Affirmative Action in India via Vertical, Horizontal, and Overlapping Reservations
aa r X i v : . [ ec on . T H ] F e b AFFIRMATIVE ACTION IN INDIA VIA VERTICAL, HORIZONTAL, ANDOVERLAPPING RESERVATIONS
TAYFUN S ¨ONMEZ AND M. BUMIN YENMEZA
BSTRACT . Sanctioned by its constitution, India is home to an elaborate affirmative actionprogram for allocation of public jobs, where historically discriminated groups are protectedwith vertical reservations implemented as “set asides,” and other disadvantaged groupsare protected with horizontal reservations implemented as “minimum guarantees.” Con-current implementation of these two policies with overlapping beneficiaries makes thisprogram more complex than others elsewhere. An allocation mechanism mandated by theSupreme Court judgement
Anil Kumar Gupta vs. Uttar Pradesh (1995) suffers from a num-ber of anomalies, including disadvantaged candidates losing positions to privileged candi-dates of lower merit, triggering countless litigations and disarray in the country. Foretellinga recent reform in India, we propose an alternative mechanism that resolves all anomalies,and uniquely characterize it with desiderata reflecting the laws of India. Subsequently rein-vented with an August 2020 High Court judgement and mandated for the state of Gujarat,our mechanism is endorsed for India with a December 2020 judgement of the SupremeCourt.
Keywords:
Market design, matching, affirmative action
JEL codes:
C78, D47
Date : First version: March 2019, this version: January 2021. This version subsumes and replacestwo distinct working papers “Affirmative Action in India via Vertical and Horizontal Reservations”(S ¨onmez and Yenmez, 2019) and “Affirmative Action with Overlapping Reserves” (S ¨onmez and Yenmez,2020).Both S ¨onmez and Yenmez are affiliated with the Department of Economics, Boston College, 140 Com-monwealth Ave, Chestnut Hill, MA, 02467. Emails: [email protected] , [email protected] . S ¨onmez ac-knowledges the research support of Goldman Sachs Gives via Dalinc Ariburnu—Goldman Sachs FacultyResearch Fund.
1. Introduction
Sanctioned by the country’s constitution, one of the world’s most comprehensive affir-mative action programs is implemented in India. Allocation of government positions andseats at publicly funded educational institutions have to comply with mandates outlinedby the landmark Supreme Court judgement
Indra Sawhney and others v. Union of India(1992) , widely known as the Mandal Commission Case . Under these mandates, an alloca-tion mechanism that relies on an objective merit list of candidates is integrated with twotypes of affirmative action policies referred to as vertical reservations and horizontal reser-vations . Of the two types of policies formulated by the constitution bench of the SupremeCourt, vertical reservations are intended as the higher-level protection policy, and as suchthey are mandated to be implemented on a “set aside” (or equivalently “over-and-above”)basis. This means that a position awarded to an individual who deserves an unreservedposition solely on the basis of her merit score does not count towards a vertically reservedposition if the individual belongs to a protected class. In the past these higher level pro-tections were exclusively intended for classes that faced historical discrimination suchas
Scheduled Castes (SC), Scheduled Tribes (ST), and
Other Backward Castes (OBC) , althoughwith a January 2019 amendment in the constitution their scope now includes
Economi-cally Weaker Sections (EWS) of the rest of the society. Horizontal reservations, on the otherhand, are intended as the lower-level protection policy, and as such they are mandated tobe implemented on a “minimum guarantee” basis. This means that a position awardedto a member of a beneficiary class for this lower-level protection policy always counts to-wards horizontally reserved positions, even if the individual receives this position solelyon the basis of her merit score. As of January 2021, persons with disabilities is the only groupin India that is eligible for horizontal reservations at the federal level. In several states,however, there are additional beneficiary groups for horizontal reservations. For exam-ple, horizontal reservations for women is mandated in several states including in Biharwith 35% (of the positions), Andhra Pradesh and Gujarat with 33 % each, and MadhyaPradesh, Uttarakhand, Chhattisgarh, Rajasthan, and Sikkim with 30% each. Other groupswho are eligible for horizontal reservations in various applications include ex-servicemen,sportsmen, and speakers of the local language .In India, beneficiary groups for vertical reservations do not overlap. An individualcan be a member of at most one (vertical) reserve-eligible category . This structure resultsin a straightforward implementation of vertical reservations in the absence of horizontal The case is available at https://indiankanoon.org/doc/1363234/ (last accessed on 01/19/2021). See the Supreme Court judgement
Union Of India & Anr vs National Federation Of The Blind & ... on 8October, 2013 , available at https://indiankanoon.org/doc/178530295/ (last accessed on 01/23/2021).
FFIRMATIVE ACTION IN INDIA 3 reservations: First open (i.e., unreserved) positions are allocated based on merit scores,and next for each of the (mutually exclusive) groups eligible for vertical reservation, po-sitions that are set aside are allocated to the remaining members of the protected groupagain based on their merit scores. Let us refer this simple allocation mechanism as the over-and-above choice rule . Assuming that beneficiary groups for horizontal reservationsdo not overlap either, the implementation of horizontal reservations is equally straight-forward: First horizontally protected positions are allocated to their intended beneficia-ries based on merit scores for each protected group, and next all unfilled positions (i.e.,open positions and unfilled horizontally protected positions) are allocated to remainingindividuals with highest merit scores. When implemented individually, these two basictypes of reservation policies are widespread throughout the world in a wide range of ap-plications including affirmative action policies in K-12 education or college admissions,H-1B visa allocation in the U.S., and most recently Covid-19 pandemic medical resourceallocation, although in some of these applications policymakers do not seem to appreciatethe distinction between the two policies. In that sense the clear distinction made betweenthe vertical and horizontal reservation policies in India at the Supreme Court level israther extraordinary. But despite the clear formulation and distinction of the these affir-mative action policies in the Indian legal framework, the country has endured thousandsof litigations on their real-life implementation. One of the driving forces for that disrup-tion is the following additional complexity in India pertaining to implementation of thesepolicies: • Unlike most applications of reservation policies in the rest of the world wherethese policies are implemented individually, vertical and horizontal reservationpolicies are implemented concurrently in India.There is one other potential complexity that further complicates the implementation ofhorizontal reservations: • In some of the applications in India, beneficiary groups for horizontal reservationsdo overlap .Even though the principles that govern implementation of vertical and horizontal reser-vation policies were clearly formulated in
Indra Sawhney (1992) , this judgement has notaddressed how to implement them. This was subsequently done in another judgement
Anil Kumar Gupta v. State of U.P. (1995) , where the Supreme Court formulated a choice rulefor this complex version of the problem by augmenting the basic over-and-above choice
S ¨ONMEZ AND YENMEZ rule with a horizontal adjustment subroutine that carries out the necessary corrections to ac-commodate the horizontal reservations. Through this subroutine, however,
Anil KumarGupta (1995) also introduced a number of anomalies to the resulting choice rule. We referthis allocation rule, federally mandated in India for twenty five years, as the
SCI-AKGchoice rule .While the aspect of overlapping horizontal reservations has received some attention inboth theoretical (Kurata et al. (2017)), and applied literature (Correa et al. (2019)), to thebest of our knowledge the concurrent implementation of vertical and horizontal reserva-tion policies has never been studied before. Moreover, the theoretical solution offered inKurata et al. (2017) for implementation of overlapping horizontal reservations is mainlyintended for a different variant of the problem where individuals have strict preferencesfor whether and which protection is invoked in securing a position. This modeling choiceallows formulating the problem as a special case of the celebrated matching with contracts model (Hatfield and Milgrom, 2005). Although Correa et al. (2019) apply this theoreticalsolution to the real-life K-12 school choice application in Chile, the resulting allocationmechanism suffers from three shortcomings presented in Section 3 largely due to the mis-match between the application and the theory developed by Kurata et al. (2017).In this paper we make contributions to microeconomic theory and also to the appliedfield of market design. Our main contributions to the field of market design are:(1) formulation of the legal framework for implementation of vertical and horizontalreservation policies in India,(2) formulation of the SCI-AKG choice rule along with its shortcomings, and docu-mentation of the the scale of its disruption in India due to some of these shortcom-ings,(3) introduction of a simple modification of the SCI-AKG choice rule that escapes allthese shortcomings, and(4) formulation of a number of related shortcomings of the Chilean choice rule forallocation of K-12 public school seats, and introduction of an alternative choicerule that escapes these shortcomings.Our main contributions to microeconomic theory, which in part overlap with our contri-butions to market design, are:(5) formulation of a simpler version of the model with overlapping horizontal reser-vations (but no vertical reservations), and in this context The case is available at https://indiankanoon.org/doc/1055016/ . See also
Rajesh Kumar Dariavs Rajasthan Public Service (2007) for a more detailed description of the procedure, available at https://indiankanoon.org/doc/698833/ . (Both cases last accessed on 01/29/2021).
FFIRMATIVE ACTION IN INDIA 5 (a) introduction of an allocation mechanism, the meritorious horizontal choice rule ,that differs from all its predecessors in its “smart” processing of reserved po-sitions,(b) characterization of the meritorious horizontal choice rule as the unique rulethat satisfies three basic axioms, and(6) formulation of a general version of the problem allowing for concurrent imple-mentation of vertical reservations with overlapping horizontal reservations, andin this context(a) introduction of an allocation mechanism, the two-step meritorious horizontal(2SMH) choice rule , that escapes all shortcomings of the SCI-AKG choice rule,and(b) characterization of the 2SMH choice rule as the unique choice rule that sat-isfies four axioms motivated by the legal framework in India as well as thechallenges faced in the country due to the limitations of the SCI-AKG choicerule.It is worthwhile to emphasize that, even though our proposed 2SMH choice rule es-capes all deficiencies of the SCI-AKG choice rule formulated and documented in our pa-per, its mechanics is surprisingly similar to this faulty mechanism. Our proposed 2SMHchoice rule can be seen as a modification of the SCI-AKG choice rule in two aspects, eachmodification eliminating some of the shortcomings.As we have highlighted earlier, the culprit behind the shortcomings of the SCI-AKGchoice rule is its horizontal adjustment subroutine , or more specifically how this subrou-tine is implemented. To illustrate the mechanics of this subroutine, consider the hori-zontal reservations for women. Whenever these protections are not satisfied for a verti-cal category v under the over-and-above choice rule, the subroutine replaces the lowestmerit-score men admitted to category- v positions with the highest merit-score unselectedwomen who are eligible for category v . The description of the subroutine is given in thelegal framework in a similar way for a single beneficiary group (such as women), but theprocedure is also well-defined and well-behaved when it is sequentially applied to multi-ple beneficiary groups provided that group memberships do not overlap. The first set ofshortcomings of the SCI-AKG choice rule, presented in detail in Section 3.1, emerge only This aspect of an allocation mechanism is generally considered to be highly plausible in the field ofmarket design, since it may potentially increase the likelihood of its acceptance in the field. Indeed, fol-lowing the March 2019 circulation of the first draft of this paper and while it was under revision for thisjournal, this prospect became increasingly more likely after two important developments, first a mandate ofa simpler version of this mechanism (defined in the absence of overlapping horizontal reservations) in theState of Gujarat in August 2020, and subsequently the endorsement of this simpler version by the SupremeCourt in December 2020. We present a detailed account of these developments in the Epilogue in Section 9.
S ¨ONMEZ AND YENMEZ when the beneficiary group memberships for horizontal reservations overlap. We refer tothis case as overlapping horizontal reservations . In this case, the sequential implementationof the subroutine • may differ in its outcome depending on the sequence of horizontal adjustments, • may provide an “under adjustment” of horizontal reservation protections, and • may result in “ineffective” adjustments by admitting needlessly low merit scoreindividuals.Fortunately, there is a clear solution to this conundrum that relies on enhancing the basicmechanics of the horizontal adjustment subroutine with a “maximal matching” proce-dure. In order to provide an intuition for this innovation, consider the following analogy.There is a gathering where lunch is served. One of the two guests, Violet, is a veg-etarian, whereas the second guest, Charlie, is flexible in his diet. Suppose there is onevegetarian and one chicken sandwich available for the two guests. If Charlie is servedhis lunch prior to Violet, it would be a blunder to offer him the only vegetarian sand-wich, since that would mean Violet has to skip her lunch. Charlie is flexible in his diet,whereas Violet is not, and serving the only vegetarian sandwich to Charlie results in wast-ing this valuable flexibility. A more careful server would have planned ahead taking intoViolet’s dietary restriction into consideration, and thus would have offered Charlie thechicken sandwich utilizing the flexibility in his diet. Now consider the accommodationof horizontal reservations in India. Suppose there is a female candidate Freya, a disabledfemale candidate Devi, one horizontally reserved position for female candidates, and onehorizontally reserved position for disabled candidates. Just as it is implausible to offerthe vegetable sandwich to Charlie in the above scenario, it is implausible to assign Devithe horizontally reserved position for female candidates and consequently deny Freya aposition since she is not qualified for the horizontally reserved position for disabled can-didates. Both horizontal reservation protections can be granted if Devi is instead assignedthe horizontally reserved position for disabled candidates. Ironically, many real-life ap-plications do not utilize the flexibility generated by candidates who qualify for multipletypes of horizontal reservations, and thus suffer from the very shortcoming we illustratein these two scenarios. In addition to our main application in India, the school choice sys-tem in Chile also suffers from the same shortcoming, precisely for the reason illustratedhere. A better design would not give up the flexibility generated by candidates whoqualify for multiple horizontal reservations, and instead it would capitalize on it. This isthe basic idea under our proposed meritorious horizontal choice rule. Indeed, not onlythis choice rule eliminates all three shortcomings of the horizontal adjustment subroutinedescribed above, it is also the only choice rule to do so (Theorem 2). FFIRMATIVE ACTION IN INDIA 7
The above-described limitations of the SCI-AKG rule may be viewed as of second orderimportance in the field, because they are are hard to verify in practice and they likely affectthe outcome rarely. The SCI-AKG choice rule, however, has two additional shortcomingsthat have been visibly disruptive in India in the last two decades. This time, the sourceof the anomaly has to do with how the horizontal adjustment subroutine is implementedrather than its mechanics. When the subroutine is applied for the open-category posi-tions, individuals from reserve-eligible categories are ruled out for potential horizontaladjustments and only individuals from the general category are deemed eligible for thisrole. That is, by announcing their eligibility for vertical reservation protections, individu-als risk losing their open-category horizontal reservation protections. Therefore, contraryto the objectives of affirmative action policies, the SCI-AKG choice rule can generate out-comes where a candidate from a disadvantaged group, despite being more meritorious,may lose a position to a candidate from a more privileged group. We refer to this irregu-larity as a failure to satisfy no justified envy (or a potential for justified envy ). In addition tothis highly implausible possibility, the SCI-AKG choice rule may also penalize candidatesfor reporting their eligibility for a vertical reservation protection, since it risks them to losetheir open-category horizontal reservation protections. In that sense, the procedure is not incentive compatible . These two anomalies, first formally introduced by (Ayg ¨un and B ´o,2016) in the context of Brazilian college admissions, are clearly against the philosophyof affirmative action. Not only they result in countless lawsuits, but they also provide aloophole in the procedure that can be used to discriminate against members of backwardclasses. In Section 8.1, we provide ample evidence that these shortcomings are responsiblefor widespread confusion in India, often resulting in legal action, inconsistent rulings, andeven defiance in some states through the implementation of better-behaved versions ofthe mandated procedure. We also provide evidence in Section 8.1.2 that, in some jurisdic-tions these shortcomings are exploited by local officials to discriminate against membersof backward classes. These litigations often result in the interruption of the recruitmentprocess, as well as reversals of recruitment decisions. Reporting a judgement by the Gu-jarat High Court, an article in
The Times of India highlights this very issue: The advertisement was issued in 2010 and recruitment took place in 2016amid too many litigations over the issue of reservation . . .
With the recent observation by the HC, the merit list will now be changedfor the third time. Those already selected and at present under trainingmight lose their jobs, and half a dozen new candidates might find their The Times of India story is available at https://timesofindia.indiatimes.com/city/ahmedabad/general-seat-vacated-by-quota-candidate-remains-general-hc/articleshowprint/57658109.cms (last accessed on 04/12/2019).
S ¨ONMEZ AND YENMEZ names on the new list. However, all appointments have been made by the HCconditionally and subject to final outcome of these multiple litigations.
