c . By (IA) and (P*) , therefore, (cid:8) h Λ b , i i (cid:9) = {h Λ c , i i} .Next, let Λ d be exactly as Λ c , except that the i in round t ∗ is replaced by i . By (PI) , {h Λ c , i i} = (cid:8) h Λ d , i i (cid:9) . Notice that the rounds in which i hires in Λ d are exactlythe same as in Λ , and as a result, (P*) and (IA) imply that the last hire made by i in h Λ d , i i is the same as in h Λ , i i . Not only that, (IA) implies that the set of workershired in the first k hires are the same, and therefore (cid:8) h Λ d , i i (cid:9) = {h Λ , i i} , implyingthat {h Λ , i i} = {h Λ , i i} , a contradiction. Case (ii): i = i . We will use three institutions in the following steps: i, i a , i b , where i = i = i , and i = i a = i b . Let Λ a be a sequence of single hires, such that | Λ a | = k , the rounds inwhich i hires in Λ are exactly the same in which i hires in Λ a , if any. Moreover, let i b be the institution hiring at the first round in which i doesn’t hire in Λ (note that thismust exist, since Λ has hirings from at least two institutions), and i a be the institutionhiring in every other rounds, if any. By (IA) and (P*) , {h Λ , i i} = {h Λ a , i i} .Next, let Λ b be exactly as Λ a , replacing the single place where i b is by i . By (PI) , {h Λ a , i i} = (cid:8) h Λ b , i b i (cid:9) .Next, let Λ c be exactly as Λ , except that every hire made by i b , if any, is madeinstead by i . Moreover, let i be the institution hiring at the first round in which i b doesn’t hire in Λ (note that this must exist, since Λ has hirings from at least twoinstitutions). Denote this round by t ∗ . Notice, therefore, that there is no hire from i b in Λ c . By (IA) and (P*) , therefore, (cid:8) h Λ b , i b i (cid:9) = {h Λ c , i b i} .Next, let Λ d be exactly as Λ c , except that the i in round t ∗ is replaced by i b . By (PI) , {h Λ c , i b i} = (cid:8) h Λ d , i i (cid:9) .Notice that the rounds in which i hires in Λ d are exactly the same as in Λ , and asa result, (P*) and (IA) imply that the last hire made by i in h Λ d , i i is the same as in h Λ , i i . Not only that, (IA) implies that the set of workers hired in the first k hires IRING FROM A POOL OF WORKERS 33 are the same, and therefore (cid:8) h Λ d , i i (cid:9) = {h Λ , i i} , implying that {h Λ , i i} = {h Λ , i i} ,a contradiction. (cid:3) Finally, let Λ = h ( i , q ) , ( i , q ) , . . . , ( i k , q k ) i be any plural sequence of hires and Φ ∗ be a rule that satisfies common top, permutation independence, and aggregationindependence. By (AI) : Φ ∗ ( W, Λ) = Φ ∗ W, h ( i , , . . . , ( i , | {z } q times , . . . , ( i k , , . . . , ( i k , | {z } q k times i That is, aggregation independence implies that each hire from an institution can besplit into single hires without changing the workers that are chosen, round by round. Our claim above implies, therefore, that the rule Φ ∗ must be single priority, finishingour proof. (cid:3) Notice that the property of aggregation independence holds for any initial matching µµ