Constitutional Implementation of Affirmative Action Policies in India

In this paper, we turn to minimalist market design (Sönmez, 2023) to formulate and propose a mechanism for the general version of the problem, involving joint implementation of VR and HR policies, overlapping protected groups (to the extent possible in India),⁴⁴4In India, VR-protected groups do not overlap. However, this aspect of the Indian reservation system faced a challenge in 2019 when a VR-protected group was introduced based on economic deprivation. The Supreme Court addressed this issue, maintaining the non-overlapping nature of VR-protected groups with a 3-2 split verdict, as discussed in Sönmez and Ünver (2022). However, HR-protected groups not only overlap with VR-protected groups but also often overlap with other HR-protected groups. and heterogeneous positions.

Minimalist market design is an institution design framework that relies on the underlying objectives of an institution, namely, its “mission”, along with the operational details of the existing institution, to formulate a reformed institution whenever the existing one falls short of meeting certain aspects of its mission. Under this approach, the root causes of the existing institution’s failures are identified, and a new institution that aligns with the mission is designed by addressing only the flawed aspects of the existing system. This is the essence of the term “minimalist” in this institution design framework.

Naturally, this paradigm is most effective in environments where the mission of the institution is clear and the root causes of institution failures can be identified. This renders our application an ideal candidate for adopting minimalist market design.

Making minimalist market design especially promising to address the failures later presented in Section 1.3, the objectives of the institution—specifically, the principles governing the joint operation of VR and HR policies—are already formulated in Saurav Yadav (2020). While the focus of this judgment pertains to scenarios with homogeneous positions, it effectively clarifies ambiguities introduced in Indra Sawhney (1992) concerning the joint implementation of VR and HR policies across all problem variations.⁵⁵5See, for example, paragraph 31 in Saurav Yadav (2020) in Appendix A.6. Our task now lies in formalizing the principles outlined in Saurav Yadav (2020)—as presented in the judgment—into formal axioms. Expanding upon their formulation in Sönmez and Yenmez (2022) for the case of homogeneous positions, in Section 2.5, we articulate these principles as formal axioms applicable to the broader context with heterogeneous positions.

One of the fundamental principles guiding India’s implementation of its reservation policy is the principle of inter se merit, which dictates that allocation should be based on merit within the same category, with exceptions outlined by HR policy.⁶⁶6The effect of HR policy on this principle became clear in India with the Saurav Yadav (2020) case. This principle, pivotal in numerous Supreme Court rulings, often sparks legal disputes in India (see, for instance, Sections A.1, A.2 and A.3 in the Online Appendix). In this section, we identify the root causes of these challenges, focusing solely on VR policy as the core issue remains independent of HR policy.

The VR policy, designed as the primary affirmative action provision in the Constitution to rectify years of caste-based discrimination, aims to uplift historically disadvantaged groups. Since no individual belongs to more than one caste (i.e., the VR-protected groups cannot overlap), the stand-alone implementation of the VR policy is straightforward with homogeneous positions: Open-category positions are allocated based on merit first, and reserved positions are allocated next to the remaining members of each VR-protected category based on inter se merit. Under this benchmark procedure, called the over-and-above choice rule, positions within each category are processed one at a time, starting with the open category.

In India, legal terminology fails to distinguish between categories of individuals and categories of positions. Although individuals not belonging to any VR-protected category are referred to as general-category candidates, they are often also called open-category candidates. Since open-category positions are not exclusively reserved for individuals in the general category, unlike positions in VR-protected categories that are exclusively reserved for their members, this terminology can lead to confusion. To help avoid any resulting confusion, the legal concept of migration emerged, signifying that when a VR-protected individual secures an open-category position based on merit, they are considered to have “migrated” to the open or general category. With this convention, open-category positions are awarded to members of the “revised” open category.

Parallel to the case of homogeneous positions, traditions of allocating open-category positions before the VR-protected ones and utilizing the concept of migration persist in the allocation of heterogeneous positions. In a typical mechanism, open-category positions across all institutions are tentatively allocated via a simple serial dictatorship (SSD). The highest-merit individual is assigned their top choice, the second-highest-merit individual is assigned their highest choice among the remaining open-category positions, and so on. Importantly, VR-protected individuals who tentatively receive an open position are granted a status called meritorious reserved candidate (MRC), one that plays a key role in several important judgments.

Once open-category positions are provisionally managed, VR-protected positions are also tentatively allocated to eligible individuals in a similar way, again using SSD in a second phase. Since an MRC can receive two distinct tentative assignments in the two phases, based on inter se merit, they migrate to the institution-category pair associated with their more preferred position. Subsequently, positions vacated by MRCs must be reallocated while still adhering to principle of merit for open-category positions and the principle of inter se merit for VR-protected positions.

Enforcing directives based on the mechanics of the above-described (and misguided) class of SSD-based mechanisms rather than the underlying principles, several key Supreme Court judgments made fundamental errors at this juncture. For example, in Union of India vs Ramesh Ram (2010),⁷⁷7The ruling can be accessed at https://indiankanoon.org/doc/1368252/, retrieved on 02/25/2024. positions vacated by MRCs were exclusively awarded to general category candidates for government jobs, while in Tripurari Sharan vs Ranjit Kumar Yadav (2018),⁸⁸8The ruling can be accessed at https://indiankanoon.org/doc/102870864/, retrieved on 02/25/2024. they were exclusively awarded to VR-protected individuals from the category of vacating MRCs. Notably, the creation of such “artificial” property rights for a specific category of individuals directly contradicts the principle of inter se merit, as individuals deserving the vacated positions can be from any category, either unmatched or tentatively holding less-preferred positions.

Thus, the root causes of the legal challenges lies in the excessive reliance on the concept of migration, further exacerbated by the practice of awarding “artificial” property rights to essentially arbitrary groups based on tentative assignments of MRCs, which itself is an ill-equipped tool for addressing applications with heterogeneous positions. This methodological failure lies at the core of the legal inconsistencies and litigations in India.⁹⁹9A simple search via Indian Kanoon, a free search engine for Indian Law, reveals that as of February 2024, there have been 869 cases related to the “migration of meritorious reserved candidates” at the Supreme Court or the high courts.

As we address these challenges, it is important to observe that it is not possible to accommodate the principles outlined in Indra Sawhney (1992) or their refinements in Saurav Yadav (2020) solely by relying on the concept of migration, unless an arbitrary number of rounds of migrations are allowed through an iterative procedure such as the individual-proposing deferred acceptance (DA) algorithm (Gale and Shapley, 1962). The reason is analogous to the necessity of iterative procedures to find stable matchings in two-sided matching markets. This observation is at the heart of our proposed resolution. Before we formulate and propose a solution in Section 1.5 utilizing the DA algorithm, we first characterize the set of outcomes that satisfy the mandates of Saurav Yadav (2020).