A simple search of the phrase “horizontal reservation” via Indian Kanoon, a free searchengine for Indian Law, reveals the scale of the litigations relating to this concept. Exclud-ing cases at lower courts, as of 01/19/2021 there are 1961 cases at the Supreme Court andState High Courts related to the implementation of horizontal reservations. The potentialfor justified envy under the SCI-AKG choice rule is the primary culprit for a significantfraction of these cases. There is, however, reason to be optimistic that this impasse maybe finally coming to an end. That is because, just as this paper was under revision, therehas been a major breakthrough on this very issue; one that has a strong potential to bringan end to this predicament.Prior to the March 2019 circulation of the first draft of our paper, the inability of the SCI-AKG choice rule to eliminate justified envy was never directly addressed by the highestcourt of India, despite the large scale disarray it has created for over two decades in lowercourts. This situation has recently changed in a decisive way with its December 2020judgement
Saurav Yadav & Ors v. State of Uttar Pradesh & Ors (2020) , where the SupremeCourt mantated elimination of justified envy in allocation of all public positions, bringingand end to the 25-years tenure of its SCI-AKG choice rule. While the Supreme Court hasnot mandated an alternative choice rule, it has endorsed the two-step minimum guaranteechoice rule , a rule that is mandated in the State of Gujarat through its August 2020 HighCourt judgement Tamannaben Ashokbhai Desai v. Shital Amrutlal Nishar (2020) . Impor-tantly, this rule that is given for the basic case of non-overlapping horizontal reservationsin the High Court’s judgement, is equivalent to our proposed 2SMH choice rule in thissimple environment. We report a detailed account of these recent developments, and howthey relate to our paper in Section 9.The rest of our paper is organized as follows. After formulating the model in Section 2,we present a single-category analysis with overlapping horizontal reservations in Section3 and our analysis for the most general version of the model in Section 4. In Section 5we present a theoretical comparison of the SCI-AKG choice rule with the 2SMH choice Not all cases on “horizontal reservation” is about disputes related to elimination of justified envy orincentive compatibility. However during our search, we observed that the terminology of “migration” wasused is some of the cases to indicate the situations where members of reserved categories were allowed forhorizontal adjustments of open-category positions, and the more refined search of “horizontal reservation,migration” generated 256 cases at the Supreme Court and High Courts. As far as we can tell, a vast majorityof these cases relate to the shortcomings on justified envy. Judgement available in https://main.sci.gov.in/supremecourt/2019/44789/44789_2019_34_1501_25207_Judgement_18-Dec-2020.pdf ,last accessed 01/26/2021. Judgement available in , last ac-cessed 01/26/2021.
FFIRMATIVE ACTION IN INDIA 9 rule. We conclude the theoretical part of our paper with Section 6 where an extensionof our model is presented allowing for heterogeneity of positions across multiple institu-tions and a detailed literature review is presented. We devote Section 7 to our applicationin Chile, and Section 8 to our primary application from India, documenting elaborateevidence on the practical relevance of our findings. We conclude with an epilogue in Sec-tion 9 and relegate all proofs and the institutional background on vertical and horizontalreservations to an Online Appendix.
2. Model and the Primitives
There exist a finite set of individuals I and q identical positions. Each individual is inneed of a single position and has a distinct merit score. Let σ ( i ) ∈ R + denote the meritscore of individual i ∈ I . While individuals with higher merit scores have higher claimsfor a position in the absence of affirmative action policies, special provisions are providedfor the members of various disadvantaged populations through two types of affirmativeaction policies, called vertical reservations (VR) and horizontal reservations (HR) . Werefer these policies as VR protections and
HR protections . Qualification for VR protections is determined through a cat-egory membership. Let R denote the set of reserve-eligible categories and g denote the general category . Each individual belongs to a single category given by a category mem-bership function ρ : I → R ∪ ∅ , where, for any individual i ∈ I , • ρ ( i ) = ∅ indicates that i is a member of the general category g , and • ρ ( i ) = c indicates that i is a member of the reserve-eligible category c ∈ R .Members of the general category do not receive any special provisions under the VRpolicies. In contrast, q c positions are “set aside” exclusively for the members of category c ∈ R under the VR policies. These positions are referred to as category-c positions .Since no position is earmarked for the members of the general category, the remaining q o = q − ∑ c ∈R q c positions are open for all individuals. These positions are referred to as open-category positions (or category-o positions ). Observe that, • in contrast to category-c positions that are exclusively reserved for the members ofthe reserve-eligible category c ∈ R , • open-category positions are not exclusively reserved for the members of the gen-eral category g . While students can have the same merit score in practice, tie-breaking rules are used to strictlyrank them. For example, the Union Public Service Commission uses age and exam scores to breakties. See (lastaccessed on 6/7/2020).0 S ¨ONMEZ AND YENMEZ
HR protections . Qualification for VR protections is determined through a cat-egory membership. Let R denote the set of reserve-eligible categories and g denote the general category . Each individual belongs to a single category given by a category mem-bership function ρ : I → R ∪ ∅ , where, for any individual i ∈ I , • ρ ( i ) = ∅ indicates that i is a member of the general category g , and • ρ ( i ) = c indicates that i is a member of the reserve-eligible category c ∈ R .Members of the general category do not receive any special provisions under the VRpolicies. In contrast, q c positions are “set aside” exclusively for the members of category c ∈ R under the VR policies. These positions are referred to as category-c positions .Since no position is earmarked for the members of the general category, the remaining q o = q − ∑ c ∈R q c positions are open for all individuals. These positions are referred to as open-category positions (or category-o positions ). Observe that, • in contrast to category-c positions that are exclusively reserved for the members ofthe reserve-eligible category c ∈ R , • open-category positions are not exclusively reserved for the members of the gen-eral category g . While students can have the same merit score in practice, tie-breaking rules are used to strictlyrank them. For example, the Union Public Service Commission uses age and exam scores to breakties. See (lastaccessed on 6/7/2020).0 S ¨ONMEZ AND YENMEZ
Let V = R ∪ { o } denote the set of vertical categories for positions . Definition 1.
Given a category v ∈ V , an individual i ∈ I is eligible for positions in category vif v = o or ρ ( i ) = v .That is, while all individuals are eligible for open-category positions, only individualswho are members of category c are eligible for category- c positions for any reserve-eligiblecategory c ∈ R . Let I v ⊆ I denote the set of individuals eligible for category v ∈ V .Given a category v ∈ V , a single-category choice rule is a function C v : 2 I → I v suchthat, for any I ⊆ I , C v ( I ) ⊆ I ∩ I v and | C v ( I ) | ≤ q v .A choice rule is a profile of single-category choice rules C = ( C v ) v ∈V such that, for anyset of individuals I ⊆ I and two distinct categories v , v ′ ∈ V , C v ( I ) ∩ C v ′ ( I ) = ∅ .In addition to the specification of the recipients of the positions, a choice rule also specifiesthe categories of these positions.Given a choice rule C = ( C v ) v ∈V , define the aggregate choice rule b C : 2 I → I as b C ( I ) = [ v ∈V C v ( I ) for any I ⊆ I .For any set of individuals, the aggregate choice rule gives the set of chosen individualsacross all categories.In the absence of horizontal reservations introduced in Section 2.2, the following threefederally-mandated principles uniquely define a choice rule, making the implementationof VR policies straightforward. First, an allocation must respect inter se merit: Given twoindividuals from the same category, if the individual with the lower merit score is givena position, then the individual with the higher merit score must also be given a position.Next, when an individual from a reserve-eligible category receives an open-category po-sition on the basis of his merit score alone, this assignment does not count against thevertical reservations for his reserve-eligible category. This is the sense in which VR posi-tions are “set aside” for members of reserve-eligible categories, regardless of who receivesopen-category positions. Finally, subject to eligibility requirements, all positions has to befilled to the extent the two principles above would allow. It is easy to see that these threeprinciples uniquely imply the following choice rule: First, individuals with the highestmerit scores are allocated the open-category positions. Next, positions reserved for the FFIRMATIVE ACTION IN INDIA 11 reserve-eligible categories are allocated to the remaining members of these categories,again based on their merit scores.
In addition to a possiblemembership of a category, each individual also has a (possibly empty) set of traits. Eachtrait represents a disadvantage in the society, and the government provides individualswho have this trait with easier access to positions. The set of traits is finite and denotedby T , and the (possibly empty) set of traits of individual i ∈ I is given by τ ( i ) ⊆ T .For any reserve-eligible category c ∈ C and any trait t ∈ T , let q ct denote the numberof category- c positions that are HR-protected for individuals from category- c with trait- t .Similarly, let q ot denote the number of open-category positions that are HR-protected forindividuals with trait- t .For each vertical category v ∈ V , we assume that the aggregate number of HR-protected positions for category v is no more than the number of positions in this category,i.e., for every category v ∈ V , ∑ t ∈T q vt ≤ q v .Unlike the VR protections which are provided on a “set-aside” basis, HR protectionsare provided within each vertical category on a “minimum guarantee” basis. Importantly,while an individual can qualify for multiple HR protections through different traits, uponadmission she counts towards the minimum guarantee only for one of them. For examplea woman with disabilities can count towards either the minimum guarantee for the HRprotections for women or the minimum guarantee for the HR protections for personswith disabilities, but not both. We refer this convention of implementing HR protectionsas one-to-one HR matching . Under an alternative one-to-all HR matching convention,an individual can accommodate the minimum guarantee for all her traits. There are tworeasons for this important modeling choice. First of all, while either convention appearsto be allowed by Indian laws, the former is more widespread in the field. The secondreason is technical. The analysis of HR policies becomes both computationally hard andalso less elegant under the one-to-all HR matching convention.
3. Single-Category Analysis with Overlapping Horizontal Reservations
Since HR policies are implemented within vertical categories, we start our analysiswith the case of a single category, and thus with a special case of our model with HRprotections only. Throughout this section, we fix a category v ∈ V . This is sometimes explicitly indicated by the allocation rule and sometimes implicitly implied by thepractice of assigning individuals to category-trait pairs. See S ¨onmez and Yenmez (2020) for an analysis with two traits.
While each individual can benefit from VR protections through at most one reserve-eligible category, she can potentially benefit from HR protections through multiple traitseven within a single vertical category. This results in a potentially overlapping structurefor HR policies, thus making it technically more involved than the analysis of VR poli-cies. This technical aspect of the overlapping HR protections is also the source of threeshortcomings of the SCI-AKG choice rule. To motivate our axioms and the meritorioushorizontal choice rule that we introduce later in this section, we first present the mechan-ics and shortcomings of the SCI subroutine responsible for handling the HR protections.
The mechanics forimplementing HR protections is described in the two Supreme Court judgements
Anil Ku-mar Gupta (1995) and
Rajesh Kumar Daria (2007) . Both judgements describe the procedurefor a single trait, although the procedure can be repeated sequentially for each trait. Inour description below, we adhere to this straightforward extension of the procedure. As we argue in Section 2.1, implementing VR protections is straightforward in the ab-sence of HR protections. First open-category positions are allocated to the highest meritscore candidates (across all categories), followed by the positions at each reserve-eligiblecategory to the highest merit score remaining candidates from these categories. This is in-deed the first step of the SCI-AKG choice rule. Once a tentative assignment is determined,the necessary adjustments are subsequently made to implement HR protections, first forthe open-category positions, and subsequently for positions at each reserve-eligible cate-gory. The adjustment process is repeated for each trait.Formally, for a given category v ∈ V of positions, let a set of individuals J ⊆ I v whoare tentatively assigned to category- v positions and a set of individuals K ⊆ I v \ J whoare eligible for horizontal adjustments at category v be such that(1) | J | = q v and(2) σ ( i ) > σ ( i ′ ) for any i ∈ J and i ′ ∈ K .Then for a given processing sequence t , t , . . . , t |T | of traits, the horizontal adjustmentprocess is carried out with the following procedure. AKG Horizontal Adjustment Subroutine (AKG-HAS)Step 1 (Trait- t adjustments) : Let r be the number of individuals in J with trait t . Case 1. r ≥ q vt The description of this mechanics in the Supreme Court judgements
Anil Kumar Gupta (1995) and
Rajesh Kumar Daria (2007) can be seen in Section C.3 of the Online Appendix.
FFIRMATIVE ACTION IN INDIA 13
Let J be the set of q vt -highest merit score individuals in J with trait t . Finalizetheir assignments as the recipients of trait- t HR-protected positions within cate-gory v . Proceed to Step 2. Case 2. r < q vt Let J m be the set of all individuals in J with trait t . Let s be the number indi-viduals in K who have trait t . Let J h be • the set of ( q vt − | J m | ) highest merit score individuals in K who have trait t if s ≥ q vt − | J m | , and • the set of all individuals in K who have trait t if s < q vt − | J m | .Let J = J m ∪ J h and finalize their assignments as the recipients of trait- t HR-protected positions within category v . Proceed to Step 2. Step k ∈ {
2, . . . , |T |} (Trait- t k adjustments) : Let r k be the number of individualsin J \ S k − ℓ = J ℓ with trait t k . Case 1. r k ≥ q vt k Let J k be the set of q vt k highest merit score individuals in J \ S k − ℓ = J ℓ with trait t k . Finalize their assignments as the recipients of trait- t k HR-protected positionswithin category v . Proceed to Step 2. Case 2. r k < q vt k Let J km be the set of all individuals in J \ S k − ℓ = J ℓ with trait t k . Let s k be the numberindividuals in K \ S k − ℓ = J ℓ with trait t k . Let J kh be • the set of ( q vt − | J km | ) highest merit score individuals in K \ S k − ℓ = J ℓ who havetrait t k if s k ≥ q vt k − | J km | , and • the set of all individuals in K \ S k − ℓ = J ℓ who have trait t k if s k < q vt k − | J km | .Let J k = J km ∪ J kh and finalize their assignments as the recipients of trait- t k HR-protected positions within category v . Proceed to Step (k+1). Step ( |T | + ) (Finalization of category- v no-trait assignments) : Let J be the setof (cid:0) q v − ∑ |T | ℓ = | J ℓ | (cid:1) highest merit score individuals in J \ S |T | ℓ = J ℓ The procedure selects the set of individuals in S |T | ℓ = J ℓ . Here J is the set of in-dividuals from the original group J who have survived the horizontal adjustmentprocess without invoking any HR protection, and J k is the set of individuals whoaccommodate trait- t k HR protections for any trait t k . While all individuals in J k accommodate trait- t k HR protections, only those who are in the set J k \ J owe their assignments to trait- t k HR protections.
When each individual has at most one trait, it is easy to see that the processing sequenceof traits becomes immaterial under the AKG-HAS, and it produces the same outcomeas the following category- v minimum guarantee choice rule C vmg (Echenique and Yenmez(2015)) applied to the set of individuals J ∪ K . Minimum Guarantee Choice Rule C vmg For every I ⊆ I v , Step 1:
For each trait t ∈ T , • choose all individuals in I with trait t if the number of trait- t individuals in I is no more than q vt , and • q vt highest merit-score individuals in I with trait t otherwise. Step 2:
For positions unfilled in Step 1, choose unassigned individuals in I withhighest merit scores.Our first result formulates this observation. Proposition 1.
Suppose that each individual has at most one trait. Let v ∈ V be any categoryof positions, J ⊆ I v be a set of individuals who are tentatively assigned category-v positions, andK ⊆ I v \ J be a set of unmatched individuals who are eligible for category-v positions. If | J | = q v and every individual in J has a higher merit score than every individual in K, then C vmg ( J ∪ K ) isthe set of individuals who are assigned to category-v positions under the AKG-HAS. While well-behaved when each individual has no more than one trait, this proceduresuffers from three shortcomings when individuals have multiple traits. We next presentthese shortcomings in two examples.
Example 1.
There is one category (say open category), four individuals i , i , i , i andtwo positions. There are two traits t , t with one HR-protected position for each trait.Individual i has both traits, individuals i , i have no trait and individual i has trait t only. Individuals are merit ranked as σ ( i ) > σ ( i ) > σ ( i ) > σ ( i ) .Prior to horizontal adjustments, the positions are tentatively awarded to individuals i and i . Since only one of the minimum guarantees can be accommodated by this group,the AKG-HAS is invoked. Nonetheless, if trait- t adjustments are carried out prior totrait- t adjustments, this allocation does not change through the AKG-HAS: Once thehighest merit score individual i accounts for the trait- t minimum guarantee, no one elsehas trait t . Hence the final set of awardees remains unaltered from the tentative oneas { i , i } . If, however, trait- t adjustments are carried out prior to trait- t adjustments, FFIRMATIVE ACTION IN INDIA 15 then the allocation changes through the AKG-HAS: This time the highest merit score indi-vidual i accounts for the trait- t minimum guarantee instead, and the trait- t minimumguarantee can be accommodated subsequently by individual i . Hence in this secondscenario, the final set of awardees is { i , i } .Two shortcomings of the AKG-HAS is revealed by Example 1:(1) The outcome of this adjustment procedure depends on the processing sequence ofthe traits, and hence the single-category choice rule induced through this subrou-tine is not well-defined.(2) Fewer than the maximum feasible number of HR protections may be accommo-dated under this procedure.Essentially, Example 1 shows that the mechanical processing sequence of traits may resultin an “under adjustment” under the AKG-HAS. Our next example reveals that, it can alsoresult in adjustments by admitting needlessly low merit score individuals. Example 2.
