Epsilon-stability in school choice

Abstract

In many school choice practices, scores, instead of ordinal rankings, are used to indicate students’ qualification. We study school choice problems where students have ordinal preference over schools while their priorities at schools are in the form of cardinal scores. The cardinality of scores allows us to measure the intensity of priority violations and hence relax stability by proposing epsilon-stability. We also propose the epsilon-EADA mechanism to select the constrained efficient matching under epsilon-stability.

Appendix

1.1 A. Proof of Theorem 1

For each $k\ge 0$, let $P^{k}$ denote the updated preferences of students who still remain in round-k DA of the $\varepsilon$-EADAM procedure, and let matching $\alpha _{k}$ denote the matching produced by round-k of the algorithm, in which the students who remain after removal in round-k are matched by round-k DA, and students removed in or before round-k are matched with the seats they are removed together with. Let $\alpha$ denote the eventual matching produced by the algorithm.

We begin by showing that the eventual matching $\alpha$ produced by the $\varepsilon$-EADAM is $\varepsilon$-stable. Consider any student i,  and suppose i is matched with an under demanded school in round-k DA of the $\varepsilon$-EADAM procedure. Suppose student j is removed earlier than i or at the same step as i, then at the step that j is removed, either j’s assignment is still in i’s preference or it has been removed from i’s preference at some earlier step. In the former case, apparently i does not desire the assignment of j since j’s assignment is under demanded, and thus j does not violate i’s priority. In the latter case, note that the assignment of j is in j’s preference but not in i’s preference. It implies that j has higher scores at this school than i, and thus j does not violate i’s priority.

If student j is removed later than i and i desires j’s assignment s, then at the step that i is removed, in the algorithm, s would have been removed from $P_{j}$ if $a_{j}^{s}<a_{i}^{s}-\varepsilon$. Therefore, j cannot violate i’s priority by more than $\varepsilon$. As a result, the matching produced by $\varepsilon$-EADAM is $\varepsilon$-stable.

Next, we show that any matching $\mu$ that Pareto dominates $\alpha$ must not be $\varepsilon$-stable. Since $\alpha$ weakly Pareto dominates all $\alpha _{k}$’s, we know that $\mu$ Pareto dominates all $\alpha _{k}$’s. For each $k\ge 0$, let $UD_{k}\subset I$ denote the set of students matched with under demanded schools at the round-k DA matching $DA(P^{k},\succ )$. Since $\mu$ Pareto dominates the round-0 outcome $\alpha _{0}$, for all $i\in UD_{0}$, $\mu (i)=\alpha _{0}(i)$.

We use the following lemma to carry out induction.

Lemma 2

Let $\mu$ be an $\varepsilon$-stable matching. Consider any round-k, $k\ge 1$, of the $\varepsilon$-EADAM. Suppose we already know that for each $0\le l\le k-1$, $\mu$ Pareto dominates $\alpha _{l}$ implies that for all $i\in UD_{l},\mu (i)=\alpha _{l}(i)$. Then $\mu$ Pareto dominates $\alpha _{k}$ implies that for all $i\in UD_{k},\mu (i)=\alpha _{k}(i)$.

Proof

Suppose for some $i\in UD_{k}$, $\mu (i)P_{i}\alpha _{k}(i)$. Since $\mu$ Pareto dominates $\alpha _{k}$ under P, $\mu$ also Pareto dominates $\alpha _{l}$ for each $0\le l\le k-1$ due to Lemma 1. By the assumption of the lemma, $\mu (i)=\alpha _{l}(i)$, for each $i\in UD_{l}$ and each $0\le l\le k-1$. Therefore, $\mu$ Pareto dominates $\alpha _{k}$ implies that for students who remain after removal in round-k, $\mu$ Pareto dominates $DA(P^{k},\succ )$ under P. So either $\mu$ also Pareto dominates $DA(P^{k},\succ )$ under $P^{k}$ or for some $i^{\prime }$ who remains in round-k, $\mu (i^{\prime })$ is removed from $P_{i^{\prime }}^{k}$ before round-k. In the former case, we know that for all $i\in UD_{k}$, $\mu (i)=DA(P^{k},\succ )(i)=\alpha _{k}(i)$. This contradicts with the assumption that $\mu (i)P_{i}\alpha _{k}(i)$ for some $i\in UD_{k}$. In the latter case, $\mu$ matches some student who remains after removal in round-k with a school removed in the earlier modifications of her preference. Again by definition of the algorithm, $\mu$ is not $\varepsilon$-stable. We have a contradiction. $\square$

Since the case of $k=1$ holds, by induction (due to lemma 2), for each $k\ge 1$, if $\mu$ Pareto dominates $\alpha _{k}$ and is $\varepsilon$-stable (under the original preference profile P), then $i\in UD_{k}$ implies $\mu (i)=\alpha _{k}(i)$.

We now prove the theorem. Suppose there exists a matching $\mu$ that Pareto dominates $\alpha$ and is $\varepsilon$-stable. Let round-K be the last round of the $\varepsilon$-EADAM, which produces the eventual outcome $\alpha =\alpha _{K}$. Since $\mu$ Pareto dominates $\alpha$, it Pareto dominates $\alpha _{0},\ldots ,\alpha _{K}$. Since $\mu$ is $\varepsilon$-stable, due to Lemma 2, for each $0\le k\le K$, if $i\in UD_{k}$, then $\mu (i)=\alpha _{k}(i)$. That is, at matching $\mu$, each student $i\in I$ is matched with the seat she is removed together with. Therefore, $\mu =\alpha$. We have a contradiction.