For scenarios with homogeneous positions, the mandates outlined in Saurav Yadav (2020) are uniquely satisfied by the 2SMH choice rule (Sönmez and Yenmez, 2022). This rule not only determines which candidates are awarded positions but also specifies the category from which they receive them. This additional structure enables us to introduce a refinement of the traditional stability axiom for our setting.

For each individual, an assignment specifies whether they receive a position and the following specifics of this position: Which institution offers the position and from which category. Given a strict preference ranking for each individual over all institutions and remaining unmatched, and a (multi-category) choice rule for each institution, an assignment is stable if:

1. (1)

   No individual prefers their institution assignment to remaining unmatched.

2. (2)

   No institution would rather remove an individual from its admitted group or change the category of the position they are offered.

3. (3)

   There exists no individual-institution pair where

   1. (a)

      the individual prefers the institution to their assignment, and

   2. (b)

      the institution would rather offer the individual a position when it is allowed to maintain some or all of the existing individuals from its admitted group.

There is only one subtle difference between our stability notion and the traditional stability notion in standard two-sided matching models, such as the matching with contracts (Hatfield and Milgrom, 2005): In the second condition, in addition to maintaining its admitted group of individuals, an institution is also expected to retain their assigned categories. Since our notion of a choice rule is multidimensional, also specifying categories, it is only natural that an extension of the standard stability to our setting takes this additional specification into consideration. Thus, we believe our notion is a natural refinement of the traditional stability notion for our more structured environment.

This notion of stability is critical for our analysis: In Theorem 1, we show that an assignment satisfies the mandates of Saurav Yadav (2020) if and only if it is stable when each institution is endowed with the 2SMH choice rule. Thus, the path to adhering to the Supreme Court’s mandates goes through a natural generalization of a well-established methodology in matching theory.

Unlike in the case of homogeneous positions, as shown by Theorem 1, the Supreme Court’s mandates do not prescribe a unique mechanism when positions are heterogeneous. Fortunately, earlier literature in two-sided matching and school choice guides us in a promising direction to formulate and propose a unique mechanism for this more general version of the problem.

Our proposed mechanism, 2SMH-DA, is based on extending the 2SMH choice rule to the general version of the problem with heterogeneous positions across multiple institutions through a joint implementation of the individual-proposing DA algorithm, where each institution is endowed with the 2SMH choice rule.

It is well-established that, in the standard school choice setting without categories (Abdulkadiroğlu and Sönmez, 2003), there exists a stable outcome that Pareto dominates any other stable outcome (Gale and Shapley, 1962; Balinski and Sönmez, 1999). Moreover, this outcome can be obtained with the individual-proposing DA algorithm (Balinski and Sönmez, 1999). Finally, not only does the resulting mechanism satisfy strategy-proofness (Dubins and Freedman, 1981; Roth, 1982), it is the only one that also satisfies stability (Alcalde and Barberà, 1994; Balinski and Sönmez, 1999). Under some technical conditions on (single-category) choice rules, these results are very robust across a wide range of settings.

Our multi-category setting is not an exception. In Theorem 2, we show that the 2SMH-DA mechanism Pareto dominates any other mechanism that satisfies the Supreme Court’s mandates in Saurav Yadav (2020), and in Theorem 3, we show that it is the only strategy-proof mechanism that satisfies these mandates.

Therefore, either one of the two fundamental principles in economic theory directly implies the 2SMH-DA mechanism when combined with Saurav Yadav (2020) refinement of the principles in Indra Sawhney (1992). Hence, we argue that 2SMH-DA is the only natural mechanism to address numerous legal challenges faced by public institutions in India due to their flawed allocation mechanisms.

In Section 2, we present the model, including the formulation of the directives of Saurav Yadav (2020) regarding the joint implementation of VR and HR policies. Section 3 summarizes the key Supreme Court judgments on VR and HR policies, identifying several inconsistencies within and between these cases. This section also identifies the root causes of these failures. In Section 4, we formulate a variant of the celebrated stability axiom from the two-sided matching literature to our more structured setting. We show that, collectively, the mandates of the Supreme Court are equivalent to stability under the 2SMH choice rule by Sönmez and Yenmez (2022). Section 5 presents our proposed mechanism 2SMH-DA and shows that, among all mechanisms that satisfy the Supreme Court’s mandates, it (i) Pareto dominates any other and (ii) is the only one that is strategy-proof. In Section 6, we discuss additional real-life applications of our model, with a particular focus on school choice in Chile. We conclude in Section 7. Legal analysis of several pivotal Supreme Court cases and all proofs of our formal results are provided in an Online Appendix.

There exist a set $\mathcal{R}$ of VR-protected (or reserved) categories and a general category $g\not\in\mathcal{R}$. Each individual belongs to a single category in $\mathcal{R}\mathbin{\scalebox{1.2}{$\cup$}}\{g\}$. Individual memberships to VR-protected categories is given by a function $\rho:\mathcal{I}\rightarrow\mathcal{R}\mathbin{\scalebox{1.2}{$\cup$}}\{\emptyset\}$. Here, $\rho(i)=\emptyset$ indicates that individual $i$ is ineligible for VR protections, and thus she is a member of the general category $g$.¹¹¹¹11To keep the notation at a minimum, we assume that (i) the set of VR-protected categories $\mathcal{R}$, (ii) the general category $g$, and (iii) the category-membership function $\rho$ are all independent of a job. This assumption is without any loss of generality and all these primitives can be made job-dependent by a simple inclusion of a job index without interfering with any aspect of our analysis.

At any given job $j\in\mathcal{J}$, there are $r^{c}_{j}\geq 0$ positions set aside exclusively for the members of category $c\in\mathcal{R}$. We refer to these positions as category-c positions or (VR-protected positions for category c). We assume that $\sum_{c\in\mathcal{R}}r^{c}_{j}\leq q_{j}$. In contrast, members of the general category do not have any positions set aside for them under the VR policy. Therefore, $r^{o}_{j}=q_{j}-\sum_{c\in\mathcal{R}}r^{c}_{j}$ positions are open for all individuals. We refer to these positions as open-category positions (or category-o positions). Let $\mathcal{V}=\mathcal{R}\mathbin{\scalebox{1.2}{$\cup$}}\{o\}$ denote the set of vertical categories for positions.

Given a VR-protected category $c\in\mathcal{R}$, an individual $i\in\mathcal{I}$ is considered eligible for positions in category $c$ if $\rho(i)=c$. Each individual $i\in\mathcal{I}$ is eligible for open-category positions. Given a category $v\in\mathcal{V}$, let $\mathcal{I}^{v}\subseteq\mathcal{I}$ denote the set of individuals who are eligible for category-$v$ positions.

The defining characteristic of the VR protections is stated as follows in the landmark Supreme Court judgment Indra Sawhney (1992):

It may well happen that some members belonging to, say Scheduled Castes get selected in the open competition field on the basis of their own merit; they will not be counted against the quota reserved for Scheduled Castes; they will be treated as open competition candidates.

