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Feature-based Uncertainty Model for School Choice

Yao Zhang, Makoto Yokoo

TL;DR

This work studies school choice under feature-based uncertainty, where a student’s ranking of colleges is a weighted sum $\sum_{f\in F} w_s^f u_s^f(c)$ with $w_s\sim\mu_s$ and deterministic college preferences. It develops a generalized deferred acceptance framework and four proposing rules—HEUF, LOCV, LOICV, and HERF—to balance stability probability (ProS) and incentive compatibility (IC), and derives fundamental complexity and impossibility results. The key findings show that maximizing ProS is NP-hard, IC-A is unattainable in general, and among the four methods, HERF achieves the best worst-case ProS (a $$(1/n)^n$$-approximation), while the others have zero worst-case guarantees; LOICV achieves IC-R for $|F|=2$, with tighter limitations for larger feature sets. For the tractable case $|F|=2$, ProS can be computed in polynomial time, whereas for $|F|\ge3$ computing ProS becomes #P-hard and several IC properties break down, motivating practical approximations and future work on data-driven uncertainty estimation and information design.

Abstract

In this work, we consider a school choice scenario where a student does not exactly know which college is better for her. Although it is hard for a student to obtain an exact preference, she can usually compare specific features of colleges, such as reputation, location, and campus facilities. Motivated by this, we propose a feature-based uncertainty model for school choice where a student's preference is based on a linear combination of her utilities over different features, and the coefficients of the combination are treated as random variables. Our main goal is to achieve a higher probability of stability (ProS) and incentive compatibility (IC) for students. Unfortunately, these two goals are incompatible in general. We show that a student-proposing deferred acceptance (DA) that prioritizes colleges with higher expected ranking can achieve a worst-case approximation ratio of $(1/n)^n$ on ProS, while a DA with a carefully defined iterated comparison vector can guarantee the strongest achievable form of IC. Finally, we provide additional results for some specific restrictions on the model.

Feature-based Uncertainty Model for School Choice

TL;DR

This work studies school choice under feature-based uncertainty, where a student’s ranking of colleges is a weighted sum with and deterministic college preferences. It develops a generalized deferred acceptance framework and four proposing rules—HEUF, LOCV, LOICV, and HERF—to balance stability probability (ProS) and incentive compatibility (IC), and derives fundamental complexity and impossibility results. The key findings show that maximizing ProS is NP-hard, IC-A is unattainable in general, and among the four methods, HERF achieves the best worst-case ProS (a -approximation), while the others have zero worst-case guarantees; LOICV achieves IC-R for , with tighter limitations for larger feature sets. For the tractable case , ProS can be computed in polynomial time, whereas for computing ProS becomes #P-hard and several IC properties break down, motivating practical approximations and future work on data-driven uncertainty estimation and information design.

Abstract

In this work, we consider a school choice scenario where a student does not exactly know which college is better for her. Although it is hard for a student to obtain an exact preference, she can usually compare specific features of colleges, such as reputation, location, and campus facilities. Motivated by this, we propose a feature-based uncertainty model for school choice where a student's preference is based on a linear combination of her utilities over different features, and the coefficients of the combination are treated as random variables. Our main goal is to achieve a higher probability of stability (ProS) and incentive compatibility (IC) for students. Unfortunately, these two goals are incompatible in general. We show that a student-proposing deferred acceptance (DA) that prioritizes colleges with higher expected ranking can achieve a worst-case approximation ratio of on ProS, while a DA with a carefully defined iterated comparison vector can guarantee the strongest achievable form of IC. Finally, we provide additional results for some specific restrictions on the model.
Paper Structure (13 sections, 17 theorems, 22 equations, 2 figures, 1 table)

This paper contains 13 sections, 17 theorems, 22 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Relationships to Other Uncertainty Models
  4. The Model
  5. Impossibilities and Intractability
  6. Methods
  7. Properties of Methods
  8. Probability of Stability
  9. Incentive Compatibility
  10. Additional Issues
  11. Restrictions of "mean=median"
  12. Computational Issues for More Features
  13. Conclusion and Future Work

Key Result

Lemma 1

If the number of features equals to 2, i.e., $F = \{f_1, f_2\}$, then for any student $s\in N$ and any two colleges $c_1, c_2\in M$, denote $\Delta_s^f(c_i,c_j) = |u_s^f(c_i) - u_s^f(c_j)|$ for any $f\in F$, and $\eta_{s}(c_i,c_j) = 1\left/\left( 1+\frac{\Delta_s^{f_1}(c_i,c_j)}{\Delta_s^{f_2}(c_i,c and the case of $\mathrm{Pr}[c_i\succeq_s^{w_s} c_j]$ can be generated by $1 -\mathrm{Pr}[c_j\succ_

Figures (2)

  • Figure 1: Box-plots of the approximation ratio of $\mathsf{ProS}$ in all sampled instances.
  • Figure 2: An example of the probability distribution of $\vec{w}_s$.

Theorems & Definitions (40)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • ...and 30 more