Affirmative Action in India with Hierarchical Reservations

1. Problem Statement

The paper addresses the complexity of resource allocation in India's affirmative action system for government jobs and university admissions. The system is characterized by two intersecting layers of constraints:

• Vertical Reservations: Quotas for specific social groups (Scheduled Castes [SC], Scheduled Tribes [ST], Other Backward Classes [OBC], and Economically Weaker Sections [EWS]).
• Horizontal Reservations: Quotas for marginalized groups (e.g., Persons with Disabilities) that cut across all vertical categories, including the open (general) category.

The Core Challenge:
The paper focuses on the interaction between these two layers when an applicant belongs to multiple horizontal categories (e.g., a candidate with both a visual impairment and a hearing impairment).

• One-to-One vs. One-to-All: Previous literature (e.g., Sönmez and Yenmez, 2022) often assumed a "one-to-one" matching policy where a candidate counts against only one horizontal quota. However, Indian law and practice often utilize a "one-to-all" policy, where a candidate counts against all horizontal quotas they are eligible for.
• Stability Issues: In a general one-to-all framework, complementarities between applicants can arise, potentially leading to the non-existence of a stable matching.
• Ambiguity: The implementation of horizontal reservations within vertical categories is ambiguous when applicants possess multiple horizontal types. The paper specifically addresses the case where horizontal reservations follow a hierarchical (nested/laminar) structure (e.g., a specific disability type is a subset of a broader disability category).

2. Methodology and Model

The authors employ Market Design and Matching Theory frameworks to model the Indian allocation problem.

• Agents and Contracts:

  • Individuals ($I$): Have strict preferences over pairs of $(Institution, Vertical Category)$. Unlike previous models that assumed indifference between open and reserved slots within an institution, this model allows candidates to express preferences over specific vertical categories (capturing potential stigma or preference for open slots).
  • Institutions ($S$): Have capacities divided into vertical categories ($q_o, q_{SC}, q_{ST}, q_{OBC}, q_{EWS}$).
  • Contracts ($X$): A contract specifies an individual, an institution, and the vertical category under which they are admitted.

• Horizontal Types ($H$):

  • Modeled via a correspondence $\rho: I \to H$.
  • Hierarchical Structure: Defined as a condition where for any two types $h, h'$, if their eligible sets overlap, one is a subset of the other ($\rho^{-1}(h) \subset \rho^{-1}(h')$ or vice versa). This "nested" structure is crucial for the existence of stable mechanisms.

• Choice Rules:

  • The authors define a Hierarchical Choice Rule ($Ch$) for each vertical category.
  • Mechanism: The rule operates sequentially. It first fills quotas for the most specific horizontal types (those not containing others), then moves to broader types, updating capacities dynamically. Finally, remaining positions are filled based on pure merit.
  • Aggregate Choice Rule: Combines the hierarchical rules across all vertical categories with a specific precedence order ($Open \to SC \to ST \to OBC \to EWS$).

• Mechanism:

  • The authors utilize the Generalized Deferred Acceptance (DA) mechanism (also known as the Cumulative Offer Mechanism) coupled with the proposed aggregate choice rules.

3. Key Contributions

1. The Hierarchical Choice Rule:

   • The paper introduces a specific choice rule designed for hierarchical horizontal reservations under a one-to-all framework.
   • Merit-Undominated: Theorem 1 proves that this rule is the unique choice rule that is "merit-undominated" among all rules satisfying horizontal reservations. This means it selects the set of applicants that is lexicographically superior in terms of merit scores compared to any other feasible set satisfying the constraints.

2. Properties of the Aggregate Choice Rule:

   • The authors prove that the aggregate choice rule (combining vertical and horizontal rules) satisfies:
     • Substitutes Property: Essential for the stability of the Deferred Acceptance mechanism.
     • Size Monotonicity: If capacity increases, the set of chosen individuals does not shrink.
     • Fairness: If an applicant is rejected, every accepted applicant must either have a higher merit score or possess a vertical/horizontal reservation status that the rejected applicant lacks.

3. Extension to Soft Reserves (OBC De-reservation):

   • The paper extends the model to handle "soft reserves" (specifically for OBC in academic admissions), where unfilled OBC seats are transferred to the open category.
   • They propose an Aggregate Choice Rule with Transfer ($Ch^T$) using a "forward transfer" scheme (filling OBC first, then transferring vacancies to Open).
   • They prove this extended rule also maintains fairness and stability properties.

4. Preference Domain Realism:

   • The model explicitly allows individuals to have preferences over the vertical category of admission, not just the institution. This captures real-world phenomena like the stigma associated with reserved categories, a feature often ignored in prior literature.

4. Main Results

• Theorem 1: The Hierarchical Choice Rule ($Ch$) is the unique merit-undominated choice rule satisfying horizontal reservations.
• Theorem 2: The Aggregate Choice Rule ($Ch$) is fair.
• Theorem 3: The Generalized Deferred Acceptance mechanism ($\Phi^h$) coupled with the aggregate choice rule is:
  • Stable: No blocking pairs exist.
  • Strategy-Proof: Individuals cannot benefit by misrepresenting their preferences.
  • Justified-Envy Free: No individual envies another's assignment unless the other has a superior merit score or a specific reservation status the first lacks.
  • Uniqueness: It is the unique mechanism satisfying these three properties.
• Theorem 4 & 5: These results extend to the Soft Reserve scenario (OBC de-reservation). The mechanism with forward transfer ($\Phi^{hT}$) remains stable, strategy-proof, and eliminates justified envy.
• Proposition 1: The paper establishes that "Elimination of Justified Envy" and "Stability with respect to fair choice rules" are independent properties, clarifying theoretical distinctions in the literature.

5. Significance and Implications

• Theoretical Advancement: The paper resolves the theoretical gap regarding the existence of stable mechanisms in "one-to-all" matching with complex constraints. It demonstrates that hierarchical constraints are a sufficient condition for stability, even when candidates count against multiple quotas.
• Policy Relevance: The findings provide a rigorous mathematical justification for the current implementation of India's affirmative action policies, specifically the hierarchical nature of disability quotas (Rights of Persons with Disabilities Act, 2016). It validates the "forward transfer" approach for OBC seats in academic admissions as a merit-maximizing and fair solution.
• Global Applicability: While focused on India, the framework applies to any resource allocation problem with layered diversity constraints (e.g., university admissions with race, income, and geographic quotas; public housing; military recruitment).
• Behavioral Realism: By modeling preferences over vertical categories, the paper offers a more realistic view of applicant behavior, acknowledging that candidates may prefer open-category admission over reserved-category admission at the same institution.

In summary, the paper successfully bridges the gap between complex legal mandates in India and market design theory, proving that a stable, fair, and strategy-proof allocation is achievable through a specific hierarchical choice rule and the Generalized Deferred Acceptance mechanism.