There is one category (say open category), four individuals i , i , i , i andtwo positions. There are two traits t , t with one HR-protected position for each trait.Individual i has both traits, individual i has no trait, individual i has only trait t , andindividual i has only trait t . Individuals are merit ranked as σ ( i ) > σ ( i ) > σ ( i ) > σ ( i ) .Prior to horizontal adjustments, the positions are tentatively awarded to individuals i and i . Since only one of the minimum guarantees can be accommodated by thisgroup, the AKG-HAS is invoked. If trait- t adjustments are carried out prior to trait- t adjustments, then the AKG-HAS replaces individual i with individual i : The highestmerit score individual i already accounts for the trait- t minimum guarantee, and sub-sequently individual i is the only remaining individual with trait t . Hence the final setof awardees is { i , i } . If, however, trait- t adjustments are carried out prior to trait- t adjustments, then the AKG-HAS instead replaces individual i with individual i : Thehighest merit score individual i this time accounts for the trait- t minimum guarantee,and subsequently individual i is the only remaining individual with trait t . Hence thefinal set of awardees is { i , i } .Example 2 reveals a third shortcoming of the AKG-HAS: Depending on the process-ing sequence of traits, this procedure may carry out its adjustments in an “ineffective”way by admitting lower merit-score individuals than it is necessary. In this example, itis not necessary to admit the lowest merit-score individual i to accommodate the HRprotections. These two examples not only motivate the axioms formulated in Section 3.2, but alsothe meritorious horizontal choice rule introduced in Section 3.3.
To formulate our single-category axioms,we use the following construction.Given a category v ∈ V and a set of individuals I ⊆ I v , construct the following two-sided category- v HR graph . On one side of the graph, there are individuals in I . On theother side, there are HR-protected positions for category v . Let H vt denote the set of trait- t HR-protected positions for category v and let H v = S t ∈T H vt . There are q vt positions in H vt and ∑ t ∈T q vt positions in H v . An individual i ∈ I and a position p ∈ H vt are connected inthis graph if and only if individual i has trait t . Definition 2.
Given a category v ∈ V and a set of individuals I ⊆ I v , a trait-matching ofindividuals in I with HR-protected positions in H v is a function µ : I → H v ∪ { ∅ } such that(1) for any i ∈ I and µ ( i ) ∈ H v , individual i is connected with position µ ( i ) , and(2) for any i , j ∈ I, µ ( i ) = µ ( j ) = ∅ = ⇒ i = j . Definition 3.
Given a category v ∈ V and a set of individuals I ⊆ I v , a trait-matching ofindividuals in I with HR-protected positions in H v has maximum cardinality in (category-v ) HR graph if there exists no other trait-matching that assigns a strictly higher number of HR-protected positions to individuals.
Let n v ( I ) denote the maximum number of category- v HR-protected positions that canbe assigned to individuals in I . Definition 4.
Given a category v ∈ V and a set of individuals I ⊆ I v , an individual i ∈ I v \ I increases (category-v ) HR utilization of I ifn v ( I ∪ { i } ) = n v (cid:0) I ) + Definition 5.
Given a category v ∈ V and a set of individuals I ⊆ I v , a set of individuals J ⊆ I maximally accommodates (category-v ) HR protections for I ifn v ( J ) = n v ( I ) . This number can be found through a number of polynomial time algorithms such as
Edmonds’ BlossomAlgorithm (Edmonds, 1965).
FFIRMATIVE ACTION IN INDIA 17
Given a category v ∈ V , a single-category choice rule C v maximally accommodates HRprotections , if for every set of individuals I ⊆ I v , the set of selected individuals C v ( I ) maximallyaccommodates HR protections for I. Our second axiom is motivated by Example 2, and it precludes unnecessary rejection ofhigher merit score individuals at the expense of lower merit score ones due to suboptimalaccounting of individuals for HR-protected positions.
Definition 6.
Given a category v ∈ V , a single-category choice rule C v satisfies no justifiedenvy , if for every I ⊆ I v , i ∈ C v ( I ) , and j ∈ I \ C v ( I ) , σ ( j ) > σ ( i ) = ⇒ n v (cid:0) ( C v ( I ) \ { i } ) ∪ { j } (cid:1) < n v ( C v ( I )) .In words, if a higher merit score individual j is rejected even though a lower merit scoreindividual i is chosen, then it must be the case that replacing i with j would decrease thenumber of HR-protected positions that can be filled with intended beneficiaries. Whenthis condition is violated, we say that there is justified envy , which means that there exista set of individuals I and two individuals i , j ∈ I such that(1) σ ( j ) > σ ( i ) ,(2) i ∈ C v ( I ) ,(3) j / ∈ C v ( I ) , and(4) n v (cid:0) ( C v ( I ) \ { i } ) S { j } (cid:1) ≥ n v ( C v ( I )) .Therefore, when there is justified envy, a lower merit score individual can be replacedwith a higher merit score one without any adverse affect on the intended HR policies.Our third axiom, standard in the analysis of choice rules, is a weak efficiency property. Definition 7.
Given a category v ∈ V , a single-category choice rule C v is non-wasteful , if forevery I ⊆ I v , | C v ( I ) | = min {| I | , q v } .Equivalently, non-wastefulness requires that an individual is rejected only when noposition remains. We are ready to introduce a single-categorychoice rule that escapes the shortcomings presented in Examples 1 and 2. The main inno-vation in this choice rule is the optimization it carries out to determine who is to accountfor each minimum guarantee when some of the individuals can account for one or an-other due to multiple traits they have. Intuitively, this choice rule exploits the flexibilityin trait-matching in order to accommodate the HR protections with higher merit-scoreindividuals.
Given a category v ∈ V and a set of individuals I ⊆ I v , the outcome of this choice ruleis obtained with the following procedure. Meritorious Horizontal Choice Rule C v M Step 1.1 : Choose the highest merit-score individual in I with a trait for a HR-protected position. Denote this individual by i and let I = { i } . If there exists nosuch individual, proceed to Step 2. Step 1.k ( k ∈ {
2, . . . , ∑ t ∈T q vt } ): Assuming such an individual exists, choose thehighest merit-score individual in I \ I k − who increases the HR utilization of I k − . Denote this individual by i k and let I k = I k − ∪ { i k } . If no such individual exists,proceed to Step 2. Step 2 : For unfilled positions, choose unassigned individuals with highest meritscores until either all positions are filled or all individuals are selected.When the number of individuals is less than q v , this procedure selects all individuals.Otherwise, if there are more than q v individuals, then it chooses a set with q v individuals. Example 3.
Consider the following economy: • I = { i , i , i , i , i , i , i } , • T = { t , t , t } , • q vt = q vt = q vt = q v = • σ ( i ) > σ ( i ) > σ ( i ) > σ ( i ) > σ ( i ) > σ ( i ) > σ ( i ) , • τ ( i ) = ∅ , τ ( i ) = { t , t , t } , τ ( i ) = ∅ , τ ( i ) = { t , t } , τ ( i ) = { t } , τ ( i ) = { t } , τ ( i ) = { t } .The HR graph for this allocation problem has one HR-protected position for each traitand three HR-protected positions in total. An individual is connected with a position ifthe individual has the corresponding trait. The HR graph is depicted in Figure 1.Let I be the set of individuals in consideration. The meritorious horizontal choice ruleworks as follows. Having at least one trait each, only individuals i , i , i , i , and i arequalified to receive an HR-protected position at the first step. At Step 1.1, individual i is selected because she is the highest merit-score individual who qualifies for a HR-protected position. At Step 1.2, individual i is selected because she is the highest merit-score individual who can simultaneously be trait-matched to a HR-protected positionalong with i . For example, i can be trait-matched with s and i can be trait-matchedwith s (see the dashed matching in Figure 2). This can be done with various computationally efficient algorithms, see, for example, the bipartitecardinality matching algorithm (Lawler, 2001, Page 195).
FFIRMATIVE ACTION IN INDIA 19 i i i i i i i s s s F IGURE
1. The HR graph of the allocation problem in Example 3. For each k ∈ {
1, 2, 3 } , node s k represents the HR-protected position for trait t k . i i i i s s s F IGURE
2. At Step 1.2, subject to matching individual i with an HR-protected position, individual i can also be matched with an HR-protectedposition, thus increasing HR utilization of { i } .At Step 1.3, individual i is selected because she is the highest merit-score individualwho can be trait-matched to a HR-protected position together with i and i . However,the implementation of such a trait matching requires that i and i are trait-matched withdifferent positions than the dashed matching shown in Figure 2. To be more precise, i can be trait-matched with s , i can be trait-matched with s , and i can be trait-matchedwith s (see the dotted matching in Figure 3). No remaining individuals can be trait-matched with a HR-protected position togetherwith i , i , and i , so we proceed to Step 2. At Step 2, individuals i and i are selected This step illustrates the necessity of keeping trait-matching flexible until the end of Step 1.0 S ¨ONMEZ AND YENMEZ
2. At Step 1.2, subject to matching individual i with an HR-protected position, individual i can also be matched with an HR-protectedposition, thus increasing HR utilization of { i } .At Step 1.3, individual i is selected because she is the highest merit-score individualwho can be trait-matched to a HR-protected position together with i and i . However,the implementation of such a trait matching requires that i and i are trait-matched withdifferent positions than the dashed matching shown in Figure 2. To be more precise, i can be trait-matched with s , i can be trait-matched with s , and i can be trait-matchedwith s (see the dotted matching in Figure 3). No remaining individuals can be trait-matched with a HR-protected position togetherwith i , i , and i , so we proceed to Step 2. At Step 2, individuals i and i are selected This step illustrates the necessity of keeping trait-matching flexible until the end of Step 1.0 S ¨ONMEZ AND YENMEZ i i i i i s s s F IGURE
3. At Step 1.3, subject to matching individuals i and i with HR-protected positions, individual i can also be matched with an HR-protectedposition, thus increasing HR utilization of { i , i } . However, matching indi-vidual i requires changing the assignment of individual i from position s (i.e. from her assignment in Figure 2) to position s , since position s is theonly position individual i is connected to.because there are two vacant positions and they have the highest merit scores among theremaining individuals. Therefore, C v M ( I ) = { i , i , i , i , i } . (cid:3) We next present our single-category results, which sug-gest that the case for the meritorious horizontal choice rule is especially strong in thisframework.Justifying the naming of this choice rule, our first main result shows that the merito-rious horizontal choice rule C v M always selects higher merit-score individuals than anyother choice rule that maximally accommodates HR protections. Theorem 1.
Given a category c ∈ V , let C v be any single-category choice rule that maximallyaccommodates HR protections. Then, for every set of individuals I ⊆ I v ,(1) | C v ( I ) | ≤ | C v M ( I ) | , and(2) for every k ≤ | C v ( I ) | , if i is the k-th highest merit-score individual in C v M ( I ) and j is thek-th highest merit-score individual in C v ( I ) , theni = j or σ ( i ) > σ ( j ) .We next present the main result of our single-category analysis, a characterization ofthe meritorious horizontal choice rule C v M . Theorem 2.
Given a category v ∈ V , a single-category choice rule(1) maximally accommodates HR protections,
FFIRMATIVE ACTION IN INDIA 21 (2) satisfies no justified envy, and(3) is non-wastefulif, and only if, it is the meritorious horizontal choice rule C v M . Observe that, the algorithm for the meritorious horizontal choice rule C v M simplifiesto the algorithm of the minimum guarantee choice rule C vmg when each individual hasat most one horizontal trait. Therefore, an immediate corollary to Theorem 2 is thefollowing characterization of the minimum guarantee choice rule C vmg . Corollary 1.
Assume that each individual has at most one horizontal trait and fix a categoryv ∈ V . A single-category choice rule(1) maximally accommodates HR protections,(2) satisfies no justified envy, and(3) is non-wastefulif, and only if, it is the minimum guarantee choice rule C vmg .
4. Multi-Category Analysis with Overlapping Horizontal Reservations
We are ready to analyze our model in its full generality.
Our first three multi-category axioms are the immediatecounterparts of their single-category versions, applied separately to each category.
Definition 8.
A choice rule C = ( C ν ) ν ∈V maximally accommodates HR protections , if forevery I ⊆ I , v ∈ V , and j ∈ I \ b C ( I ) who is eligible for category v,n v ( C v ( I )) = n v ( C v ( I ) ∪ { j } ) .In words, an individual who is not awarded a position (at any category) should not beable to increase the utilization of HR protections for any category where she has eligibility. Definition 9.
A choice rule C = ( C ν ) ν ∈V satisfies no justified envy , if for every I ⊆ I , v ∈ V ,i ∈ C v ( I ) , and j ∈ I \ b C ( I ) who is eligible for category v, σ ( j ) > σ ( i ) = ⇒ n v (cid:0) ( C v ( I ) \ { i } ) ∪ { j } (cid:1) < n v ( C v ( I )) .In words, for any category v , • if an individual i receives a position at category v , • while a higher merit-score and category- v eligible individual j is declined a posi-tion from all categories (including category v ), More precisely, Step 1 of both algorithms produce the same outcome when each individual has at mostone horizontal trait. • then it must be the case that replacing individual j with individual i results in adecreased utilization of HR protections at category v . Definition 10.
A choice rule C = ( C ν ) ν ∈V is non-wasteful if, for every I ⊆ I , v ∈ V , andj ∈ I \ b C ( I ) who is eligible for category v, | C v ( I ) | = q v .In words, if there is an idle position at a category v while an individual j remains un-matched (thus being declined a position from all categories), then it must be the case thatindividual j is not eligible for category v .Our next axiom has no counterpart in the single-category framework for it regulatesthe relation between the recipients of open category positions and reserved-category po-sitions. Since reserved-category positions are mandated to be allocated on a “set aside”basis under Indra Sawhney (1992) , in the absence of HR protections this axiom would sim-ply require the merit score of each recipient of an open-category position to be higher thanthe merit score of any recipient of a reserved-category position. We extend this conditionto our more general model as follows:
Definition 11.
A choice rule C = ( C ν ) ν ∈V complies with VR protections if, for every I ⊆ I ,c ∈ R , and i ∈ C c ( I ) ,(1) | C o ( I ) | = q o ,(2) for every j ∈ C o ( I ) , σ ( j ) < σ ( i ) = ⇒ n o (cid:0) C o ( I ) (cid:1) > n o (cid:0) ( C o ( I ) \ { j } ) ∪ { i } (cid:1) , and(3) n o (cid:0) C o ( I ) ∪ { i } (cid:1) = n o (cid:0) C o ( I ) (cid:1) .Here the first two conditions formulate the idea of a vertical reservation `a laIndra Sawhney (1992), and they are directly suggested by the concept of “set aside.” Foran individual to receive a position set aside for a reserve-eligible category, it must be thecase that each open-category position is either allocated to a higher merit-score individ-ual or to an individual who accommodates an HR protection. The third condition, whilenatural, is extra, and it further requires that • not only open-category positions should be allocated to higher merit-score indi-viduals than the recipients of VR-protected positions, • but also the open-category horizontal adjustments must be carried out with thehighest merit-score individuals who are eligible for these adjustments. We areready to formulate and propose a choice rule for our model in its full generality. The
FFIRMATIVE ACTION IN INDIA 23 following choice rule uses the meritorious horizontal choice rule first to allocate open-category positions, and next in parallel for each reserve-eligible category to allocatevertically-reserved positions.
Two-Step Meritorious Horizontal (2SMH) Choice Rule C s M = ( C s , ν M ) ν ∈V For every I ⊂ I , C s , o M ( I ) = C o M ( I ) , and C s , c M ( I ) = C c M (cid:0) { i ∈ I \ C o M ( I ) : ρ ( i ) = c } (cid:1) for any c ∈ R .We now present the main result for our multi-category analysis, a characterization ofthe 2SMH choice rule. Theorem 3.
A choice rule(1) maximally accommodates HR protections,(2) satisfies no justified envy,(3) is non-wasteful, and(4) complies with VR protectionsif, and only if, it is the 2SMH choice rule C s M . Observe that, since the two choice rules C vmg and C v M are equivalent for any category v ∈ V when each individual has at most one horizontal trait, our proposed 2SMH choicerule is equivalent to the following two-step minimum guarantee (2SMG) choice rule underthis condition. Two-Step Minimum Guarantee (2SMG) Choice Rule C smg = ( C s , ν mg ) ν ∈V For every I ⊆ I , C s , omg ( I ) = C omg ( I ) , and C s , cmg ( I ) = C cmg (cid:0) { i ∈ I \ C o M ( I ) : ρ ( i ) = c } (cid:1) for any c ∈ R .Therefore, an immediate corollary of Theorem 3 is the following result: Corollary 2.