When there is a single job and the VR policy is the only affirmative action policy, the interpretation of this statement becomes airtight: If a VR-protected individual deserves an open position on the basis of her merit score only, she should be awarded an open position and not use up a VR-protected position set aside for her category. In this sense, VR protections are implemented on an “over-and-above” basis, a feature which makes this policy the “higher level” affirmative action policy. Unfortunately, Indra Sawhney (1992) formulation of the VR policy given above loses its clarity when it is implemented jointly with the HR policy which is formally introduced next in Section 2.2.

In addition to the categories in $\mathcal{R}$ that are associated with the primary VR protections, there is a finite set $\mathcal{T}$ of traits associated with the secondary HR protections. Each individual has a (possibly empty) subset of traits, given by the function $\tau:\mathcal{I}\rightarrow 2^{\mathcal{T}}$.

HR protections are provided in the form of “minimum guarantees” within each vertical category $v\in\mathcal{V}$.¹²¹²12Provision of HR protections within each vertical category is not a federal mandate in India but rather a formal recommendation by the Supreme Court judgment Anil Kumar Gupta (1995). The vast majority of the institutions in India follows this recommendation and implement the HR policy in this form, which is also referred to as interlocking reservations or compartmentalized horizontal reservations. For any job $j\in\mathcal{J}$, VR-protected category $c\in\mathcal{R}$, and trait $t\in\mathcal{T}$, subject to the availability of qualified individuals, a minimum of $r^{c,t}_{j}$ category-$c$ positions are to be assigned to individuals from category $c$ with trait $t$. If there are not enough individuals from category $c$ with trait $t$ to fill these positions, then the remaining empty seats are to be allocated to other individuals from category $c$. We refer to these positions as category-c HR-protected positions for trait t. Similarly, for any trait $t\in\mathcal{T}$ and subject to the availability of individuals with trait $t$, a minimum of $r^{o,t}_{j}$ open-category positions are to be assigned to individuals with trait $t$. If there are not enough individuals with trait $t$ to fill these positions, then the remaining empty seats are to be allocated to other individuals. We refer to these positions as open-category HR-protected positions for trait t.

For each job $j\in\mathcal{J}$ and vertical category $v\in\mathcal{V}$, we assume that the total number of category-$v$ HR-protected positions is no more than the number of positions in category $v$. That is, for each job $j\in\mathcal{J}$ and category $v\in\mathcal{V}$, we assume $\sum_{t\in\mathcal{T}}r^{v,t}_{j}\leq r^{v}_{j}.$

As we already indicated, in contrast to VR protections, which are provided on an “over-and-above” basis, HR protections are provided within each vertical category on a “minimum guarantee” basis. This means that positions obtained without invoking any HR protection still accommodate the HR protections.

In India, each position is classified by its job, vertical category (including the open category), and the associated trait (or its absence). Therefore, to describe an outcome, it may be compelling to assign individuals to a triple consisting of a job, a vertical category, and a trait or its absence. However, we will take a different approach for the reasons we elaborate below.

An outcome needs to indicate the job assignments of individuals because they have strict preferences over jobs. While the category assignment is not important for individual preferences, it is crucial for the implementation of the VR policy. This is because the laws clearly specify who should receive the open positions and who should receive the VR-protected positions. Therefore, at a minimum, an outcome needs to specify the job assignment and the category assignment of each individual who receives a position.

The specification of a trait assignment (or its absence), on the other hand, offers some flexibility in terms of modeling an outcome. While an outcome can explicitly specify the trait (or its absence) for a position received, this modeling choice results in immaterial multiplicities under our axioms, which formulate affirmative action legislation in India. For example, if there is a minimum guarantee of two positions for women in the open category of a given job, and five women receive open-category positions, under Indian laws, there is no meaningful way to specify which two of these five women receive the HR-protected positions. Therefore, to avoid any arbitrary conditions that fail to capture Indian legislation, in our model, an outcome simply assigns individuals to job-vertical category pairs or leaves them unassigned.

An implicit trait assignment will still be important to verify that the HR protections are honored to the extent possible, and it will be captured in our model through a secondary assignment introduced in Section 2.4.¹³¹³13This modeling choice allows us to relegate any immaterial multiplicities to a secondary assignment within the primary assignment.

An assignment specifies the job offering the position received by each individual, if any, and the vertical category through which it is obtained.

Given an assignment $\alpha\in\mathcal{A}$, let

denote the set of individuals who receive a position at job $j$.

Since individual preferences are defined over $\mathcal{J}\mathbin{\scalebox{1.2}{$\cup$}}\{\emptyset\}$, rather than over $\big{(}\mathcal{J}\times\mathcal{V}\big{)}\mathbin{\scalebox{1.2}{$\cup$}}\{\emptyset\}$, we trivially extend the domain $\mathcal{P}$ of the preferences to $\big{(}\mathcal{J}\times\mathcal{V}\big{)}\mathbin{\scalebox{1.2}{$\cup$}}\mathcal{J}\mathbin{\scalebox{1.2}{$\cup$}}\{\emptyset\}$.

For any preference profile $\succ_{\mathcal{I}}\in\mathcal{P}$, individual $i\in\mathcal{I}$, job $j\in\mathcal{J}$, and category $v\in\mathcal{V}$, we have

or equivalently

Therefore, an individual has preferences over jobs only and is indifferent between categories of the same job.

Given a mechanism $\varphi$ and a profile $\succ_{\mathcal{I}}$ of preferences, the assignment for individual $i\in\mathcal{I}$ is denoted by $\varphi(\mathcal{\succ_{\mathcal{I}}})(i)$. Likewise, the set of individuals assigned to job-category pair $(j,v)\in\mathcal{J}\times\mathcal{V}$ is denoted by $\varphi^{-1}(\succ_{\mathcal{I}})(j,v)$, and the set of individuals assigned to job $j$ is denoted by $\varphi^{-1}(\succ_{\mathcal{I}})(j)$.

In order to manage the secondary HR policy, in this section, we discuss a technical tool called trait-matching (Sönmez and Yenmez, 2022). Given an assignment $\alpha\in\mathcal{A}$, a trait-matching can be thought of as a secondary assignment of HR-protected individuals to traits within any given pair $(j,v)\in\mathcal{J}\times\mathcal{V}$, and its size provides us with a natural metric to assess the extent to which the HR protections are honored at pair $(j,v)$.

Fix an assignment $\alpha\in\mathcal{A}$, a job $j\in\mathcal{J}$, and a category $v\in\mathcal{V}$. Let $I\subseteq\mathcal{I}$ be the set of individuals who receive category-$v$ positions at job $j$ under assignment $\alpha$. Hence, $I=\alpha^{-1}(j,v)$.

