Remark 1 — justified max envy is justified envy under $\succ^r$; stability is classical stability for $(P,\succ^r)$

Throughout, $I$ is a finite set of students, $S$ a finite set of schools with capacities $q=(q_s)_{s\in S}$, $P=(P_i)_{i\in I}$ strict preferences over $S\cup\{\emptyset\}$ ($\emptyset$ the no-school option, $R_i$ the weak version of $P_i$), $\succ=(\succ_s)_{s\in S}$ strict priorities over $I$, and $r=(r(i))_{i\in I}$ minimal-access rights, $r(i)\subseteq S$.

Let $\succ^r$ be the adjusted priorities. For every matching $\mu$ and every student $i$:

1. $i$ has justified max envy at $\mu$ with respect to $(P,\succ,r)$ if and only if $i$ has justified envy at $\mu$ with respect to $(P,\succ^r)$;
2. $\mu$ is stable with respect to $(P,\succ,r)$ if and only if $\mu$ is classically stable with respect to $(P,\succ^r)$.

This reduces stability in the extended model to stability in a classical school choice problem, which is what lets the paper import the classical results on deferred acceptance.

Differences from the source:

• encoding: finite sets I, S; no-school option ∅; strict preferences P_i over S ∪ {∅}; strict priorities ≻_s over I → I = Fin n, S = Fin m, ∅ = none; P i : Option (Fin m) ≃ Fin (m+1) and pri s : Fin n ≃ Fin n rank bijections, rank 0 most preferred / highest priority; reason: finite relabelling; every strict order is exactly one rank bijection, so the problems are the same
• encoding: adjusted priorities ≻^r given by (1)–(3) of Remark 1 → keys adjKey pri r s i = (if s ∈ r i then 0 else n) + pri s i, i ≻^r_s j iff adjKey … i < adjKey … j; reason: the keys realise exactly (1)–(3): ranks are < n, so right-holders come first and ≻_s orders each group
• generalization: µ a matching → µ any assignment (capacities not required); reason: neither side of either equivalence uses the capacity bound; the statement for matchings is the special case
• generalization: capacities q_s ∈ N, positive (cf. q_s ≥ 1, p.29) → q : Fin m → ℕ, q_s = 0 allowed; reason: the statement for positive capacities is the special case; the argument never uses positivity