Suppose each individual has at most one horizontal trait. A choice rule(1) maximally accommodates HR protections,(2) satisfies no justified envy,(3) is non-wasteful, and With the August 2020 judgement
Tamannaben Ashokbhai Desai (2020) of the High Court of Gujarat, the2SMH choice rule has been recently mandated in the state of Gujarat. Moreover, with its December 2020judgement
Saurav Yadav (2020) , this rule is endorsed by the the Supreme Court for the entire country. Ourintroduction and advocacy of this rule predates both of these important judgements. (4) complies with VR protectionsif, and only if, it is the 2SMG choice rule C smg . All four axioms that uniquely characterize the 2SMH choice rule (or the 2SMG choicerule for the case of non-overlapping HR protections) are motivated by the legal frame-work in India along with the shortcomings of the SCI-AKG choice rule. Our first threeaxioms are fairly benign, and they are formulated to implement HR protections in themost meritorious way (as implied by our Theorem 1). The first two conditions of our lastaxiom compliance with VR protections are also both necessary due to the federally man-dated “set aside” nature of the VR protections. The third condition of this axiom furtherrequires carrying out the necessary horizontal adjustments with the highest merit-scoreindividuals eligible for these adjustments, and while natural, it is not federally mandated.Hence, in our view it is the only condition that can be dropped while still staying true tothe essence of
Indra Sawhney (1992) .
5. SCI-AKG Choice Rule vs. 2SMH Choice Rule
In Section 3.1 we show that the AKG horizontal adjustment subroutine is the source ofthree shortcomings of the SCI-AKG choice rule when an individual can qualify for mul-tiple HR protections, and we introduce the meritorious horizontal choice rule in Section3.3 as a remedy. We also show that, the AKG-HAS escapes these shortcomings when anindividual can qualify for no more than one HR protection. This special case of the of theallocation problem where AKG-HAS is well-behaved is important in practice, because theonly federally mandated HR protections in India is for persons with disabilities . The moregeneral case where an individual can qualify for multiple HR protections is of practicalrelevance in some of the states only. The SCI-AKG choice rule, however, suffers from two additional shortcomings evenwhen there is a single HR protection. Moreover, unlike the limitations of the AKG-HASthat largely escape public scrutiny, these two shortcomings are highly visible and theyare responsible from thousands of litigations disrupting recruitment processes through-out India as presented in Section 8.1. To formulate these shortcomings, we need someadditional notation. For the simpler version of the problem with non-overlapping HR protections, the choice rule C horite givenin an earlier draft of this paper (S ¨onmez and Yenmez, 2019) fails the third condition of compliance with VRprotections, but otherwise it satisfies all other conditions that uniquely characterize the 2SMG choice rule. While affirmative action in India is our main application due to its concurrent implementation of VRand HR policies, it is not the only real-life application where an individual can qualify for multiple HRprotections. For example Section 7 for K-12 school choice in Chile, where a student can qualify for anysubset of the HR protections for financially disadvantaged, special needs, and high-achieving students.
FFIRMATIVE ACTION IN INDIA 25
Given a set of individuals I ∈ I , let I g = { i ∈ I : ρ ( i ) = ∅ } be the set of individuals in I who are members of the general category, and I c = { i ∈ I : ρ ( i ) = c } be the set of individuals in I who are members of category c for a given reserve-eligiblecategory c ∈ R . We are ready to formulate the choice rule that is responsiblefrom thousands of litigations in India in the last two decades. SCI-AKG Choice Rule C SCI
For every I ⊆ I , Step 1 (Open-category tentative assignment) : • If | I | ≤ q o then assign all individuals in I to open-category positions and ter-minate the procedure. In this case C SCI , o ( I ) = I and C SCI , c ( I ) = ∅ for anyreserve-eligible category c ∈ R . • Otherwise, if | I | > q o then tentatively assign the highest merit-score q o indi-viduals in I to open-category positions. Let J o denote the set of individualswho are tentatively assigned to open-category positions in this case. Step 2 (Finalization of open-category positions) : The set of individuals eligiblefor open-category horizontal adjustments is I g \ J o . Apply the AKG-HAS • to the set J o of tentative recipients of open-category positions • with the set of individuals in I g \ J o who are eligible for open-category hori-zontal adjustmentsto finalize the set of recipients C SCI , o ( I ) of open-category positions. Step 3 (Reserve-eligible category tentative assignment) : For any reserve-eligiblecategory c ∈ R , • If | I c \ C SCI , o ( I ) | ≤ q c then assign all individuals in I c \ C SCI , o ( I ) to category- c positions and terminate the procedure. In this case C SCI , c ( I ) = I c \ C SCI , o ( I ) . • Otherwise, if | I c \ C SCI , o ( I ) | > q c then tentatively assign the highest merit-score q c individuals in I c \ C SCI , o ( I ) to category- c positions. Let J c denote theset of individuals who are tentatively assigned to category- c positions in thiscase. The description of the SCI-AKG choice rule in the Supreme Court judgements
Anil Kumar Gupta (1995) and
Rajesh Kumar Daria (2007) can be seen in Section C of the Online Appendix.
Step 4 (Finalization of reserve-eligible category positions) : For any reserve-eligible category c ∈ R , the set of individuals eligible for category- c horizontaladjustments is I c \ ( C SCI , o ( I ) ∪ J c ) . For any reserve-eligible category c ∈ R , applythe AKG-HAS • to the set J c of tentative recipients of category- c positions • with the set of individuals in I c \ ( C SCI , o ( I ) ∪ J c ) who are eligible for category- c horizontal adjustmentsto finalize the set of recipients C SCI , c ( I ) of category- c positions.The outcome of the SCI-AKG choice rule is C SCI ( I ) = (cid:0) C SCI , v ( I ) (cid:1) v ∈V . We have already presented in Section 3.1 that hor-izontal adjustments through AKG-HAS is problematic when an individual can qualifyfor multiple HR protections, an issue which can be corrected by using the meritorioushorizontal choice rule for each vertical category. If we go though this adjustment, the na-tive SCI-AKG choice rule transforms into an amended choice rule that is closely relatedto our proposed 2SMH choice rule. We need the following terminology to present thisassociation.Given a set of individuals I ∈ I , define the set of meritorious reserved candidates I m as the set of individuals in I , each of whom(1) is a member of a reserve-eligible category in R , and(2) has a merit score among the q o -highest merit scores of all individuals in I .The SCI-AKG choice rule takes the following form when its horizontal adjustment pro-cess is amended: Choice Rule C SCI M = ( C SCI , ν M ) ν ∈V For every I ⊆ I , C SCI , o M ( I ) = C o M ( I m ∪ I g ) , and C SCI , c M ( I ) = C c M (cid:0) { i ∈ I \ C o M ( I ) : ρ ( i ) = c } (cid:1) for any c ∈ R .Observe that the only difference between the two choice rules C SCI M and C s M is, • while all individuals with relevant traits are considered for open-category HR pro-tections under C s M , • only the general category individuals and meritorious reserved candidates areconsidered for open-category HR protections under C SCI M . FFIRMATIVE ACTION IN INDIA 27
This observation immediately reveals an important conflict for individuals who bothqualify for VR protections through their reserve-eligible categories and also for HR pro-tections through their traits: With the exception of meritorious reserved candidates, theseindividuals lose their qualification for open-category HR protections by claiming their VRprotections. It is important to emphasize that this conflict exists regardless of how manyhorizontal traits each individual can have, and therefore it is prevalent under both thenative and also the amended version of the SCI-AKG choice rule.This conflict reflects itself in the following two deficiencies that go against the philoso-phy of affirmative action under both versions of the SCI-AKG choice rule:(1)
Failure to satisfy no justified envy : For example, a woman from the VR-protectedcategory Scheduled Castes may remain unassigned while a lower merit-scorewoman from the higher-privilege general category receives a position throughopen-category HR protections for women.(2)
Failure to satisfy incentive compatibility : For example, a woman from Sched-uled Castes may remain unassigned by declaring her membership for ScheduledCastes, but she can receive an open-category HR-protected position for women bywithholding her Scheduled Castes membership, thus benefitting from not declar-ing this information.We have already formulated a more general version of the first property in Section 4.1. Wenext formulate the second one, first introduced by (Ayg ¨un and B ´o, 2016) in their analysisof the affirmative action policies in Brazilian college admissions:
Definition 12.
An individual withholds some of her reserve-eligible privileges if she doesnot declare either her reserve-eligible category membership or some of her traits (or both).
In India, individuals are not required to declare their reserve-eligible privileges.
Definition 13.
A choice rule C is incentive compatible if, for every I ⊆ I and i ∈ I, ifindividual i is selected from I under the aggregate choice rule b C by withholding some of her reserve-eligible privileges, then individual i is also selected from I under the aggregate choice rule b C bydeclaring all her reserve-eligible privileges.
In other words, privileges that are intended to protect an individual do not instead hurther upon declaring eligibility for these privileges (as one would normally expect) underan incentive compatible choice rule.Failure of incentive compatibility is implausible both from a normative perspectivesince it is against the philosophy of affirmative action, and also from a strategic perspec-tive since it forces individuals to chose between their VR protections and open-category
HR protections. As we present clear evidence in Section 8.1.2, it also creates an additionalissue in our main application in India. Eligibility for VR protections typically dependson applicant’s caste membership. While this information is supposed to be private in-formation, it often can be inferred by the central planner due to indications such as theapplicant’s last name. Central planner can also obtain this information through docu-ments such as graduation diploma. Hence eligibility for VR protections may not be trulyprivate information, and the lack of incentive compatibility of a choice rule may enable amalicious central planner to use this information to deny an applicant her open-categoryHR protections. As documented in Section 8.1.2, this type of misconduct not only hasbeen widespread in parts of India, but it even appears to be centrally organized by thelocal governing bodies in some of its jurisdictions.We have already explained that the native version of the SCI-AKG choice rule failsto satisfy both no justified envy and also incentive compatibility in a rather visible way.Since an amendment via meritorious horizontal choice rule addresses completely inde-pendent shortcomings, the amended version too fails both desiderata. As for our pro-posed 2SMH choice rule C s M , we know from Theorem 3 that it satisfies no justified envy.Our next result shows that this choice rule also satisfies incentive compatibility. Proposition 2.
The 2SMH choice rule C s M satisfies incentive compatibility. In light of our characterization in Theorem3 and Proposition 2, we believe a compelling case can be made for the 2SMH choicerule C s M as a better alternative to the SCI-AKG choice rule C SCI . Not only C s M escapesall five shortcomings of C SCI presented in Sections 3.1 and 5.2, it does so through twosimple modifications on the mechanics and implementation of the technical horizontaladjustment subroutine AKG-HAS, thereby keeping the main ideas of the SCI-AKG choicerule intact.Naturally, the outcomes of the two choice rules may be different in general. We nextshow that, compared to C SCI M (1) the outcome of the C s M is less favorable for individuals from the general category,and(2) assuming there are at least as many individuals from each reserve-eligible categoryas the number of positions they are eligible for, the outcome of the C s M is also morefavorable overall for individuals from reserve-eligible categories.The comparison is made with the amended version of the SCI-AKG choice rule ratherthan its native version, because the latter is not well-defined when individuals potentiallyqualify for multiple HR protections. FFIRMATIVE ACTION IN INDIA 29
Proposition 3.
For every I ⊆ I , b C s M ( I ) ∩ I g ⊆ b C SCI M ( I ) ∩ I g , and assuming | I c | ≥ q o + q c for each reserve-eligible category c ∈ R , ∑ c ∈R (cid:12)(cid:12) b C s M ( I ) ∩ I c (cid:12)(cid:12) ≥ ∑ c ∈R (cid:12)(cid:12) b C SCI M ( I ) ∩ I c (cid:12)(cid:12) .
6. Extension to Centralized Allocation of Positions Across Multiple Institutions andthe Related Literature
Mainly motivated by the shortcomings of the SCI-AKG choice rule, our main focus hasbeen allocation of identical positions at a single institution. Over the last fifteen years,the celebrated individual-proposing deferred acceptance algorithm (Gale and Shapley, 1962)has become the mechanism of choice for priority-based allocation with heterogenous po-sitions across multiple institutions, where the policies of the institutions are capturedthrough the choice rules that are used in conjunction with this algorithm. For this jointimplementation to be well-defined, it is sufficient that each individual has strict pref-erences over the institutions (but otherwise indifferent between positions of any giveninstitution) and the choice rule at each institution satisfies the following two properties:
Definition 14. (Kelso and Crawford, 1982) A choice rule C satisfies the substitutes condition,if, for every I ⊆ I , i ∈ b C ( I ) and j ∈ I \ { i } = ⇒ i ∈ b C ( I \ { j } ) . Definition 15. (Ayg ¨un and S¨onmez, 2013) A choice rule C satisfies the irrelevance of rejectedindividuals condition, if, for every I ⊆ I ,i ∈ I \ b C ( I ) = ⇒ b C ( I \ { i } ) = b C ( I ) .Our next result states that the 2SMH choice rule C s M satisfies both properties. Proposition 4.
The 2SMH choice rule C s M satisfies the substitutes condition and the irrelevanceof rejected individuals condition. Therefore, a natural (and straightforward) extension of our model involves a joint im-plementation of 2SMH choice rule with the individual-proposing deferred acceptancealgorithm when there are multiple institutions introducing heterogeneity in positions. This is also the case for our K-12 school choice application in Chile presented in Section 7.0 S ¨ONMEZ AND YENMEZ
The 2SMH choice rule C s M satisfies the substitutes condition and the irrelevanceof rejected individuals condition. Therefore, a natural (and straightforward) extension of our model involves a joint im-plementation of 2SMH choice rule with the individual-proposing deferred acceptancealgorithm when there are multiple institutions introducing heterogeneity in positions. This is also the case for our K-12 school choice application in Chile presented in Section 7.0 S ¨ONMEZ AND YENMEZ
Our theoretical analysis of reservation policies differs from itspredecessors in the following two ways:(1) concurrent implementation of vertical and horizontal reservation policies and(2) potentially overlapping structure of horizontal reservations.While there is a rich literature on affirmative action policies in India and elsewhere, ourpaper is the first one to formally analyze vertical and horizontal reservation policies whenthey are implemented concurrently. The aspect of overlapping horizontal reservationshas received some attention in the literature (Kurata et al. (2017)), albeit for a differentvariant of the problem where individuals have strict preferences for whether and whichprotection is invoked in securing a position. When applied in an environment where in-dividuals are indifferent between all positions, choice rules recommended in Kurata et al.(2017) result in all three shortcomings presented in Section 3. This observation is the ba-sis of our proposed reform for the K-12 school choice application in Chile, presented inSection 7.There are a number of recent papers on reservation policies, most in the context ofschool choice. Abdulkadiro ˘glu and S ¨onmez (2003) study affirmative action policies thatlimit the number of admitted students of a given type through hard quotas. Kojima(2012) shows that a policy of limiting the number of majority students through hard quo-tas can hurt minority students, the intended beneficiaries. To overcome the detrimentaleffect of affirmative action policies based on majority quotas, Hafalir et al. (2013) intro-duce affirmative action policies based on minority reserves . In the absence of overlappingreservations, Echenique and Yenmez (2015) present the first axiomatic characterization ofthe minimum guarantee choice rule C vmg . Most recently Pathak et al. (2020b) consider ageneral model of reservation policies to balance various ethical principles for pandemicmedical resource allocation, although their model is not equipped to analyze concurrentimplementation of vertical and overlapping horizontal reservation policies.A few papers study implementation of vertical or (non-overlapping) horizontal reser-vations individually in various real-life applications. These include Dur et al. (2018) forschool choice in Boston, Dur et al. (2020) for school choice in Chicago, and Pathak et al.(2020a) for H-1B visa allocation in the US. All these models are applications of themore general model in Kominers and S ¨onmez (2016), where the authors introduce amatching model with slot-specific priorities . In contrast, our model is independent thanKominers and S ¨onmez (2016). Three additional papers on reservation policies includeAyg ¨un and Turhan (2016, 2017), where the authors study admissions to engineering col-leges in India, and Ayg ¨un and B ´o (2016), where the authors study admissions to Brazilian FFIRMATIVE ACTION IN INDIA 31 public universities. While the application in Ayg ¨un and Turhan (2016, 2017) is closely re-lated to ours, their analysis is independent because not only horizontal reservations areassumed away altogether in these papers, but also analysis in these papers largely ab-stract away from the legal requirements in India. In contrast, our model and analysiscompletely build on Indian laws on reservation policies, and all shortcomings we formu-late disappear in the absence of horizontal reservations. The Brazilian affirmative actionapplication studied by Ayg ¨un and B ´o (2016) relates to ours in that it also includes multi-dimensional reservation policies, but unlike our models their application is a special caseof Kominers and S ¨onmez (2016). There is, however, one important element in our paperthat directly builds on Ayg ¨un and B ´o (2016). Not only the two desiderata that play animportant role in our proposed reform in India, no justified envy and incentive compatibility are originally introduced by Ayg ¨un and B ´o (2016), but also evidence from aggregate datais presented in this paper that the presence of justified envy is widespread in Brazil. As inAyg ¨un and B ´o (2016), we also present extensive evidence of justified envy in the field, butin addition we also document the large scale disruption this anomaly creates in the field.Other less related papers on reservation policies include Westkamp (2013), Ehlers et al.(2014), Kamada and Kojima (2015), and Fragiadakis and Troyan (2017).More broadly, our paper contributes to the field of market design, whereeconomists are increasingly taking advantage of advances in technology to de-sign new or improved allocation mechanisms in applications as diverse as entry-level labor markets (Roth and Peranson, 1999), school choice (Balinski and S ¨onmez,1999; Abdulkadiro ˘glu and S ¨onmez, 2003), spectrum auctions (Milgrom, 2000), kid-ney exchange (Roth et al., 2004, 2005), internet auctions (Edelman et al., 2007; Varian,2007), course allocation (S ¨onmez and ¨Unver, 2010; Budish, 2011), cadet-branch match-ing (S ¨onmez and Switzer, 2013; S ¨onmez, 2013), assignment of airline arrival slots(Schummer and Vohra, 2013; Schummer and Abizada, 2017), and refugee matching(Jones and Teytelboym, 2017; Delacr´etaz et al., 2016; Andersson, 2017).