First, consider a simpler version of the problem where each individual has at most one trait. For any trait $t\in\mathcal{T}$, the set of individuals in $I$ who have trait $t$ is given by the set $\{i\in I:t\in\tau(i)\}$. Therefore, within category $v$ of job $j$, trait-$t$ HR protections are fully honored if $\left|\{i\in I:t\in\tau(i)\}\right|\geq r^{v,t}_{j}$, whereas $(r^{v,t}_{j}-\left|\{i\in I:t\in\tau(i)\}\right|)$ of them are left dishonored otherwise.

For the latter case, an individual $i\in\mathcal{I}^{v}\setminus I$ can object to the allocation of category-$v$ positions at job $j$ under assignment $\alpha$, provided that she has trait $t$ and desires to receive a position at job $j$. Also observe that maximum number of HR-protected positions that can be honored within category $v$ at job $j$ by the set of individuals $I$ is given by

Hence, any individual $i\in\mathcal{I}^{v}\setminus I$ can object to the allocation of category-$v$ positions at job $j$ under assignment $\alpha$, provided that $n_{j}^{v}\big{(}I\mathbin{\scalebox{1.2}{$\cup$}}\{i\}\big{)}>n_{j}^{v}(I)$ and they desire to receive a position at job $j$. This observation plays a key role in several of our formal axioms, later introduced in Section 2.5.

The same idea can also be extended to the more general version of the problem when individuals can have multiple traits. However, the secondary assignment of individuals to traits requires additional care in this case.

For example, suppose there is a single HR-protected position for women and another single HR-protected position for persons with disabilities. In such a scenario, a disabled woman can receive positive discrimination for either of the two HR-protected positions. However, if the only other individual possessing either of the two traits is a disabled man, it would be implausible to award the HR-protected position for persons with disabilities to the disabled woman and consequently deny an HR-protected position to the disabled man. Both HR-protected positions can be honored by assigning the HR-protected position for women to the disabled woman and the HR-protected position for persons with disabilities to the disabled man. We will now build on this simple observation to extend the above-given function $n_{j}^{v}$ to the general version of the problem.

Fix a job $j\in\mathcal{J}$, a category $v\in\mathcal{V}$, and a set of individuals $I\subseteq\mathcal{I}^{v}$. Let $H_{j}^{v,t}$ denote the set of HR-protected positions for trait-$t$ within category $v$ at job $j$, and $H^{v}_{j}=\bigcup_{t\in\mathcal{T}}H_{j}^{v,t}$ denote the set of all HR-protected positions within category $v$ at job $j$. Construct the following bipartite HR graph: Individuals in $I$ are on one side of the graph and positions in $H^{v}_{j}$ are on the other side. For any trait $t\in\mathcal{T}$, an individual $i\in I$ and a position $p\in H_{j}^{v,t}$ are connected in this graph if and only if individual $i$ has trait $t$.

Let $n^{v}_{j}(I)$ denote the maximum number of job-$j$ category-$v$ HR-protected positions in $H^{v}_{j}$ that can be assigned to individuals in $I$.¹⁴¹⁴14This number can be found in polynomial time by the famous Hungarian maximum matching algorithm, which is originally published in Kuhn (1955) and based on the earlier work of the Hungarian mathematicians Dénes König and Jenö Egerváry. For any job $j\in\mathcal{J}$ and category $v\in\mathcal{V}$, this number identifies how many of its HR-protected positions are honored when category-$v$ positions of job $j$ are awarded to individuals in $I$. As such, it serves as a key summary statistic on compliance with the HR policy,¹⁵¹⁵15This observation is the main reason why it it not necessary to explicitly include a trait matching in our formulation of an assignment. reflected in three of our formal axioms introduced next in Section 2.5.

For illustration, consider the case where each individual has at most one trait. In addition to singleton nodes associated with individuals who have no trait, the rest of the HR graph for this case consists of $\left|\mathcal{T}\right|$ disjoint components, one for each trait. Moreover, each component is a complete bipartite graph. That is, each node on one side of any component is connected to each node on the other side. Thus, given a job $j\in\mathcal{J}$ and a category $v\in\mathcal{V}$, for any given component associated with trait $t\in\mathcal{T}$, the size of the maximum cardinality trait-matching is equal to the size of the smaller of the two sides of the component; i.e., $\min\Big{\{}\big{|}{i\in I:t\in\tau(i)}\big{|},r^{v,t}_{j}\Big{\}}$. Therefore, just as we found before, the maximum number of job-$j$ category-$v$ HR-protected positions that can be honored can be derived as

In this section, we introduce our primary axioms on assignments and mechanisms. Our main objective in formulating these axioms is giving a mathematically precise meaning to the mandates of the Supreme Court of India on VR and HR policies. We will organize our axioms into two groups.

1. (1)

   In the first group, we have two axioms that are so benign that, although they were implicitly intended, they are not explicitly discussed in the court rulings.¹⁶¹⁶16We are unaware of any practical mechanism in India where either of these two axioms fails or any court ruling that pertains to these axioms.

2. (2)

   In the second group, we have three core axioms that formulate the Supreme Court’s explicit mandates on the concurrent implementation of VR and HR policies. As far as we can tell, the inability to design mechanisms that satisfy these axioms, and the legislative confusion on their formulation are the primary reasons for the challenges in India. The formulation of these three axioms in Saurav Yadav (2020) finally clears the second of these reasons; i.e., the legal confusion on their formulation.

With a slight abuse of terminology, we refer to the five axioms in these two groups as Saurav Yadav (2020) axioms.

Our first axiom states that no position should be awarded to an individual who has no desire to receive this position.

Our second axiom states that, a position can be left idle only if no individual who desires to receive it is eligible for the position.

Our third axiom formulates the positive discrimination given to HR-protected individuals. It states that an individual who remains unassigned cannot be denied a position at any job-category pair $(j,v)\in\mathcal{J}\times\mathcal{V}$, if her recruitment increases the number of HR-protected positions honored at pair $(j,v)$.

Our fourth axiom formulates the following equity principle given in the paragraph 31 of the Supreme Court judgment Saurav Yadav (2020):

[…] Subject to any permissible reservations i.e. either Social (Vertical) or Special (Horizontal), opportunities to public employment and selection of candidates must purely be based on merit.

Any selection which results in candidates getting selected against Open/General category with less merit than the other available candidates will certainly be opposed to principles of equality. There can be special dispensation when it comes to candidates being considered against seats or quota meant for reserved categories and in theory it is possible that a more meritorious candidate coming from Open/General category may not get selected. But the converse can never be true and will be opposed to the very basic principles which have all the while been accepted by this Court. Any view or process of interpretation which will lead to incongruity as highlighted earlier, must be rejected.

