Achieving Balanced Representation in School Choice with Diversity Goals

TL;DR

The paper addresses fair student placement under diverse quotas in a one-to-one convention, where each student belongs to multiple types but is counted once. It introduces balanced representation to mitigate imbalances across type combinations and proposes a unique choice function that achieves maximal diversity, non-wastefulness, justified envy-freeness, and balanced representation. To realize this, the authors develop a compact flow-network formalization alongside efficient algorithms based on both rank-maximal matching and flow techniques, with practical implementations for single and multi-school settings. The approach improves computational efficiency and fairness guarantees, and it highlights limitations of substitutability when extending to multi-school generalized Deferred Acceptance. Overall, the work advances rigorous, implementable mechanisms for diversity-aware school choice that respect priorities while promoting balanced, equitable representation across groups and their combinations.

Abstract

Student placements under diversity constraints are a common practice globally. This paper addresses the selection of students by a single school under a \emph{one-to-one convention}, where students can belong to multiple types but are counted only once based on one type. While existing algorithms in economics and computer science aim to help schools meet diversity goals and priorities, we demonstrate that these methods can result in significant imbalances among students with different type combinations. To address this issue, we introduce a new property called \emph{balanced representation}, which ensures fair representation across all types and type combinations. We propose a straightforward choice function that uniquely satisfies four fundamental properties: maximal diversity, non-wastefulness, justified envy-freeness, and balanced representation. While previous research has primarily focused on algorithms based on bipartite graphs, we take a different approach by utilizing flow networks. This method provides a more compact formalization of the problem and significantly improves computational efficiency. Additionally, we present efficient algorithms for implementing our choice function within both the bipartite graph and flow network frameworks.

Figures (4)

• Figure 1: An interpretation of minimum and maximum quotas for type $t$ as two ranks of quotas is as follows. Rank 1 corresponds to the scenario where the number of matched students of type $t$ does not exceed the minimum quota. Rank 2 applies when the number of matched students surpasses the minimum quota but does not exceed the maximum quota.
• Figure 2: Three types and four groups in Example \ref{['example:issue']}.
• Figure 3: A ranked reservation graph for Example \ref{['example:instance']}. Create one reserved seat $v_{t_1, 1}^1$ of rank $1$ for type $t_1$ and two reserved seats $v_{t_0, 1}^2$ and $v_{t_0, 2}^2$ of rank $2$ for type $t_0$. Solid lines incident to vertex $v_{t_1, 1}^1$ have rank $1$ and dashed lines incident to vertices $v_{t_0, 1}^2$ and $v_{t_0, 2}^2$ have rank $2$.
• Figure 4: The flow network corresponding to Example \ref{['example:instance']} consists of four layers, plus a source node and a terminal node. Each edge is labeled with $(c, w)$ where $c$ denotes the edge capacity and $w$ denotes the cost. Only edges between Layer 2 and Layer 3 have non-zero costs, representing their ranks. For example, dashed lines represent a cost of 2.

Theorems & Definitions (40)

• Example 1
• Definition 1: Matching
• Definition 2: Signature
• Definition 3: Rank Maximality
• Example 2
• Definition 4: Flow
• Definition 5: Minimum Cost Maximum Flow
• Example 3
• Theorem 1
• proof