7. Application: School Inclusion Law in Chile
With the promulgation of the
School Inclusion Law in Chile in 2015, a centralized schoolchoice system has been adopted in Chile, following a similar series of reforms throughoutthe world (Correa et al., 2019). The system is the product of an ongoing collaborationbetween the Chilean Ministry of Education and a team of researchers from economicsand operations research, and it covers all grades prior to higher education (i.e., Pre-K tograde 12). The system was first implemented in 2016 as a pilot program in the smallest of See also the discussion of Indian college admissions in (Echenique and Yenmez, 2015, Appendix C.1). the sixteen regions of Chile, and it has been adopted in all regions but the MetropolitanArea of Santiago by 2019, where over 274,000 students applied to more than 6400 schools.As many of its predecessors, the Chilean school choice system is based on the celebratedindividual-proposing deferred acceptance algorithm, and the following three features inits design make it a perfect application of our model:(1) To promote diversity, the School Inclusion Law includes affirmative action policiesfor financially disadvantaged students and children with special needs. Underthe new system, these policies are implemented through reserved seats at eachschool. In addition, a number of schools are allowed to reserve seats for high-achieving students. Hence, using our terminology there are three traits,
Financiallydisadvantaged, Special needs, High-achieving , where a student potentially can haveany subset of these traits, possibly including none of them. Students with none ofthe three traits are called
Regular .(2) While a student with multiple traits (say a financially disadvantaged student whois also high-achieving) is eligible for reserved seats for each of her traits, parallel toour modeling choice of overlapping horizontal reservations she “consumes” onlyone of the reserved seats in case she receives a seat. This feature in Chilean designeliminates potential complementarities between the regular students and studentswith multiple traits.(3) Reserved seats at each school are implemented in the form of a minimum guaran-tee.As also emphasized in the Introduction, a subtle implication of the second design fea-ture is that it allows the model to be interpreted as an application of the matching withcontracts model of Hatfield and Milgrom (2005), where the contractual term between aschool and a student specifies which of the four types of seats (i.e., open seats, reservedseats for financially disadvantaged students, reserved seats for special needs students,and reserved seats for high-achieving students) the student receives, an approach that istaken in Kurata et al. (2017). However, the theory of matching with contracts is developedunder the assumption that students have strict preferences over all their contracts, whichin this context corresponds to them having strict preferences on the specific type of seatsthey receive at each school. Since students have preferences over only schools, a fixed tie-breaking rule is used to construct student preferences over specific type of seats at eachschool. In Correa et al. (2019), the designers emphasize that the choice of a tie-breakingrule is not straightforward, and it has distributional consequences. In order to implementthe reserves in the form of a minimum guarantee, they break ties in a way each student isassumed to prefer reserved seats for any of their traits to open seats. When each student
FFIRMATIVE ACTION IN INDIA 33 has at most one trait, this construction assures that the reserves are implemented as a min-imum guarantee (Hafalir et al. (2013), S ¨onmez and Yenmez (2019)). However, analogousto phenomena presented in Examples 1 and 2 in Section 3.1, interpreting this problemas an application of matching with contracts and relying on a fixed tie-breaking betweenreserved seats results in a number of shortcomings including an “underutilization” ofreserved positions as well as their “ineffective” implementation through admitting need-lessly low baseline priority students. Therefore, a better approach would be using themeritorious horizontal choice rule for each school.
8. Application: Affirmative Action in India
Background on the legal framework for implementation of vertical and horizontalreservation policies in India is presented in Section C of the Online Appendix, and aformal comparison of the SCI-AKG choice rule with our proposed 2SMH choice rule ispresented in Section 5. In this section we present extensive evidence on the disarraycaused by the shortcomings of the SCI-AKG choice rule in India, and articulate how thepotential disruptive effects of this allocation rule can be expected to be amplified in thepresence of a new vertical reservation category introduced by a 2019 amendment of theConstitution of India.
As we have argued in Section 5.2, theSCI-AKG choice rule allows for justified envy. Moreover, it also fails incentive compati-bility due to backward class candidates losing their open-category HR protections uponclaiming their VR protections by declaring their backward class status.The failure of SCI-AKG choice rule to satisfy no justified envy is fairly straightforwardto observe. All it takes is a rejected backward class candidate to realize that her meritscore is higher than an accepted general-category candidate, even though she qualifiesfor all the HR protections the less-deserving (but still accepted) candidate qualifies for.Since the primary role of the reservation policy is positive discrimination for candidateswith more vulnerable backgrounds, this situation is very counterintuitive, and it oftenresults in legal action. Focusing on complications caused by either anomaly, we nextpresent several court cases to document how they handicap concurrent implementationof vertical and horizontal reservation policies in India.
High Court Cases Related to Justified Envy . Much of our analysis and the High Court judgements we present in this section parallels the argumentsand the decision of the December 2020 Supreme Court case
Saurav Yadav v State of Uttar Pradesh (2020) . Ouranalysis predates this important judgement, and it was already presented in an earlier draft of this paperin S ¨onmez and Yenmez (2019).
The possibility of justified envy under the SCI-AKG choice rule has resulted in nu-merous court cases throughout India for more than two decades, and since the presenceof justified envy in the system is highly implausible, these legal challenges often result incontroversial rulings. In addition, there are also cases where authorities who implement abetter-behaved version of the choice rule, one that does not suffer from this shortcoming,are nonetheless challenged in court, on the basis that their adopted choice rules differfrom the one mandated by the Supreme Court. These court cases are not restricted tolower courts, and include several cases in state high courts. Even at the level of statehigh courts, the judgements on this issue are highly inconsistent, largely due to the dis-array created by the possibility of justified envy under the SCI-AKG choice rule. We nextpresent four representative cases from high courts, each from a different state:(1)
Ashish Kumar Pandey And 24 Others vs State Of U.P. And 29 Others on 16 March,2016 , Allahabad High Court . This lawsuit was brought to Allahabad HighCourt by 25 petitioners, disputing the mechanism employed by the State of UttarPradesh—the most populous state in India with more than 200 million residents—to apply the provisions of horizontal reservations in their allocation of more than4000 civil police and platoon commander positions. Of these positions, 27%, 21%,2% are each vertically reserved for backward classes OBC, SC, and ST, respectively,and 20%, 5%, and 2% are each horizontally reserved for women, ex-servicemen,and dependents of freedom fighters, respectively. While only 19 women are se-lected for open-category positions based on their merit scores, the total number offemale candidates is less than even the number of open-category horizontally re-served positions for women, and as such all remaining women are selected. How-ever, instead of assigning them positions from their respective backward class cat-egories (as it is mandated under the SCI-AKG choice rule), all of them are assignedpositions from the open category. Similarly, backward class candidates are deemedeligible to use horizontal reservations for dependents of freedom fighters and ex-serviceman as well. The counsel for the petitioners argues that not only did theState of U.P. make an error in their implementation of horizontal reservations, butalso that the error was intentional. The following quote is from the court case:
Per contra, learned counsel appearing for the petitioners wouldsubmit that fallacy was committed by the Board deliberately, andwith malafide intention to deprive the meritorious candidates theirrightful placement in the open category. The candidates seekinghorizontal reservations belonging to OBC and SC category were wrongly The case is available at https://indiankanoon.org/doc/74817661/ (last accessed on 03/07/2019).
FFIRMATIVE ACTION IN INDIA 35 adjusted in the open category, whereas, they ought to have beenadjusted in their quota provided in respective social category. Theaction of the Board is not only motivated, but purports to takeforward the unwritten agenda of the State Government to accommodateas many number of OBC/SC candidates in the open category.
The judge of the case sides with the petitioners, and rules that the State of UttarPradesh must correct their erroneous application of the provisions of horizontalreservations. The judge further emphasizes that the State has played foul, stating:
There is merit in the submission of the learned counsel for thepetitioners that the conduct of the members of the Board appearsnot only mischievous but motivated to achieve a calculated agendaby deliberately keeping meritorious candidates out of the selectlist. The Board and the officials involved in the recruitmentprocess were fully aware of the principle of horizontal reservationsenshrined in Act, 1993 and Government Orders which were beingfollowed by them in previous selections of SICP and PC (PAC), butin the present selection they chose to adopt a principle againsttheir own Government Orders and the statutory provisions which werebinding upon them...I am constrained to hold that both the State and the Board haveplayed fraud on the principles enshrined in the Constitution withregard to public appointment.
What is especially surprising is, despite the heavy tone of this judgement, the Stategoes on to appeal in another Allahabad High Court case
State Of U.P. And 2 Ors.vs Ashish Kumar Pandey And 58 Ors, 29 July, 2016 , in an effort to continue using itspreferred method for implementing horizontal reservations. Perhaps unsurpris-ingly, this appeal was denied by the High Court.This particular case clearly illustrates that there is a strong resistance in at leastsome of the states for implementing the provisions of horizontal reservations asmandated under the SCI-AKG choice rule. While this resistance most likely re-flects the political nature of this debate, the arguments of the counsel for the stateto maintain their preferred mechanism are mostly based on the presence of justi-fied envy under the SCI-AKG choice rule. The case is available at https://indiankanoon.org/doc/71146861/ (last accessed on 03/07/2019). (2)
Asha Ramnath Gholap vs President, District Selection Committee & Ors. onMarch 3rd, 2016 , Bombay High Court . In this case, there are 23 pharmacist po-sitions to be allocated; 13 of these positions are vertically reserved for backwardclasses and the remaining ten are open for all candidates. In the open category,eight of the ten positions are horizontally reserved for various groups, includingthree for women. The petitioner, Asha Ramnath Gholap, is an SC woman, andwhile there is one vertically reserved position for SC candidates, there is no hor-izontally reserved position for SC women. Under the SCI-AKG choice rule, sheis not eligible for any of the horizontally reserved women positions at the opencategory. Nevertheless, she brings her case to the Bombay High Court based on aninstance of justified envy, described in the court records as follows:
It is the contention of the petitioner that Respondent Nos. 4 & 5have received less marks than the petitioner and as such, both werenot liable to be selected. The petitioner has, therefore, approachedthis court by invoking the writ jurisdiction of this court underArticle 226 of the Constitution of India, seeking quashment of theselect list to the extent it contains the names of Respondent Nos.4and 5 against the seats reserved for the candidates belonging to openfemale category.
Under the federal law, there is no merit to this argument, because the SCI-AKGchoice rule allows for justified envy. However, the judges side with the petitioneron the basis that a candidate cannot be denied a position from the open categorybased on her backward class membership, essentially ruling out the possibility ofjustified envy under a Supreme Court-mandated choice rule, which is designed toallow for positive discrimination for the vulnerable groups in the society. Theirjustification is given in the court records as follows:
We find the argument advanced as above to be fallacious. Once itis held that general category or open category takes in its sweepall candidates belonging to all categories irrespective of theircaste, class or community or tribe, it is irrelevant whether thereservation provided is vertical or horizontal. There cannot be twointerpretations of the words ‘open category’ . . . (3)
Smt. Megha Shetty vs State Of Raj. & Anr on 26 July, 2013 , Rajasthan HighCourt . In contrast to
Asha Ramnath Gholap (2016) where the judges have been The case is available at https://indiankanoon.org/doc/178693513/ (last accessed on 03/08/2019). The case is available at https://indiankanoon.org/doc/78343251/ (last accessed on 10/08/2019).
FFIRMATIVE ACTION IN INDIA 37 erroneous siding with petitioners whose lawsuits are based on instances of justi-fied envy, in this case a petitioner who is a member of the general category seekslegal action against the state on the basis that several horizontally reserved open-category women positions are allocated to women from OBC who are not eligiblefor these positions (unless they receive it without invoking the benefits of hori-zontal reservation). While all these OBC women have higher merit scores thanthe petitioner and the state have apparently used a better behaved procedure, thepetitioner’s case has merit because SCI-AKG choice rule allows for justified envyin those situations. In an earlier lawsuit, the petitioner’s lawsuit was already de-clined by a single judge of the same court based on an erroneous interpretation of
Indra Sawhney (1992) . The petitioner subsequently appeals this erroneous decisionand brings the case to a larger bench of the same court. However, the three judgesside with the earlier judgement, thus erroneously dismissing the appeal. Theirdecision is justified as follows:
The outstanding and important feature to be noticed is that it is notthe case of the appellant-petitioner that she has obtained more marksthan those 8 OBC (Woman) candidates, who have been appointed againstthe posts meant for General Category (Woman), inasmuch as, while theappellant is at Serial No.184 in the merit list, the last OBC (Woman)appointed is at Serial No.125 in the merit list. The controversyraised by the appellant is required to be examined in the context andbackdrop of these significant factual aspects.
As seen from this argument, many judges have difficulty perceiving that theSupreme Court-mandated procedure could possibly allow for justified envy.(4)
Arpita Sahu vs The State Of Madhya Pradesh on 21 August, 2012
MadhyaPradesh High Court . The petitioner files a lawsuit based on an instance of jus-tified envy, however in contrast to
Asha Ramnath Gholap (2016) , the judges havecorrectly dismissed the petition in this case.
Wrongful Implementation and Possible Misconduct . It is bad enough that theSupreme Court-mandated SCI-AKG choice rule is not incentive compatible, forcing somecandidates to risk losing their open-category HR protections by claiming their VR pro-tections. To make matters worse, in some cases candidates are denied access to open-category HR protections even when they do not submit their backward class status, giv-ing up their VR protections. Therefore, even when the candidate applies for a position asa general-category candidate without claiming the benefits of VR protections, the central The case is available at https://indiankanoon.org/doc/102792215/ (last accessed on 10/10/2019). planner processes the application as if the backward class status was claimed, denyingthe candidate’s eligibility for open-category HR protections. The central planners are of-ten able to do this, because last names in India are, to a large extent, indicative of a castemembership. This type of misconduct seems to be fairly widespread in some jusistictions,and it is the main cause of the lawsuit in dozens of cases such as the two Bombay HighCourt cases
Shilpa Sahebrao Kadam vs The State Of Maharashtra (2019) and
Vinod KadubalRathod vs Maharashtra State Electricity (2017) . Indeed, this type of misconduct is some-times intentional and systematic. The following statement is from
Shilpa Sahebrao Kadam(2019) : According to Respondent - Maharashtra Public Service Commission, inview of the Circular dated 13.08.2014, only the candidates belongingto open (Non-reserved) category can be considered for open horizontallyreserved posts meaning thereby, the reserved category candidates cannotbe considered for open horizontally reserved post. Reference is made toa communication issued by the Additional Chief Secretary (Service) of theState of Maharashtra dated 26.07.2017, whereunder it is prescribed thata female candidate belonging to any reserved category, even if tendersapplication form seeking employment as an open category candidate, the nameof such candidate shall not be recommended for employment against a opencategory seat.
Moreover, not all decisions in these lawsuits are made in accordance with the SCI-AKRchoice rule, which allows candidates to forego their VR (or HR) protections. This is thecase both for the first lawsuit and the last one listed above. For example, in the last lawsuitgiven above, two petitioners each applied for a position without declaring their backwardclass membership, with an intention to benefit from open-category HR protections. Fol-lowing their application, these petitioners were requested to provide their school leavingcertificates, which provided information on their backward class status. Upon receivingthis information, the petitioners were declined eligibility for open-category HR protec-tions, even though they never claimed their VR protections. Hence, they filed the fourthlawsuit given above. Remarkably, their petition was declined on the basis of their back-ward class membership. Here we have a case where the authorities not only go to greatlengths to obtain the backward class membership of the candidates, and wrongfully de-cline their eligibility for open category HR protections, but they also manage to get their The cases are available at https://indiankanoon.org/doc/89017459/ and https://indiankanoon.org/doc/162611497/ (last accessed on 03/09/2019).
FFIRMATIVE ACTION IN INDIA 39 lawsuits dismissed. The mishandling of this case is consistent with the concerns indicatedin the February 2006 issue of
The Inter-Regional Inequality Facility policy brief: Another issue relates to the access of SCs and STs to the institutions ofjustice in seeking protection against discrimination. Studies indicatethat SCs and STs are generally faced with insurmountable obstacles intheir efforts to seek justice in the event of discrimination. The officialstatistics and primary survey data bring out this character of justiceinstitutions. The data on Civil Rights cases, for example, shows that only1.6% of the total cases registered in 1991 were convicted, and that thishad fallen to 0.9% in 2000.
