Top trading cycles with reordering: improving match priority in school choice

Abstract

School districts commonly ration public school seats based on students’ preferences and schools’ priorities. Priorities reflect the school districts’ objectives for reducing busing costs (walk-zone priority) or utilizing siblings’ learning spillovers (sibling priority). I develop a simple modification of the well-studied Top Trading Cycles mechanism that matches schools to higher priority students while preserving the mechanism’s desirable efficiency and incentives properties.

Appendix A

1.1 A.1 Proof of Lemma 1

It is sufficient to restrict attention to one school $s \in S$ and show that the desired permutation on $\{1,2,\ldots ,q_s\}$ can be achieved by a product of transpositions, where a transposition is a permutation that changes the position of only two elements.

Let $\lambda$ be an arbitrary permutation of $(1,2,\ldots ,q_s)$, e.g. one corresponding to the restriction of an arbitrary reordering $\pi$ on school s. It is sufficient to find permutations $(\lambda _{j})_{j=1}^{q_s}$ such that for each $n \in \{1,2,3,\ldots ,q_s\}$ the following conditions are satisfied:

• 1. $\lambda _n$ is a product of transpositions,

• 2. $\lambda _{n}^{-1}(k) = \lambda ^{-1}(k)$ for all $k \in \{q_s - n + 1, q_s -n + 2, \ldots , q_s\}$,

• 3. for each $m, l \in L_n := \big \{ \lambda ^{-1}(1), \lambda ^{-1}(2), \ldots , \lambda ^{-1} (q_s - n) \big \}$ we have

  $$\begin{aligned} m \le l \iff \lambda _{n}(m) \le \lambda _{n}(l), \end{aligned}$$

  i.e., $L_n$ preserves order under $\lambda _n$.

We construct such permutations inductively. First, observe that $m_1 := \lambda ^{-1}(q_s) \le q_s$. If $m_1 = q_s$, then setting the $\lambda _n$ being the identity permutation satisfies the desired conditions. Now assume $m_1 < q_s$. Consider the set of transpositions $(\lambda ^{j,1})_{j = 0}^{q_s - m_1-1}$ given by $\lambda ^{j,1}(m_1+j) = m_1+j+1$⁵. Let $\lambda _{1} := \times _{j =0}^{q_s - m_1-1} \lambda ^{j,1}$. Then, $\lambda _{1}$ satisfies the desired conditions. Now suppose $(\lambda _j)_{j=1}^{n-1}$ satisfy the desired conditions. Let $m_{n} := \lambda ^{-1}(q_s - n+1)$ and let $l := \lambda _{n-1} (m_n)$. Since the elements in $L_{n-1}$ preserve order under $\lambda _{n-1}$, we get $l \le \lambda _{n-1}^{-1}(q_s - n +1)$. Consider transpositions $(\lambda ^{j,n})_{j=0}^{q_s - l -n}$ given by $\lambda ^{j,n}(l+j) = l+j + 1$. Then, $\lambda _{n} = \lambda _{n-1} \times \big ( \times _{j =0}^{q_s - l -n} \lambda ^{j,n} \big )$ satisfies the desired conditions.