That is, if an individual $i$ receives a category-$v$ position at job $j$ while another individual $i^{\prime}$ receives a less-desired assignment, it is either because individual $i$ has a higher merit score under $\sigma_{j}$ than individual $i^{\prime}$ or because replacing individual $i$ with individual $i^{\prime}$ decreases the number of HR-protected positions that are honored.

When applied to open-category positions where every individual is eligible, the axiom of no justified envy transforms into a condition known as the principle of merit in India. Similarly, when applied to positions in a VR-protected category $c\in\mathcal{R}$ where only category-$c$ individuals are eligible, the axiom of no justified envy transforms into a condition known as the principle of inter se merit. These principles were originally established in the landmark Supreme Court judgment Indra Sawhney (1992). However, the precise role of HR protections for open positions under the principle of merit only became clear with the recent judgment Saurav Yadav (2020).¹⁷¹⁷17In particular, the role of individuals who are both HR and VR protected for the open positions were unclear prior to Saurav Yadav (2020). See, for example, the following quote from the 01/25/2021 The Leaflet article “Supreme Court strikes down policy of excluding the reserved community from competing for general and open category.” “Until now, the specific question of whether female candidates belonging to any of the vertically reserved categories can be selected on ‘merit’ against the vacancies horizontally reserved for general/open category was a res integra before the Supreme Court.” The story is available in https://tinyurl.com/z6y7wwfn, last accessed on 02/09/2022. The failure of no justified envy is one of the primary reasons for litigations in India regarding the implementation of VR and HR policies.

Our fifth axiom formulates the defining characteristic of VR protections as the “higher-level” reservation policy. It states that VR-protected positions shall not be awarded to individuals who “deserve an open-category position based on merit.” Instead, they should be left for VR-protected individuals who are truly in need of positive discrimination. Critically, with the clarification in Saurav Yadav (2020), an individual may deserve an open-category position not only due to their merit score but also because they may increase the number of honored open-category HR protected positions.¹⁸¹⁸18For the exact statement of this clarification, refer to Paragraph 36 from Saurav Yadav (2020), provided in Section A.6 of the Online Appendix.

For an individual $i$ to receive a VR-protected position at job $j$, a prerequisite is that they do not merit an open-category position at job $j$. That means:

1. (1)

   There is no idle open-category position left at job $j$.

2. (2)

   Each individual $i^{\prime}$ who received an open-category position at job $j$ either has a higher merit score than $i$ or admitting $i^{\prime}$ rather than $i$ increases the number of honored open-category HR protected positions at job $j$.

3. (3)

   Replacing individual $i$ with any recipient of an open-category position at job $j$ does not increase the number of honored open-category HR protected positions at job $j$.

If any of the three conditions fail, with the clarification in Saurav Yadav (2020) that formulates how “open-category merit” accounts for HR protections, individual $i$ would deserve an open-category position at job $j$ based on merit.

In Sections A.2 and A.3 in the Online Appendix, we argue that not only do the allocation mechanisms employed by various Indian institutions have important shortcomings, but also the Supreme Court judgments on these mechanisms have a number of inconsistencies. In this section, we argue that the root cause of these difficulties lies in the excessive reliance on the concept of migration to solve more complex versions of the problem, further exacerbated by the introduction of the status of an MRC as an especially ill-equipped tool to facilitate a solution for applications with heterogeneous positions.

Since open positions are allocated prior to VR-protected positions when positions are homogeneous, decision-makers may be tempted to adopt a similar approach and allocate them via SSD in heterogeneous scenarios. However, unlike in the homogeneous case, an open-category position secured by a VR-protected individual may not be preferred over a reserved position they can obtain within their VR-protected category. Consequently, this widespread practice has led to the establishment of the MRC status in affirmative action jurisprudence and the formulation of the two questions posed in Section 3.1.3 to manage their assignments. It is critical to note that these questions focus not on the fundamentals of the problem but rather on the mechanics of a specific class of mechanisms.

The root cause of the challenges faced by MRC-based mechanisms can be traced to the following observation: once an MRC vacates an open position to secure a more-preferred position reserved for their VR-protected category, the next deserving candidate can be:

1. (1)

   a member of the general category who either holds a less-preferred open position from earlier phases or remains unassigned,

2. (2)

   another MRC who holds a less-preferred position from earlier phases, or

3. (3)

   another member of a reserved category who remains unassigned from earlier phases.

Thus, the widespread practice of tentatively allocating open positions in the first phase results in the creation of an artificial interim allocation, one that is often given undue weight despite being a technical construct. This, in turn, leads to the exclusive awarding of the “property rights” of a vacated open position to members of a specific category, creating an open invitation for litigation. This misguided and artificial construction of property rights is the primary source of dispute in the vast majority of legal conflicts involving MRCs.

Indeed, the judges of the Central Administrative Tribunal, Chennai Bench (CAT-CB), correctly identified this root cause of challenges in a lower court decision preceding the Supreme Court judgment in Ramesh Ram (2010). The judges of the CAT-CB included the following statement in their ruling:

In doing so, the respondents also would notice that the steps taken by them in accordance with the Rules 16 (3)(-)(5) are redundant once they issue the result of recruitment in one phase, instead of two as they have become primary cause for the litigation and avoidable confusion in the minds of the candidates seeking recruitment.

Therefore, the judges directed the Union of India to announce their outcome in one phase in a manner that respects inter se merit, without relying on the artificial concept of migration.²²²²22See Online Appendix A.5 for a comprehensive quote from this case. However, despite being accurate, this ruling was ignored by the Union of India, and the case moved all the way to the Supreme Court. One possible explanation for the Union of India’s refusal to follow the decision of CAT-CB may be their technical inability to construct a mechanism that complies with the court’s order.

With the subsequent clarification on the joint implementation of VR and HR policies provided by Saurav Yadav (2020), in Section 5, we will formulate a particularly compelling mechanism that complies with the tribunal’s orders.

Having identified the root cause of the failures in Supreme Court judgments on the allocation of heterogeneous positions in Section 3.2, we turn to minimalist market design (Sönmez, 2023) as a framework to address them.

Think of an institution like a machine with many parts or a body with many organs, each meant to perform specific jobs. Sometimes, these parts break or fail to work well together, causing the machine or body to fail in its tasks. To address these issues, a design economist can learn from how experts in other fields, such as medicine, handle similar problems. For instance, when a doctor encounters a problem in the body, they identify the cause, such as removing diseased tissue or organs, fixing different body systems, or performing organ transplants. A design economist operates analogously within minimalist market design, which involves identifying the root causes of failure and making minimal changes to rectify them. In simpler terms, the paradigm works like a “minimally invasive” procedure performed by a physician.

Under this paradigm, three main tasks emerge:

1. (1)

   Identify the institution’s mission: What are the primary objectives of policymakers, system operators, and other stakeholders?

2. (2)

   Determine whether the existing institution satisfies these primary objectives: If it doesn’t, there’s potential for policy impact with a compelling alternative design. To realize this potential, the root causes of the failures should be identified.