Loss of Access to HR protections without any Access to VR protections . The mainjustification offered in various Supreme Court cases for denying backward class memberstheir open-category HR protections is avoiding a situation where an excessive number ofpositions are reserved for members of these classes. In several cases, however, membersof these classes are denied access to open-category HR protections even when the numberof VR-protected positions is zero for their reserve-eligible vertical category. This is thecase in the following two court cases:(1)
Tejaswini Raghunath Galande v. The Chairman, Maharashtra Public Service Commissionand Ors. on 23 January 2019 , Writ Petition Nos. 5397 of 2016 & 5396 of 2016, HighCourt of Judicature at Bombay. (2) Original Application No. 662/2016 dated 05.12.2017, Maharashtra AdministrativeTribunal, Mumbai. In both cases, while the petitioners claimed their VR protections, there was no VR-protected position for their class. Yet in both cases petitioners lost their open-categoryHR protections. In the first case, the petitioners’ lawsuit to benefit from open-categoryHR protections was initially declined by a lower court, resulting in the appeal at the HighCourt. The lower court’s decision was overruled in the High Court, and her request wasgranted. On the other hand, the second petitioner’s similar request was declined by theMaharashtra Administrative Tribunal. What is more worrisome in the second case is that,while initially three positions were VR-protected for the petitioner’s backward class, afterthe petitioners application these VR-protected positions were withdrawn. Therefore, the The policy brief is available at (last accessed 03/09/2019). The case is available at (lastaccessed on 03/09/2019). The case is available at https://mat.maharashtra.gov.in/Site/Upload/Pdf/O.A.662%20of%202016.pdf (last accessed on 03/09/2019).0 S ¨ONMEZ AND YENMEZ
Tejaswini Raghunath Galande v. The Chairman, Maharashtra Public Service Commissionand Ors. on 23 January 2019 , Writ Petition Nos. 5397 of 2016 & 5396 of 2016, HighCourt of Judicature at Bombay. (2) Original Application No. 662/2016 dated 05.12.2017, Maharashtra AdministrativeTribunal, Mumbai. In both cases, while the petitioners claimed their VR protections, there was no VR-protected position for their class. Yet in both cases petitioners lost their open-categoryHR protections. In the first case, the petitioners’ lawsuit to benefit from open-categoryHR protections was initially declined by a lower court, resulting in the appeal at the HighCourt. The lower court’s decision was overruled in the High Court, and her request wasgranted. On the other hand, the second petitioner’s similar request was declined by theMaharashtra Administrative Tribunal. What is more worrisome in the second case is that,while initially three positions were VR-protected for the petitioner’s backward class, afterthe petitioners application these VR-protected positions were withdrawn. Therefore, the The policy brief is available at (last accessed 03/09/2019). The case is available at (lastaccessed on 03/09/2019). The case is available at https://mat.maharashtra.gov.in/Site/Upload/Pdf/O.A.662%20of%202016.pdf (last accessed on 03/09/2019).0 S ¨ONMEZ AND YENMEZ candidate declared her backward class status, giving up her open-category HR protec-tion, presumably to gain access to VR-protected positions set aside her reserve-eligibleclass, only to learn that she had given up her eligibility for nothing.
In a highly debated reform on the reservation system, the January2019
One Hundred and Third Amendment of the Constitution of India provides up to 10% VRprotections to the economically weaker sections (EWS) in the general category.In a case that is pending as of January 2021, the One Hundred and Third Amendmentwas immediately challenged at the Supreme Court and it was referred to a larger five-judge bench of the Supreme Court in August 2020. Despite the challenge at the SupremeCourt, the EWS reservation has already been adopted by federal institutions throughoutIndia as well as by most states at their state-run public institutions. If implemented jointlywith the SCI-AKG choice rule, the EWS reservation can be expected to amplify the legalchallenges formalized in Section 5.2 and documented in Section 8.1. Especially in stateswith a strong presence of horizontal reservations (such as states with 30-35% horizon-tal women reservation), legal challenges based on justified envy may become the normrather than an exception if the SCI-AKG choice rule is implemented with a 10% verticalEWS reservation. That is because, any candidate who applies both for the vertical EWSreservation and any HR protections lose access to open-category HR protections underthe SCI-AKG choice rule. To weight in what this would mean in the field, let us makesome simple ”back-of-the-envelope” calculations.It is estimated that around 26% of the population in India do not belong to the OtherBackward Classes (OBC), Scheduled Castes (SC), and Scheduled Tribes (ST) categories. Therefore, prior to the January 2019 amendment of the Constitution, approximately 26%of the population belonged to the general category. While the amendment is intendedonly for the economically weaker sections of the general category, according to most es-timates more than 80% of the members of this group satisfy the eligibility criteria for theEWS reservation. This means, with the introduction of the EWS reservation, the fraction See for the pending Supreme Court case
Youth for Equality v. Union of India . See the 01/07/2017-dated
Hindustan Times story “Quota for econom-ically weak in general category could benefit 190 mn,” available at ,last accessed on 04/14/2019. See the 01/08/2019 dated
Business Today story “In-depth: Who is el-igible for the new reservation quota for general category?” available at ,(last accessed on 04/14/2019).
FFIRMATIVE ACTION IN INDIA 41 of the population who are ineligible for any VR protections reduces to a mere 5-6% of thepopulation of India. Therefore, the “new general category,” those members of the societywho are ineligible for any VR protections, shrinks to approximately 5-6% of the wholepopulation. A key implication of this observation is the following: Under the SCI-AKGchoice rule, only this “elite” 5-6% of the population qualifies for the adjustments for open-category HR protections, which could easily be more than 10% of all positions in stateswith extensive provision of HR protections. Thus, had the Supreme Court not aban-doned the SCI-AKG choice rule in an important December 2020 judgement, maintainingthe EWS reservation would have likely increased litigations due to justified envy con-siderably throughout India, especially in states such as Bihar, Gujarat, Andhra Pradesh,Madhya Pradesh, Rajasthan, Uttarakhand, Chhattisgarh, Sikkim, all with 30 −
35% HRprotections for women.
9. Epilogue: December 2020 Supreme Court of India Resolution on Elimination ofJustified Envy and the Demise of the SCI-AKG Choice Rule
As our paper was under revision for this journal, a December 2020 Supreme Courtjudgement in
Saurav Yadav v State of Uttar Pradesh (2020) became headline news in India.Using arguments parallel to our analysis presented in Sections 5.2 and 8.1.1, a three-judgebench of the highest court reached some of the same conclusions we have reached in thispaper. Most notably, similar to our policy recommendations, with this judgement(1) all allocation rules for public recruitment are federally mandated to eliminate jus-tified envy , and thereby(2) the SCI-AKG choice rule, mandated for 25 years, loses its legality.Using several of the same judgements we present in Section 8.1, the judges have alsohighlighted the inconsistencies between several High Court judgements in relation totheir approach to possibility of justified envy in allocation. The Supreme Court judgesalso declared that while the “first view” that enforces no justified envy by the High Courtjudgements of Rajasthan, Bombay, Gujarat, and Uttarakhand is “correct and rational,” the“second view” that allows for justified envy by the High Court judgements of Allahabadand Madhya Pradesh is not. While the axiom of no justified envy is federally enforced with
Saurav Yadav v Stateof Uttar Pradesh (2020) , unlike in
Anil Kumar Gupta (1995) no explicit procedure is fed-erally mandated with this judgement. However, through its August 2020 judgement It is important to emphasize that, prior to this ruling, the second view—now deemed incorrect andirrational—was the one that is in line with the SCI-AKG choice rule, whereas the first view—now deemedcorrect and rational—deviated from the previously mandated choice rule.
Tamannaben Ashokbhai Desai v. Shital Amrutlal Nishar (2020) , the High Court of Gujaratmandated the use of the two-step minimum guarantee choice rule for the state of Gujarat. Themandated choice rule in Gujarat is described for a single group of beneficiaries (women)for horizontal reservations under this High Court ruling, and therefore not only it isequivalent to our proposed 2SMH choice rule in this special case as explained in Section4.2, but also it is the only choice rule that satisfies our four axioms as presented in Corol-lary 2. While the Supreme Court has not enforced any specific rule in its December 2020judgement, it has endorsed the two-step minimum guarantee choice rule given in TamannabenAshokbhai Desai v. Shital Amrutlal Nishar (2020) .
36. Finally, we must say that the steps indicated by the High Courtof Gujarat in para 56 of its judgment in Tamannaben Ashokbhai Desaicontemplate the correct and appropriate procedure for consideringand giving effect to both vertical and horizontal reservations. Theillustration given by us deals with only one possible dimension.There could be multiple such possibilities. Even going by the presentillustration, the first female candidate allocated in the vertical columnfor Scheduled Tribes may have secured higher position than the candidateat Serial No.64. In that event said candidate must be shifted from thecategory of Scheduled Tribes to Open / General category causing a resultantvacancy in the vertical column of Scheduled Tribes. Such vacancy must thenenure to the benefit of the candidate in the Waiting List for ScheduledTribes - Female.The steps indicated by Gujarat High Court will take care of every suchpossibility. It is true that the exercise of laying down a procedure mustnecessarily be left to the concerned authorities but we may observe thatone set out in said judgment will certainly satisfy all claims and will notlead to any incongruity as highlighted by us in the preceding paragraphs.
Since neither the Supreme Court’s nor the Gujarati High Court’s judgement involves is-sues that pertain to overlapping horizontal reservations, these decisions are parallel toour recommendation, albeit in a simpler environment. While the primary objective ofthese judgements are eliminating justified envy, they also restored the incentive compat-ibility of the system and eliminated a major discord with the One Hundred and ThirdAmendment of the Constitution of India. See Section C.4 in the Online Appendix for the description of the procedure in
Tamannaben AshokbhaiDesai (2020)
FFIRMATIVE ACTION IN INDIA 43
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Appendix A. Mathematical Preliminaries
In this appendix, we provide some preliminary results that we use in our proofs. First,we introduce some graph-theoretic terminology.Consider a category v ∈ V and a set of individuals I ⊆ I v . Let G be the category- v HRgraph for I . The vertices of G are individuals in I and reserved positions in H v . Thereexists an edge between an individual i ∈ I and a position reserved for trait t ∈ T if i hastrait t . A matching is a set of edges without common vertices. A matching covers a vertexif it has an edge adjacent to that vertex. Lemma 1 (Dulmage-Mendelsohn Theorem) . Consider the HR graph for v ∈ V and I ⊆ I v .Suppose that there exist a matching that covers individuals in J ⊆ I and a matching that coversreserved positions in S ⊆ H v . Then there exists a matching that covers both J and S. See Theorem 4.1 in Lawler (2001, Page 191) for a proof of this lemma.An alternating path between matching M and matching M is a path of connectededges that starts at a vertex covered by M but not by M and ends at a vertex covered by M but not by M such that edges of the path belong alternately to M and M . Lemma 2 (Alternating Path) . Let M and M be two distinct matchings that cover the same setof positions in a HR graph. Suppose that there exists a vertex i covered by M but not by M .Then there exists an alternating path between matching M and matching M that starts at i.Proof. Let i ≡ i and ( i , s ) be the edge that covers i in M . Since M and M cover thesame set of positions, there exists an edge ( i , s ) in M . If i is not covered by M , then weare done. Otherwise, i is covered by both M and M . Let ( i , s ) be the edge in M thatcovers i . Since M and M cover the same set of positions, there exists an edge ( i , s ) in M . If i is not covered by M , then we are done. Otherwise, i is covered by both M and M . Continue this construction. Since there exists a finite number of vertices, thisconstruction ends in finite time at a vertex i k covered by M but not by M . This finishesthe construction of an alternating path starting at i . See Figure 4 for an illustration of thealternating path that is constructed. (cid:3) For the next result, we extend the definition of the substitutes condition and the irrele-vance of rejected individuals condition to single-category choice rules.
Lemma 3.
Let v ∈ V . C v M satisfies the substitutes condition and the irrelevance of rejectedindividuals condition. FFIRMATIVE ACTION IN INDIA 47 i i i k . . . i k − s s . . . s k − F IGURE
4. The alternating path between M and M constructed in theproof of Lemma 2. The edges in M are solid and the edges in M aredashed. Proof.
The irrelevance of rejected individuals condition is satisfied trivially by the con-struction of C v M . We show that C v M also satisfies the substitutes condition.Let I ⊆ I v , i ∈ C v M ( I ) , and j ∈ I \ i . To prove the substitutes condition, we show that i ∈ C v M ( I \ { j } ) . If j / ∈ C v M ( I ) , then C v M ( I \ { j } ) = C v M ( I ) because C v M satisfies the irrelevanceof rejected individuals condition, so i ∈ C v M ( I \ { j } ) .For the rest of the proof assume that j ∈ C v M ( I ) . If i is chosen before j in the constructionof C v M ( I ) , then i ∈ C v M ( I \ { j } ) since in the construction of C v M ( I \ { j } ) all the steps will bethe same as in the construction of C v M ( I ) until individual j is considered. Now assume that j is chosen before i in the construction of C v M ( I ) . Before we consider two separate casesbelow, we introduce the following notation. For any I ′ ⊆ I v , let C ( I ′ ) ⊆ I ′ denote theset of individuals chosen at Step 1 in the construction of C v M ( I ′ ) and C ( I ′ ) ⊆ I ′ denotethe set of individuals chosen at Step 2 in the construction of C v M ( I ′ ) . If j ∈ C ( I ) , then i ∈ C v M ( I \ { j } ) follows because at the second step individuals with the highest meritscores are chosen. For the following cases, assume that j ∈ C ( I ) . Case 1 ( n v ( I \ { j } ) = n v ( I ) − ) : We claim that(1) C ( I \ { j } ) = C ( I ) \ { j } and(2) C ( I \ { j } ) ⊇ C ( I ) \ { j } .To prove the first displayed equation note that by Dulmage-Mendelsohn Theorem C ( I ) \{ j } and C ( I \ { j } ) can be matched with the same set of positions in the HR reservationgraph. Therefore, n v ( C ( I ) \ { j } ) = n v ( I ) − n v ( C ( I \ { j } ) ∪ { j } ) = n v ( I ) . WhenTheorem 1 is applied to the set of individuals I \ { j } when the number of positions forcategory v is n v ( I ) −
1, we get that the individual with the k -th highest merit score in C ( I \ { j } ) has a weakly higher merit score than the individual with the k -th highest meritscore in C ( I ) \ { j } for every k ∈ {
1, . . . , n v ( I ) − } . Likewise, when Theorem 1 is applied to the set of individuals I when the number of positions for category v is n v ( I ) , we getthat the individual with the k -th highest merit score in C ( I ) has a weakly higher meritscore than the individual with the k -th highest merit score in C ( I \ { j } ) ∪ { j } for every k ∈ {
1, . . . , n v ( I ) } . The last two sets of inequalities imply that C ( I \ { j } ) = C ( I ) \ { j } since individuals have distinct merit scores. The second displayed equation follows fromthe first one since at the second step unassigned individuals with the highest merit scoresare chosen.The first displayed equation implies that if i ∈ C ( I ) , then i ∈ C ( I \ { j } ) . The seconddisplayed equation implies that if i ∈ C ( I ) , then i ∈ C ( I \ { j } ) . Therefore, i ∈ C v M ( I \{ j } ) . Case 2: ( n v ( I \ { j } ) = n v ( I )) : As in the previous case, we get that C ( I \ { j } ) =( C ( I ) \ { j } ) ∪ { j ′ } where j ′ ∈ I \ C ( I ) by Dulmage-Mendelsohn Theorem and Theo-rem 1. Therefore, if i ∈ C ( I ) , then i ∈ C ( I \ { j } ) . Furthermore, if j ′ / ∈ C ( I ) , then C ( I \ { j } ) = C ( I ) . Otherwise, if j ′ ∈ C ( I ) , then C ( I \ { j } ) ⊇ C ( I ) \ { j ′ } . Therefore,regardless of whether j ′ is in C ( I ) or not, i ∈ C ( I ) implies j ′ = i or i ∈ C ( I \ { j } ) . As aresult, i ∈ C v M ( I \ { j } ) . (cid:3) Appendix B. Proofs
In this section, we present the main proofs.
Proof of Proposition 1.
