# All stable matchings in the hospital/residents problem with incomplete lists

## