. 2020 Feb 6;82(5):1136–1188. doi: 10.1007/s00453-019-00636-y

Table 5.

Preference lists of women and men in the proof of Theorem 10

$L (x) = (\hat{y}),$
$L (y) = (\hat{y}),$
$L (s_{j}) = ({\hat{p}}_{j}^{3}, \hat{y}),$
$L (p_{j}^{1}) = ({\hat{p}}_{j}^{1}, t_{j}),$
$L (p_{j}^{h}) = ({\hat{p}}_{j}^{h - 1}, {\hat{p}}_{j}^{h})$	for $h \in {2, 3}$ ,
$L (q_{j}) = ({\hat{q}}_{j}, t_{j}),$
$L (a_{i, j}) = ({\hat{b}}_{i, j}, {\hat{p}}_{j}^{h})$	for the unique $h \in {1, 2, 3}$ that satisfies $i = i n d (j, h)$ ,
$L (b_{i, j}) = ({\hat{b}}_{i, j}, c_{i}),$
$L (\hat{y}) = (y, s_{1}, s_{2}, \dots, s_{m}, x),$
$L (t_{j}) = (p_{j}^{1}, q_{j}),$
$L ({\hat{p}}_{j}^{h}) = (p_{j}^{h}, a_{i, j}, p_{j}^{h + 1})$	for $h \in {1, 2}$ and $i = i n d (j, h)$ ,
$L ({\hat{p}}_{j}^{3}) = (p_{j}^{3}, a_{i, j}, s_{j})$	where $i = i n d (j, 3)$ ,
$L ({\hat{q}}_{j}) = (q_{j}),$
$L ({\hat{b}}_{i, j}) = (b_{i, j}, a_{i, j}),$
$L (c_{i}) = ([B_{i}])$	where $B_{i} = {b_{i, j} ∣ u_{i} \in S_{j}}$ and $[B_{i}]$ is some fixed ordering of $B_{i}$

We denote by $i n d (j, h)$ the index i for which $u_{i}$ is the h-th element in $S_{j}$ . When not stated otherwise, indices take all possible values