Let I = J ∪ K and I ′ be the set of individuals assigned to category- v positions by AKG-HAS. We first show that(1) | I ′ | = q v ,(2) there exists no instance of justified envy involving an individual in I ′ and an indi-vidual in I \ I ′ ,(3) I ′ maximally accommodates category- v HR protections for I .Then the proof follows from Corollary 1.Proof of (1): | I ′ | = q v follows because at Step |T | + i ∈ I ′ and j ∈ I \ I ′ such that σ ( j ) > σ ( i ) . Since j / ∈ I ′ , either j doesnot have a trait or there are at least q vt individuals in I ′ where t is j ’s only trait. If j doesnot have a trait, then i must have a trait t ′ such that the number of individuals in I ′ whohas trait t ′ is min { q vt ′ , |{ i ′ ∈ I : t ′ ∈ τ ( i ′ ) }|} . Then n (( I ′ \ { i } ) ∪ { j } ) = n ( I ′ ) −
1, whichmeans that there is no instance of justified envy involving j and i . If j has trait t , then itmust be that i does not have trait t , there are at least q vt individuals with trait t in I ′ , and i must have a trait t ′ = t such that the number of individuals in I ′ who have trait t ′ is FFIRMATIVE ACTION IN INDIA 49 min { q vt ′ , |{ i ′ ∈ I : t ′ ∈ τ ( i ′ ) }|} . Then, as before, n (( I ′ \ { i } ) ∪ { j } ) = n ( I ′ ) −
1, whichmeans that there is no instance of justified envy involving j and i .Proof of (3): For every trait t , there is a corresponding step of AKG-HAS so that thenumber of individuals in I ′ who has trait t is min { q vt , |{ i ′ ∈ I : t ∈ τ ( i ′ ) }|} . Since eachindividual has at most one trait, this implies that n v ( I ′ ) = n v ( I ) . (cid:3) Proof of Theorem 1.
Let I ⊆ I v be a set of individuals. To show part (1), note that | C v M ( I ) | = min { q v , | I |} . Furthermore, for single-category choice rule C v , C v ( I ) ⊆ I and | C v ( I ) | ≤ q v . Therefore, | C v ( I ) | ≤ min { q v , | I |} = | C v M ( I ) | .We show part (2) by mathematical induction on parameters ( q v , ( q vt ) t ⊆T ) . We show theclaim that for an ordering of agents in C v M ( I ) \ C v ( I ) and C v ( I ) \ C v M ( I ) that the k -th agentin C v M ( I ) \ C v ( I ) has a higher priority than the k -th agent in C v ( I ) \ C v M ( I ) , which impliespart (2).For the base case when there are no reserved positions, statement (2) holds because C v M chooses all individuals at Step 2 according to the merit score ranking. Now suppose thatthe claim holds for all parameters bounded above by ( q v , ( q vt ) t ⊆T ) . Consider parameters ( q v , ( q vt ) t ⊆T ) . If all individuals in C v M ( I ) \ C v ( I ) are chosen at Step 2, then the claim holdsas in the base case because individuals in C v ( I ) \ C v M ( I ) are available at Step 2 in theconstruction of C v M ( I ) .Consider the situation when there exists at least one individual in C v M ( I ) \ C v ( I ) cho-sen at Step 1. Let i be the individual with the highest priority in C v M \ C v ( I ) chosen atStep 1 and t be the trait of the position that she is matched with. By Lemma 4, C v M maxi-mally accommodates HR protections, so in the HR graph, there exists a matching M thatmatches C v M ( I ) to a set of reserved positions S with maximum cardinality n v ( I ) . Since C v also maximally accommodates HR protections, by Dulmage-Mendelsohn Theorem (seeLemma 1) there exists another matching M that matches C v ( I ) to S both of which havecardinality n v ( I ) . By Lemma 2, there exists an alternating path that starts at i and ends atan individual i ′ ∈ C v ( I ) \ C v M ( I ) . Therefore, individual i can be replaced with individual i ′ in C v M ( I ) without changing the set of positions covered in the HR graph for I . Hence,by construction of C v M ( I ) , σ ( i ) > σ ( i ′ ) because i ′ is available when i is chosen at Step 1.Now consider the reduced market when capacity q v and trait- t reservation q vt areboth reduced by one and the set of individuals is I \ { i , i ′ } . In this reduced market, C v M ( I \ { i , i ′ } ) is equal to C v M ( I ) \ { i } for the original market because i ′ / ∈ C v M ( I ) and theconstruction of C v M ( I \ { i , i ′ } ) chooses individuals in the same order as they are chosenin C v M ( I ) . In particular, the set of individuals chosen before i at C v M ( I ) are chosen in the0 S ¨ONMEZ AND YENMEZ
Let I ⊆ I v be a set of individuals. To show part (1), note that | C v M ( I ) | = min { q v , | I |} . Furthermore, for single-category choice rule C v , C v ( I ) ⊆ I and | C v ( I ) | ≤ q v . Therefore, | C v ( I ) | ≤ min { q v , | I |} = | C v M ( I ) | .We show part (2) by mathematical induction on parameters ( q v , ( q vt ) t ⊆T ) . We show theclaim that for an ordering of agents in C v M ( I ) \ C v ( I ) and C v ( I ) \ C v M ( I ) that the k -th agentin C v M ( I ) \ C v ( I ) has a higher priority than the k -th agent in C v ( I ) \ C v M ( I ) , which impliespart (2).For the base case when there are no reserved positions, statement (2) holds because C v M chooses all individuals at Step 2 according to the merit score ranking. Now suppose thatthe claim holds for all parameters bounded above by ( q v , ( q vt ) t ⊆T ) . Consider parameters ( q v , ( q vt ) t ⊆T ) . If all individuals in C v M ( I ) \ C v ( I ) are chosen at Step 2, then the claim holdsas in the base case because individuals in C v ( I ) \ C v M ( I ) are available at Step 2 in theconstruction of C v M ( I ) .Consider the situation when there exists at least one individual in C v M ( I ) \ C v ( I ) cho-sen at Step 1. Let i be the individual with the highest priority in C v M \ C v ( I ) chosen atStep 1 and t be the trait of the position that she is matched with. By Lemma 4, C v M maxi-mally accommodates HR protections, so in the HR graph, there exists a matching M thatmatches C v M ( I ) to a set of reserved positions S with maximum cardinality n v ( I ) . Since C v also maximally accommodates HR protections, by Dulmage-Mendelsohn Theorem (seeLemma 1) there exists another matching M that matches C v ( I ) to S both of which havecardinality n v ( I ) . By Lemma 2, there exists an alternating path that starts at i and ends atan individual i ′ ∈ C v ( I ) \ C v M ( I ) . Therefore, individual i can be replaced with individual i ′ in C v M ( I ) without changing the set of positions covered in the HR graph for I . Hence,by construction of C v M ( I ) , σ ( i ) > σ ( i ′ ) because i ′ is available when i is chosen at Step 1.Now consider the reduced market when capacity q v and trait- t reservation q vt areboth reduced by one and the set of individuals is I \ { i , i ′ } . In this reduced market, C v M ( I \ { i , i ′ } ) is equal to C v M ( I ) \ { i } for the original market because i ′ / ∈ C v M ( I ) and theconstruction of C v M ( I \ { i , i ′ } ) chooses individuals in the same order as they are chosenin C v M ( I ) . In particular, the set of individuals chosen before i at C v M ( I ) are chosen in the0 S ¨ONMEZ AND YENMEZ same order in C v M ( I \ { i , i ′ } ) . Furthermore, after i is chosen the set of updated parametersare exactly the same. Therefore, the same set of individuals are chosen in the same orderafter i is chosen in C v M ( I ) . In addition, C v ( I ) \ { i ′ } maximally accommodates HR protec-tions and i / ∈ C v ( I ) \ { i ′ } . By the induction hypothesis, the individuals in C v M ( I \ { i , i ′ } ) and C v ( I ) \ { i ′ } can be ordered with the required property, which implies the hypothesis.Therefore, the hypothesis holds for every set of parameters ( q v , ( q vt ) t ⊆T ) . (cid:3) Proof of Theorem 2.
We first show that C v M satisfies the stated properties in several lem-mas and then show that the unique category- v choice rule satisfying these properties is C v M . Lemma 4. C v M maximally accommodates HR protections.Proof. Suppose, for contradiction, that C v M does not maximally accommodate HR protec-tions. Hence, there exists I ⊆ I v such that C v M ( I ) does not maximally accommodateHR protections for I . Therefore, in the HR graph for C v M ( I ) , the maximum cardinalitythat can be attained by a matching is strictly less than n v ( I ) . Let ¯ I ⊆ C v M ( I ) be the setof individuals who are chosen at Step 1 in the construction of C v M ( I ) . By assumption, | ¯ I | = n v ( C v M ( I )) < n v ( I ) . Now consider a maximum matching for the HR graph for I . Let S be the set of positions matched, so | S | = n v ( I ) . By Dulmage-Mendelsohn Theorem (seeLemma 1), there exists a matching that assigns every individual in ¯ I and every reservedposition in S in the HR graph of I . But this is a contradiction to the construction of C v M ( I ) ,as there exists an individual who increases HR utilization of ¯ I . (cid:4) Lemma 5. C v M satisfies no justified envy.Proof. Suppose, for contradiction, that C v M has justified envy. Therefore, there exist aset of individuals I ⊆ I v , individuals i ∈ C v M ( I ) , j ∈ I \ C v M ( I ) with σ ( j ) > σ ( i ) and n v (cid:0) ( C v M ( I ) \ { i } ) ∪ { j } (cid:1) ≥ n v ( C v M ( I )) . Consider category- v choice rule C v such that C v ( I ′ ) = C v M ( I ′ ) , if I ′ = I ( C v M ( I ) \ { i } ) ∪ { j } , if I ′ = I .Since C v M maximally accommodates HR protections, C v ( I ′ ) maximally accommodates HRprotections for I ′ whenever I ′ = I because C v ( I ′ ) = C v M ( I ′ ) . Furthermore, n v ( C v ( I )) = n v (cid:0) ( C v M ( I ) \ { i } ) ∪ { j } (cid:1) ≥ n v ( C v M ( I )) and the fact that C v M maximally accommodates HR protections by Lemma 4 (i.e., n v ( C v M ( I )) = n v ( I ) ) imply that n v ( C ( I )) = n v ( I ) because of the fact that n v ( I ) is themaximum cardinality. Hence, C v ( I ) maximally accommodates HR protections for I . By FFIRMATIVE ACTION IN INDIA 51
Theorem 1, for every k ≤ | C v ( I ) | , the individual with the k -th highest priority in C v M ( I ) has a weakly higher priority than the individual with the k -th highest priority in C v ( I ) .This is a contradiction to the construction of C v because C v ( I ) = ( C v M ( I ) \ { i } ) ∪ { j } and σ ( j ) > σ ( i ) . (cid:4) Lemma 6. C v M is non-wasteful.Proof. C v M is non-wasteful because at the second step all the unfilled positions are filledwith the unmatched individuals until all positions are filled or all individuals are assignedto positions. (cid:4) Lemma 7.
Let v ∈ V . If a category-v choice rule maximally accommodates HR protections,satisfies no justified envy, and is non-wasteful, then it has to be C v M .Proof. Let C v be a category- v choice rule that maximally accommodates category- v HRprotections, satisfies no justified envy, and is non-wasteful. Suppose, for contradiction,that C v = C v M . Therefore, there exists I ⊆ I v such that C v ( I ) = C v M ( I ) . Since both choicerules are non-wasteful | C v ( I ) | = | C v M ( I ) | .Since C v ( I ) = C v M ( I ) , this equation implies that (cid:12)(cid:12) C v M ( I ) \ C v ( I ) (cid:12)(cid:12) = (cid:12)(cid:12) C v ( I ) \ C v M ( I ) (cid:12)(cid:12) > n v ( I ) . Case 1: If n v ( I ) =
0, then no individual in I has a trait that has a positive reservation.Therefore, C v M ( I ) consists of min {| I | , q v } individuals with the highest merit score in I .This is a contradiction to the assumption that C v ( I ) satisfies no justified envy becauseany individual i ∈ C v M ( I ) \ C v ( I ) = ∅ has a higher merit score than any individual j ∈ C v ( I ) \ C v M ( I ) = ∅ and n v (( C v ( I ) \ { j } ) ∪ { i } ) ≥ n v ( C v ( I )) =
0. Hence, there is aninstance of justified envy for C v ( I ) involving i ∈ I \ C v ( I ) and j ∈ C v ( I ) , which is acontradiction. Case 2:
Let n v ( I ) = n >
0. Therefore, there are n individuals chosen at Step 1 of C v M ( I ) . For 1 ≤ k ≤ n , let i k be the k -th individual chosen at Step 1 of C v M ( I ) . Considera maximum matching M in the HR graph for C v M ( I ) that matches I ≡ { i , . . . , i n } . Weshow that C v ( I ) ⊇ I . Let S be the set of positions that are matched in M . Since C v maximally accommodates HR protections, there exists a maximum matching in the HRgraph for C v ( I ) that has cardinality n . Furthermore, by Dulmage-Mendelsohn Theorem(see Lemma 1), there exists a matching of a subset of C v ( I ) to positions in S . Let I ⊆ C v ( I ) be the set of these individuals and M be this matching. Suppose, for contradiction, that I \ C v ( I ) = ∅ . Let i k be the individual with the lowest index in I \ C v ( I ) . By Lemma2, there exists an alternating path between M and M that starts at i k and ends at avertex j covered by M but not by M . Therefore, i k and j can be replaced with eachother in both M and M without decreasing the maximum cardinality. By constructionof C v M , σ ( i k ) > σ ( j ) because j is available when i k is chosen. But this is a contradictionto the assumption that C v ( I ) satisfies no justified envy because j ∈ C v ( I ) , i k ∈ I \ C v ( I ) , σ ( i k ) > σ ( j ) , and n v ( C v ( I )) = n v (( C v ( I ) \ { j } ) ∪ { i k } ) . Therefore, I ⊆ C v ( I ) .By construction of C v M ( I ) , every individual in C v M ( I ) \ I is chosen at Step 2. Therefore,these individuals have a higher merit score than any individual in I \ C v M ( I ) . Let j ∈ C v ( I ) \ C v M ( I ) , which is non-empty by assumption. Therefore, j ∈ I \ C v M ( I ) , which meansthat any individual i ∈ C v M ( I ) \ C v ( I ) has a strictly higher merit score than j . This is acontradiction to the assumption that C v ( I ) satisfies no justified envy because j ∈ C v ( I ) , i ∈ I \ C v ( I ) , σ ( i ) > σ ( j ) , and n v ( C v ( I )) = n = n v (( C v ( I ) \ { j } ) ∪ { i } ) where the lastequation follows from I ⊆ ( C v ( I ) \ { j } ) ∪ { i } and the fact that n v ( I ) = n . (cid:4) This finishes the proof of Theorem 2. (cid:3)
Proof of Theorem 3.
Let C = ( C v ) v ∈V be a choice rule that complies with VR protections,maximally accommodates HR protections, satisfies no justified envy, and is non-wasteful.We show this result using the following lemmas. Lemma 8. C o = C s , o M .Proof. We prove that C o maximally accommodates category- o HR protections, satisfies nojustified envy, and is non-wasteful.First, we show that C o maximally accommodates category- o HR protections. Suppose,for contradiction, that n o ( C o ( I )) < n o ( I ) for some I ⊆ I . Then there exists i ∈ I \ C o ( I ) such that n o ( C o ( I ) ∪ { i } ) = n o ( C o ( I )) +
1. If i ∈ I \ b C ( I ) , then we get a contradiction withthe assumption that C maximally accommodates HR protections. Otherwise, if i ∈ C c ( I ) where c ∈ R , then we get a contradiction with the assumption that C complies with VRprotections. Therefore, C o maximally accommodates category- o HR protections.Next, we show that C o satisfies no justified envy. Let i ∈ C o ( I ) and j ∈ I \ C o ( I ) suchthat σ ( j ) > σ ( i ) . If j ∈ I \ b C ( I ) , then n o (( C o ( I ) \ { i } ) ∪ { j } ) < n o ( C o ( I )) because C satisfies no justified envy. However, if i ∈ C c ( I ) for category c ∈ R , then n o (( C o ( I ) \ { i } ) ∪ { j } ) < n o ( C o ( I )) FFIRMATIVE ACTION IN INDIA 53 because C complies with VR protections. Therefore, C o satisfies no justified envy.Now, we show that C o is non-wasteful, which means that | C o ( I ) | = min {| I | , q o } forevery I ⊆ I . If there exists an individual i ∈ I such that i / ∈ b C ( I ) , then | C o ( I ) | = q o because C is non-wasteful. If there exists an individual i ∈ I such that i ∈ C c ( I ) where c = ρ ( i ) ∈ R , then | C o ( I ) | = q o because C complies with VR protections. If these twoconditions do not hold, then all the individuals are allocated open-category positions, i.e., I = C o ( I ) . Therefore, under all possibilities, we get | C o ( I ) | = min {| I | , q o } , which meansthat C o is non-wasteful.Since C o maximally accommodates category- o HR protections, satisfies no justifiedenvy, and is non-wasteful, we get C o = C o M (Theorem 2), and hence C o = C s , o M . (cid:4) Let c ∈ R , I ⊆ I , and ¯ I c = { i ∈ I \ C o M ( I ) | ρ ( i ) = c } . Lemma 9. C c ( I ) maximally accommodates category-c HR protections for ¯ I c .Proof. Suppose, for contradiction, that n c ( C c ( I )) < n c ( ¯ I c ) . This is equivalent to n c ( C c ( I )) < n c ( ¯ I c ) = n c (cid:16) C c ( I ) ∪ { i ∈ I \ b C ( I ) | ρ ( i ) = c } (cid:17) ,which implies that there exists i ∈ I \ b C ( I ) who is eligible for category c such that n c ( C c ( I ∪ { i } )) = n c ( C c ( I )) + C maximally accommodates HR protec-tions. Therefore, C c ( I ) maximally accommodates category- c HR protections for ¯ I c . (cid:4) Lemma 10. C c ( I ) satisfies no justified envy for ¯ I c .Proof. Let i ∈ C c ( I ) and j ∈ ¯ I c \ C c ( ¯ I c ) be such that σ ( j ) > σ ( i ) . Note that i ∈ ¯ I c . Since C satisfies no justified envy, we have n c ( C c ( I )) > n c (( C c ( I ) \ { j } ) ∪ { i } ) .Hence, C c satisfies no justified envy for ¯ I c . (cid:4) Lemma 11. | C c ( I ) | = min {| ¯ I c | , q c } .Proof. We consider two cases. First, if C c ( I ) = ¯ I c , then | C c ( I ) | = min {| ¯ I c | , q c } because | C c ( I ) | ≤ q c . Otherwise, if C c ( I ) = ¯ I c , then there exists i ∈ ¯ I c \ C c ( I ) . Therefore, i ∈ I \ b C ( I ) . Since C is non-wasteful, we get | C c ( I ) | = q c Since i ∈ ¯ I c \ C c ( I ) and | C c ( I ) | = q c , | ¯ I c | > q c . Therefore, | C c ( I ) | = q c = min {| ¯ I c | , q c } . (cid:4) Therefore, C c ( I ) maximally accommodates category- c HR protections for ¯ I c , C c ( I ) sat-isfies no justified envy for ¯ I c , and C c ( I ) is non-wasteful for ¯ I c . By Theorem 2, C c ( I ) = C c M ( ¯ I c ) and, thus, C c ( I ) = C c M ( ¯ I c ) = C c M ( { i ∈ I \ C o M ( I ) | ρ ( i ) = c } ) = C s , c M ( I ) . (cid:3) Proof of Proposition 2.