3. (3)

   Address the failures of the deficient institution by interfering only with its flawed features.

As discussed in Section 3.1.1, the primary objectives of VR policy are rigorously and fully described in Indra Sawhney (1992) for the basic version of the problem in the absence of HR protections. Moreover, the primary root cause of the failures in subsequent legislation and real-life institutions for the case of heterogeneous positions is identified in Section 3.2. One possibility would be to completely ignore HR policy and deploy minimalist market design to address the challenges identified in MRC-based mechanisms (e.g., those in Sections A.2, A.3 and A.4 of the Online Appendix), using the root cause of the failures of these mechanisms identified in Section 3.2 and the objectives mandated in Indra Sawhney (1992).

While this may be plausible for purely academic purposes, it generally will not produce mechanisms that could be implemented in India because virtually all current applications involve HR protections. This is because the Union Of India vs National Federation Of the Blind (2013) case grants HR protections to persons with disabilities throughout India.²³²³23The ruling is available at https://indiankanoon.org/doc/178530295/, retrieved on 02/26/2024. Additionally, other groups, notably women, are granted HR protections in many states through high court decisions. Examples include Bihar and Madhya Pradesh with 35% of positions each, Andhra Pradesh, Gujarat and Telangana with 33% each, and Madhya Pradesh, Uttarakhand, Chhattisgarh, Rajasthan, and Sikkim with 30% each.

Fortunately, as we already discussed in Section 2.5, the primary objectives of VR and HR policies and how they must be integrated are rigorously described in Saurav Yadav (2020). Thus, in Sections 4 and 5, we will deploy minimalist market design for the most general version of the problem with VR and HR policies and heterogeneous positions.

To establish this link, we first formulate three related solution concepts for a single job, as introduced in Sönmez and Yenmez (2022).

Assuming individual rationality and that a mechanism only relies on preferences over acceptable positions, the set of applicants contains all the necessary information given in a preference profile when all positions are identical. That is because, individuals who prefer remaining unmatched over receiving a position can be ignored, and the remaining individuals all have the same preference relation. Therefore, the domain of single-job solution concepts can be given as $2^{\mathcal{I}}$ rather than $\mathcal{P}$.

For any set of applicants, a single-category choice rule specifies which applicants are chosen by a given job-category pair.²⁴²⁴24The inverse mapping of a single-category choice rule is a single-job and single-category mechanism.

For any set of applicants, multi-category choice rule specifies which applicants are chosen for each of its categories by a given job.

For any set of individuals, the aggregate choice rule yields the set of chosen individuals across all categories.

The following single-category choice rule is introduced in Sönmez and Yenmez (2022).

Consider a job $j\in\mathcal{J}$ and a category $v\in\mathcal{V}$. Let $I\subseteq\mathcal{I}^{v}$ be a set of individuals who are eligible for category $v$.

Meritorious Horizontal Choice Rule $C^{v}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}$

Step 1.1 Assuming such an individual exists, let $i_{1}$ be the the highest merit-score individual (with respect to $\sigma_{j}$) in $I$ who has a trait for an HR-protected position. Choose individual $i_{1}$ for an HR-protected position. Let $I_{1}=\{i_{1}\}$, and proceed with Step 1.2. If no such individual exists, proceed to Step 2.

Step 1.k ($k\in\{2,\ldots,\sum_{t\in\mathcal{T}}r^{v,t}_{j}\}$) Assuming such an individual exists, let $i_{k}$ be the the highest merit-score individual in $I\setminus I_{k-1}$ with

$$n^{v}_{j}(I_{k-1}\mathbin{\scalebox{1.2}{$\cup$}}\{i_{k}\})=n^{v}_{j}(I_{k-1})+1.$$

Choose individual $i_{k}$ for an HR-protected position. Let $I_{k}=I_{k-1}\mathbin{\scalebox{1.2}{$\cup$}}\{i_{k}\}$, and proceed with Step 1.(k+1). If no such individual exists, proceed to Step 2.

Step 2. For unfilled positions, choose unassigned individuals with highest merit scores until either all positions are filled or all individuals are selected.

The following multi-category choice rule uses the meritorious horizontal single-category choice rule multiple times; first, to allocate open-category positions, and next for each VR-protected category to allocate VR-protected positions.

2-Step Meritorious Horizontal (2SMH) Choice Rule $\vec{C}^{2s}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}=(C^{2s,v}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j})_{v\in\mathcal{V}}$

For each set of individuals $I\subseteq\mathcal{I}$,

	$\displaystyle C^{2s,o}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}(I)=C^{o}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}(I),\mbox{ and}$
	$\displaystyle C^{2s,c}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}(I)=C^{c}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}\Big{(}\big{(}I\setminus C^{o}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}(I)\big{)}\mathbin{\scalebox{1.2}{$\cap$}}{\mathcal{I}}^{c}\Big{)}\quad\mbox{ for any }c\in\mathcal{R}.$

The following axiom is a generalization of the celebrated stability axiom (Gale and Shapley, 1962; Hatfield and Milgrom, 2005) for two-sided matching markets.

Note that, while in the same spirit, our concept of stability diverges from its conventional definition in the two-sided matching literature due to the additional specification of position categories in assignments. In traditional two-sided matching literature (e.g., Hatfield and Milgrom, 2005), stability is understood within a single-category context. However, in our model, stability takes into account the significance of position categories, thereby assigning different roles to positions of varying categories.

Essentially, when a job “blocks” an outcome (potentially along with other agents), all its positions assume symmetric roles in the standard definition. In contrast, in our model, the roles of positions from different categories differ, reflecting affirmative action legislation in India.

In particular, the “job-and-category rationality” condition renders our stability notion stronger than its standard counterpart. This condition requires that a job not only maintains all its assigned individuals, as per standard stability, but also ensures the preservation of the categories of these individuals within the assignment. Otherwise, when there is a single category, our notion of stability is equivalent to its conventional definition.

Our first result establishes an important equivalence.

Thus, given the 2SMH choice rule for each institution, collectively, the Saurav Yadav (2020) axioms are equivalent to a natural refinement of the celebrated stability axiom for two-sided matching problems.

When all positions are identical at a single job, there is a unique assignment–one that can be obtained with the inverse mapping of the 2SMH choice rule–that satisfies the Saurav Yadav (2020) axioms (Sönmez and Yenmez, 2022). As we show in the next example, this is not the case when positions are heterogenous.

Interestingly, Example 1 reveals that, in the absence of additional considerations, the HR policy may be detrimental to the very groups it is supposed to help. Specifically, if assignment $\beta$ is chosen by a central planner (potentially motivated by maximizing the number of honored HR-protected positions), the positive discrimination given to individual $a$ for the position at job $x$ due to their trait $t_{1}$ not only ends up harming individual $b$ but also individual $a$ themselves. Indeed, as individual $a$ has the highest merit score for both jobs, they do not require positive discrimination. Therefore, an excessive effort to honor HR-protected positions without considering individual preferences can be harmful to the very groups the HR policy is meant to protect. Fortunately, there is another assignment that not only satisfies Saurav Yadav (2020) axioms but also serves the best interests of all individuals, despite not honoring any of the HR-protected positions.