Suppose that i is chosen by b C s M when she withholds some ofher reserve-eligible privileges. If i is chosen by C o M for an open-category position, then i will still be chosen by declaring all her reserve-eligible privileges because C o M does notuse the category information of individuals and an individual can never benefit from notdeclaring some of her traits under C o M because she will have more edges in the category- o HR graph. Otherwise, if i is chosen by C c M where ρ ( i ) = c ∈ R then she must havedeclared her reserve-eligible category c . In addition, by declaring all her traits she willstill be chosen by C c M if she is not chosen before for the open-category positions becauseshe will have more edges in the HR graph for category- c positions. (cid:3) Proof of Proposition 3.
Let I ⊆ I be a set of individuals and I m ⊆ I be the set of reserve-eligible individuals considered at Step 1 of b C SCI M when I is the set of applicants.Let i ∈ b C s M ( I ) ∩ I g . Then i ∈ C o M ( I ) ∩ I g because b C s M ( I ) ∩ I g = C o M ( I ) ∩ I g . Since C o M satisfies the substitutes condition (Lemma 3), i ∈ C o M ( I m ∪ I g ) because i ∈ I g and i ∈ C o M ( I ) . Therefore, i ∈ C o M ( I m ∪ I g ) ∩ I g , which implies i ∈ b C SCI M ( I ) ∩ I g because b C SCI M ( I ) ∩ I g = C o M ( I m ∪ I g ) ∩ I g . Therefore, we conclude that b C s M ( I ) ∩ I g ⊆ b C SCI M ( I ) ∩ I g .The assumption that | I c | ≥ q o + q c , for each reserve-eligible category c ∈ R , impliesthat all category- c positions are filled under C s M and C SCI M . In addition, the first part ofthe proposition implies that there are weakly more individuals with reserved categoriesassigned to open-category positions under C s M than under C SCI M . Therefore, ∑ c ∈R (cid:12)(cid:12) b C s M ( I ) ∩ I c (cid:12)(cid:12) ≥ ∑ c ∈R (cid:12)(cid:12) b C SCI M ( I ) ∩ I c (cid:12)(cid:12) . (cid:3) Proof of Proposition 4.
To show the substitutes condition, let I ⊆ I , i ∈ b C s M ( I ) , and j ∈ I \ { i } . Since i ∈ b C s M ( I ) , either i ∈ C s , o M ( I ) or i ∈ C s , c M ( I ) where ρ ( i ) = c ∈ R . If i ∈ C s , o M ( I ) = C o M ( I ) , then i ∈ C o M ( I \ { j } ) = C s , o M ( I \ { j } ) since C o M satisfies the substitutescondition (Lemma 3). Now consider the other possibility that i ∈ C s , c M ( I ) where ρ ( i ) = c ∈ R . Let ¯ I c = { i ′ ∈ I \ C o M ( I ) : ρ ( i ′ ) = c } FFIRMATIVE ACTION IN INDIA 55 and ( I \ { j } ) c = { i ′ ∈ ( I \ { j } ) \ C o M (( I \ { j } )) : ρ ( i ′ ) = c } .The assumption that i ∈ C s , c M ( I ) and C s , c M ( I ) = C c M ( ¯ I c ) imply that i ∈ C c M (( I \ { j } ) c ) whenever i / ∈ C o M ( I \ { j } ) because C c M satisfies the substitutes condition (Lemma 3) and¯ I c ⊇ ( I \ { j } ) c since C o M satisfies the substitutes condition (Lemma 3). Therefore, i ∈ b C s M ( I \ { j } ) , which means that C s M satisfies the substitutes condition.To show the irrelevance of rejected individuals condition, let I ⊆ I and i ∈ I \ b C s M ( I ) .Since i ∈ I \ b C s M ( I ) , i ∈ I \ C o M ( I ) which implies that C o M ( I ) = C o M ( I \ { i } ) because C o M satisfies the irrelevance of rejected individuals condition (Lemma 3). Fix c ∈ R and let¯ I c = { j ∈ I \ C o M ( I ) : ρ ( j ) = c } and ( I \ { i } ) c = { j ∈ ( I \ { i } ) \ C o M (( I \ { i } )) : ρ ( j ) = c } .If i / ∈ ¯ I c , then ¯ I c = ( I \ { i } ) c because C o M ( I ) = C o M ( I \ { i } ) , and hence C c M ( ¯ I c ) = C c M (( I \ { i } ) c ) , which is equivalent to C s , c M ( I ) = C s , c M ( I \ { i } ) . Otherwise, if i ∈ ¯ I c ,then ¯ I c = ( I \ { i } ) c ∪ { i } because C o M ( I ) = C o M ( I \ { i } ) . Furthermore, i / ∈ b C s M ( I ) im-plies that i / ∈ C c M ( ¯ I c ) . As a result, since C c M satisfies the irrelevance of rejected individualscondition (Lemma 3), C c M ( ¯ I c ) = C c M ( ¯ I c \ { i } ) = C c M (( I \ { i } ) c ) , which is equivalent to C s , c M ( I ) = C s , c M ( I \ { i } ) . We conclude that b C s M ( I ) = b C s M ( I \ { i } ) , so C s M satisfies the irrele-vance of rejected individuals condition. Appendix C. Institutional Background on Vertical and Horizontal Reservations
In this appendix, we present(1) the description of the concepts of vertical reservation and horizontal reservation asthey are quoted in the Supreme Court judgements
Indra Sawhney (1992) and
RajeshKumar Daria (2007) in Sections C.1 and C.2,(2) the main quotes from the Supreme Court judgements
Anil Kumar Gupta (1995) and
Rajesh Kumar Daria (2007) that allows us to formulate the SCI-AKG choice rule inSection C.3, and(3) the description of the choice rule that is mandated in the State of Gujarat as it isquoted in the August 2020 High Court of Gujarat judgement
Tamannaben AshokbhaiDesai (2020) in Section C.4.
C.1. Indra Sawhney (1992): Introduction of Vertical and Horizontal Reservations.
Theterms vertical reservation and horizontal reservation are coined by the Constitution bench ofthe Supreme Court of India, in the historical judgement
Indra Sawhney (1992) , where • the former was formulated as a policy tool to accommodate the higher-level pro-tective provisions sanctioned by the Article 16(4) of the Constitution of India, and • the latter was formulated as a policy tool to accommodate the lower-level protec-tive provisions sanctioned by the Article 16(1) of the Constitution of India.The description of these two affirmative action policies, and how they are intended tointeract with each other is given in the judgement with following quote: A little clarification is in order at this juncture: all reservations arenot of the same nature. There are two types of reservations, which may,for the sake of convenience, be referred to as ‘vertical reservations’and ‘horizontal reservations’. The reservation in favour of scheduledcastes, scheduled tribes and other backward classes [under Article 16(4)]may be called vertical reservations whereas reservations in favour ofphysically handicapped [under clause (1) of Article 16] can be referredto as horizontal reservations. Horizontal reservations cut across thevertical reservations -- what is called interlocking reservations. Tobe more precise, suppose 3% of the vacancies are reserved in favour ofphysically handicapped persons; this would be a reservation relatable toclause (1) of Article 16. The persons selected against his quota will beplaced in the appropriate category; if he belongs to SC category he willbe placed in that quota by making necessary adjustments; similarly, ifhe belongs to open competition (OC) category, he will be placed in thatcategory by making necessary adjustments.
It is further emphasized in the judgement that vertical reservations in favor of backwardclasses SC, ST, and OBC (which the judges refer to as reservations proper ) are “set aside”for these classes.
In this connection it is well to remember that the reservations underArticle 16(4) do not operate like a communal reservation. It may wellhappen that some members belonging to, say Scheduled Castes get selected inthe open competition field on the basis of their own merit; they will notbe counted against the quota reserved for Scheduled Castes; they will betreated as open competition candidates.
C.2. Rajesh Kumar Daria (2007): The Distinction Between Vertical Reservation andHorizontal Reservation.
The distinction between vertical reservations and horizontalreservations, i.e. the “over-and-above” aspect of the former and the “minimum guar-antee” aspect of the latter, is further elaborated in the Supreme Court judgement
RajeshKumar Daria (2007) . FFIRMATIVE ACTION IN INDIA 57
The second relates to the difference between the nature of verticalreservation and horizontal reservation. Social reservations in favour ofSC, ST and OBC under Article 16(4) are ’vertical reservations’. Specialreservations in favour of physically handicapped, women etc., underArticles 16(1) or 15(3) are ’horizontal reservations’. Where a verticalreservation is made in favour of a backward class under Article 16(4), thecandidates belonging to such backward class, may compete for non-reservedposts and if they are appointed to the non-reserved posts on their ownmerit, their numbers will not be counted against the quota reserved forthe respective backward class. Therefore, if the number of SC candidates,who by their own merit, get selected to open competition vacancies, equalsor even exceeds the percentage of posts reserved for SC candidates, itcannot be said the reservation quota for SCs has been filled. The entirereservation quota will be intact and available in addition to thoseselected under Open Competition category. [Vide - Indira Sawhney (Supra),R. K. Sabharwal vs. State of Punjab (1995 (2) SCC 745), Union of Indiavs. Virpal Singh Chauvan (1995 (6) SCC 684 and Ritesh R. Sah vs. Dr.Y. L. Yamul (1996 (3) SCC 253)]. But the aforesaid principle applicableto vertical (social) reservations will not apply to horizontal (special)reservations. Where a special reservation for women is provided within thesocial reservation for Scheduled Castes, the proper procedure is first tofill up the quota for scheduled castes in order of merit and then find outthe number of candidates among them who belong to the special reservationgroup of ’Scheduled Castes-Women’. If the number of women in such listis equal to or more than the number of special reservation quota, thenthere is no need for further selection towards the special reservationquota. Only if there is any shortfall, the requisite number of scheduledcaste women shall have to be taken by deleting the corresponding numberof candidates from the bottom of the list relating to Scheduled Castes.To this extent, horizontal (special) reservation differs from vertical(social) reservation. Thus women selected on merit within the verticalreservation quota will be counted against the horizontal reservation forwomen.
C.3. Anil Kumar Gupta (1995): Implementation of Horizontal Reservations Com-partmentalized within Vertical Reservations.
While horizontal reservations can be im-plemented either as overall horizontal reservations for the entire set of positions, or as compartment-wise horizontal reservations within each vertical category including the open category (OC), the Supreme Court recommended the latter in their judgement of
AnilKumar Gupta (1995) : We are of the opinion that in the interest of avoiding any complicationsand intractable problems, it would be better that in future the horizontalreservations are comparmentalised in the sense explained above. In otherwords, the notification inviting applications should itself state notonly the percentage of horizontal reservation(s) but should also specifythe number of seats reserved for them in each of the social reservationcategories, viz., S.T., S.C., O.B.C. and O.C.
The procedure to implement compartmentalized horizontal reservation is described in
Anil Kumar Gupta (1995) as follows:
The proper and correct course is to first fill up the O.C. quota (50%)on the basis of merit: then fill up each of the social reservationquotas, i.e., S.C., S.T. and B.C; the third step would be to find outhow many candidates belonging to special reservations have been selectedon the above basis. If the quota fixed for horizontal reservations isalready satisfied - in case it is an over-all horizontal reservation- no further question arises. But if it is not so satisfied, therequisite number of special reservation candidates shall have to be takenand adjusted/accommodated against their respective social reservationcategories by deleting the corresponding number of candidates therefrom.(If, however, it is a case of compartmentalised horizontal reservation,then the process of verification and adjustment/accommodation as statedabove should be applied separately to each of the vertical reservations.
The adjustment phase of the procedure for implementation of horizontal reservation isfurther elaborated in the Supreme Court judgement
Rajesh Kumar Daria (2007) as follows:
If 19 posts are reserved for SCs (of which the quota for women is four), 19SC candidates shall have to be first listed in accordance with merit, fromout of the successful eligible candidates. If such list of 19 candidatescontains four SC women candidates, then there is no need to disturb thelist by including any further SC women candidate. On the other hand, ifthe list of 19 SC candidates contains only two woman candidates, then thenext two SC woman candidates in accordance with merit, will have to beincluded in the list and corresponding number of candidates from the bottomof such list shall have to be deleted, so as to ensure that the final 19selected SC candidates contain four women SC candidates. [But if the listof 19 SC candidates contains more than four women candidates, selected onown merit, all of them will continue in the list and there is no question
FFIRMATIVE ACTION IN INDIA 59 of deleting the excess women candidate on the ground that ‘SC-women’ havebeen selected in excess of the prescribed internal quota of four.]
C.4. Tamannaben Ashokbhai Desai (2020): High Court Mandate on Adoption of theTwo-Step Minimum Guarantee Choice Rule in the State of Gujarat.
With its Au-gust 2020 High Court judgement
Tamannaben Ashokbhai Desai (2020) , the two-step min-imum guarantee choice rule (2SMG) is now mandated for allocation of state publicjobs in the State of Gujarat. While the choice rule is given in the judgement only fora single horizontal trait (women), it is also well-defined and well-behaved for multi-ple (but non-overlapping) traits as presented in Corollary 2. Originally introduced inS ¨onmez and Yenmez (2019) prior to the judgement of the High Court of Gujarat, in De-cember 2020 the choice rule is endorsed by the Supreme Court judgement Saurav Ya-dav (2020) for the entire country. Paragraph 56 of the High Court of Gujarat judgement
Tamannaben Ashokbhai Desai (2020) describes the mandated procedure as follows:
For the future guidance of the State Government, we would like to explainthe proper and correct method of implementing horizontal reservation forwomen in a more lucid manner.PROPER AND CORRECT METHOD OF IMPLEMENTING HORIZONTAL RESERVATION FOR WOMEN...Step 1: Draw up a list of at least 100 candidates (usually a list of morethan 100 candidates is prepared so that there is no shortfall of appointeeswhen some candidates don’t join after offer) qualified to be selected inthe order of merit. This list will contain the candidates belonging to allthe aforesaid categories.Step 2: From the aforesaid Step 1 List, draw up a list of the first 51candidates to fill up the OC quota (51) on the basis of merit. This listof 51 candidates may include the candidates belonging to SC, ST and SEBC.Step 3: Do a check for horizontal reservation in OC quota. In the Step2 List of OC category, if there are 17 women (category does not matter),women’s quota of 33% is fulfilled. Nothing more is to be done. If thereis a shortfall of women (say, only 10 women are available in the Step2 List of OC category), 7 more women have to be added. The way to dothis is to, first, delete the last 7 male candidates of the Step 2 List.Thereafter, go down the Step 1 List after item no. 51, and pick the first7 women (category does not matter). As soon as 7 such women from Step 1 The two-step minimum guarantee choice rule is referred to as C hor s in S ¨onmez and Yenmez (2019).0 S ¨ONMEZ AND YENMEZ