While our next axiom does not correspond to any desiderata formulated by the Supreme Court, it is among the most fundamental principles in economic theory.

It is well-established that, even in the absence of VR and HR protections, the axioms of individual rationality, non-wastefulness, and no-justified envy are incompatible with Pareto efficiency (Alcalde and Barberà, 1994; Balinski and Sönmez, 1999). However, this incompatibility disappears when the merit ranking of individuals is identical across all institutions (Balinski and Sönmez, 1999). Example 4 in Section B of the Online Appendix establishes an analogous but stronger incompatibility between the Saurav Yadav (2020) axioms and Pareto efficiency in our more general model – one that does not disappear even when all institutions have the same merit ranking of individuals.

While Pareto efficiency is incompatible with individual rationality, non-wastefulness, and no-justified envy in the absence of VR and HR policies, of all outcomes that satisfy these axioms, there exists one that Pareto dominates any other (Gale and Shapley, 1962; Balinski and Sönmez, 1999). We next show that this result extends to our model, thus motivating our proposed mechanism.

We are ready to present our proposed mechanism, which extends the (inverse mapping of the) 2SMH choice rule–defined for identical positions–to the general version of the problem with heterogenous positions using the celebrated individual-proposing DA algorithm (Gale and Shapley, 1962).

2SMH-DA Mechanism $\varphi^{2s}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

For each preference profile $\succ_{\mathcal{I}}\;\in\mathcal{P}$, the outcome $\varphi^{2s}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(\succ_{\mathcal{I}})$ of the 2SMH-DA mechanism is obtained as follows:

Step 1. Assuming such a job exists, each individual $i$ applies to her most preferred acceptable job under $\succ_{i}$. Let $I_{j}^{1}$ be the set of individuals who apply to job $j$. Each job $j$ tentatively assigns individuals in $\widehat{C}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}^{2s}(I_{j}^{1})$ to its categories based on $\vec{C}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}^{2s}(I_{j}^{1})$, and (permanently) rejects any remaining applicants. If there is no rejection by any job, then the procedure is terminated and the tentative assignments are finalized. Otherwise, proceed to Step 2.

Step k. Assuming such a job exists, each individual $i$ who is rejected in Step $(k-1)$ applies to her next preferred acceptable job under $\succ_{i}$. For any job $j$, let $I_{j}^{k}$ be the set of new applicants in Step $k$ along with individuals who are tentatively assigned to categories of job $j$ in Step $(k-1)$. Each job $j$ tentatively assigns individuals in $\widehat{C}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}^{2s}(I_{j}^{k})$ to its categories based on $\vec{C}_{\leavevmode\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35455pt\lower-2.35455pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.15456pt}{0.0pt}\pgfsys@curveto{2.15456pt}{1.18994pt}{1.18994pt}{2.15456pt}{0.0pt}{2.15456pt}\pgfsys@curveto{-1.18994pt}{2.15456pt}{-2.15456pt}{1.18994pt}{-2.15456pt}{0.0pt}\pgfsys@curveto{-2.15456pt}{-1.18994pt}{-1.18994pt}{-2.15456pt}{0.0pt}{-2.15456pt}\pgfsys@curveto{1.18994pt}{-2.15456pt}{2.15456pt}{-1.18994pt}{2.15456pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{-1.28331pt}{-0.95665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{M}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},j}^{2s}(I_{j}^{k})$, and (permanently) rejects any remaining applicants. If there is no rejection by any job, then the procedure is terminated and the tentative assignments are finalized. Otherwise, proceed to Step (k+1).

Since there is a finite number of jobs and individuals, this mechanism terminates at a finite round for every preference profile.

We next present two additional results. The general message of these results is that while the Saurav Yadav (2020) axioms do not single out a mechanism when positions are heterogeneous, they come very close. Either one of two fundamental axioms in economic theory leads to a unique prescription.

The primary objective of the Indian reservation system is to enhance the social and educational status of underprivileged communities. Given Theorem 2, we believe that 2SMH-DA is the only plausible mechanism to pursue that objective.

While the Pareto principle is perhaps the most fundamental axiom in economic theory, it is not the only condition that bridges the gap between the Saurav Yadav (2020) axioms and our proposed mechanism, 2SMH-DA. The following highly sought-after incentive compatibility condition, defined exclusively for mechanisms (and not for assignments), also serves a similar role.

Truthful preference revelation is always weakly more preferred than reporting any other preference ranking for every individual under a strategy-proof mechanism.

To the best of our knowledge, our paper is the first to propose a mechanism that complies with the Supreme Court’s mandates for the joint implementation of VR and HR policies in India when positions are heterogeneous. Our analysis builds upon Sönmez and Yenmez (2022), which addressed a simpler version of the problem with identical positions. While our work shares motivation with Ehlers and Morrill (2020), which also considers legal desiderata, our formulation directly relies on explicit mandates outlined in Saurav Yadav (2020).

Theorem 1, which establishes an equivalence between assignments satisfying the Saurav Yadav (2020) axioms and those satisfying a refinement of the traditional “two-sided matching” axiom of stability, has no antecedent in the literature.

The existence of an “individual-optimal” stable outcome is a fairly robust result under a wide range of assumptions in two-sided matching and school choice (Gale and Shapley, 1962; Balinski and Sönmez, 1999). Theorem 2 generalizes this result to our model under our more refined definition of stability.

Theorem 3 generalizes characterizations in Sönmez and Yenmez (2022) derived for homogeneous positions only, as well as the basic characterizations in Alcalde and Barberà (1994) and Balinski and Sönmez (1999), which were derived in the absence of any form of reservation policy. At a broader level, Theorem 3 is also in the same spirit as the “similar” results in Hirata and Kasuya (2017) and Hatfield et al. (2021), which characterize stable and strategy-proof mechanisms for the matching with contracts model.

Thus, one might be compelled to invoke these results by formulating our model as an application of the matching with contracts model, where each contract specifies an individual, a job, a vertical category, and a trait (or its absence). However, we cannot do this for two reasons. First, whereas the single-category choice rules in Hirata and Kasuya (2017) and Hatfield et al. (2021) are exogenous, our multi-category choice rule 2SMH is endogenous to the Supreme Court’s mandates. Additionally, since many of the theoretical constructs and results in the matching with contracts literature, including Hirata and Kasuya (2017) and Hatfield et al. (2021), assume strict preferences for individuals over the set of contracts, we cannot directly derive our results through this approach.

Interestingly, building on Kurata et al. (2017), the team that designed the K-12 school choice mechanism for Chile adopted this approach (Correa et al., 2019). As we discuss in Section 6.1, their practical application is a special case of our model with HR policy only. In order to invoke the “cumulative offer mechanism” from the matching with contracts literature with “minimum guarantee” reserves, they imposed arbitrary preferences over different traits for students. In Examples 5 and 6, presented in Section B of the Online Appendix, we show that this approach results in not only losing no justified envy but also maximal accommodation of HR protections.

Subsequent to our paper, Doğan et al. (2022) discuss a generalization of our single-category analysis to broader distributional constraints beyond HR policy, where distributional objectives are given by rank functions of some matroids. In our single-category analysis, the function that gives the number of HR-protected positions that can be filled corresponds to the rank function of a transversal matroid. In particular, Doğan et al. (2022) apply their general theory to the school choice in Chile like we do in Section 6.1, but they also accommodate policies that guarantee enrollment for returning students.

Prior to our paper, others have also proposed mechanisms based on the DA algorithm for practical applications in India. For instance, Thakur (2018) adopts a similar approach for allocating government positions by the Union Public Service Commission, while both Baswana et al. (2019a) and Aygün and Turhan (2022) utilize similar methods for seat allocation at engineering colleges. However, unlike our paper, none of these prior works properly account for the HR policy. In Thakur (2018) and Aygün and Turhan (2022), the HR policy is entirely disregarded. In Baswana et al. (2019a), the authors detail their design and implementation of a seat allocation process for several technical universities in India. Despite incorporating both VR-protected and HR-protected groups into their design, they failed to distinguish between the two policies within their mechanism. Consequently, their design is in direct violation of Indra Sawhney (1992).²⁵²⁵25The mechanism devised by Baswana et al. (2019a, b) treats persons with disabilities (PwD) as a VR-protected group, despite them being explicitly granted HR protections by the Supreme Court’s judgment in the National Federation Of The Blind (2013) case. For instance, a candidate from the general category with a disability is first considered for an open-category position, and only afterward for a PwD position in the open category. This approach implies that a position awarded to a disabled individual does not count toward the HR-protected positions for PwD, contradicting the fundamental mandate outlined in the Indra Sawhney (1992) case regarding the implementation of HR protections.

Abstracting away from the specific laws in India, there’s extensive research on priority-based resource allocation mechanisms that rely on the DA algorithm. Several papers in this field, such as those by Hafalir et al. (2013), Ehlers et al. (2014), Echenique and Yenmez (2015), Dur et al. (2018), Dur et al. (2020), and Abdulkadiroğlu and Grigoryan (2021), are particularly relevant to our work as they incorporate various reservation policies.

Finally, with our focus on a concrete real-life resource allocation problem, we contribute to the literature in market design. Studies in this vein include those on entry-level labor markets (Roth and Peranson, 1999), school choice (Abdulkadiroğlu and Sönmez, 2003; Erdil and Ergin, 2008; Kesten, 2010; Chen and Kesten, 2017; Hafalir et al., 2022; Reny, 2022), spectrum auctions (Milgrom, 2000), kidney exchange (Roth et al., 2004, 2005), liver exchange (Ergin et al., 2020; Yilmaz et al., 2023), internet auctions (Edelman et al., 2007; Varian, 2007), cadet-branch matching (Sönmez and Switzer, 2013; Sönmez, 2013; Greenberg et al., 2023), assignment of airport arrival slots (Schummer and Vohra, 2013; Schummer and Abizada, 2017), rent control (Andersson and Svensson, 2014), refugee resettlement (Jones and Teytelboym, 2017; Andersson, 2019; Andersson and Ehlers, 2020; Hadad and Teytelboym, 2022; Delacrétaz et al., 2023), and pandemic medical resource allocation (Pathak et al., 2020).

Since 2020, Chile has implemented a centralized school choice system following the enactment of the School Inclusion Law in 2015 (Correa et al., 2019). This system, covering Pre-K to grade 12, is a collaboration between the Chilean Ministry of Education and a team of economists and operations researchers. Similar to its predecessors, the Chilean school choice system is based on the individual-proposing DA algorithm. The following three features in its design make it a special case of our model, with only open category and overlapping HR protections:

1. (1)

   Affirmative action is granted for financially disadvantaged students and children with special needs, implemented through reserved seats at each school. In addition, some schools are allowed to reserve seats for high-achieving students.

2. (2)

   Reserved seats are implemented in the form of a “minimum guarantee”. Therefore, in our terminology, there are three “traits”: Financially disadvantaged, Special needs, and High-achieving. A student can have any subset of these traits, or none at all. Students with none of the three traits are referred to as Regular.

3. (3)

   Students with multiple traits are eligible for reserved seats corresponding to each trait, but, as in India, they only count towards one reserved seat upon acceptance.

For this special case, our axiom of compliance with VR protections becomes vacuous, and the 2SMH choice rule reduces to the meritorious horizontal choice rule. Otherwise, our entire formal analysis holds. Let us refer to the resulting special case of the 2SMH-DA mechanism as the MH-DA mechanism. Given Theorems 2 and 3, naturally, our proposed mechanism for Chilean school choice is the MH-DA mechanism. However, this is not the mechanism used in Chile. Adopting a methodology developed by Kurata et al. (2017), the Chilean school choice mechanism is equivalent to using the DA algorithm together with a choice rule different from the meritorious horizontal choice rule.

Following Kurata et al. (2017), the team which designed the Chilean school choice mechanism formulated their problem as an application of the matching with contracts model of Hatfield and Milgrom (2005), where the contractual term between a school and a student specifies which of the four types of seats (i.e., open seats, reserved seats for financially disadvantaged students, reserved seats for special needs students, and reserved seats for high-achieving students) the student receives. However, the theory of matching with contracts is developed under the assumption that students have strict preferences over all their contracts, which in this context corresponds to them having strict preferences on the specific type of seats they receive at each school. Since students declare preferences over only schools in Chile, a tie-breaking rule is used to construct student preferences over specific type of seats at each school. In Correa et al. (2019), the designers emphasize that the choice of a tie-breaking rule is not straightforward, and it has distributional consequences. In order to implement the reserves in the form of a minimum guarantee, a design requirement by the policy makers, they break ties in a way each student is assumed to prefer reserved seats for any of their traits to open seats. When each student has at most one trait, this construction assures that the reserves are implemented as a minimum guarantee (Hafalir et al., 2013; Dur et al., 2020).

Given Theorem 3, the Chilean school choice mechanism must violate some of our axioms. In Section B of the Online Appendix, we present two simple examples demonstrating that the Chilean school choice mechanism fails to satisfy no justified envy and maximal accommodation of HR protections. Example 5 reveals that the Chilean school choice mechanism violates no justified envy, while Example 6 shows that it fails to satisfy maximal accommodation of HR protections.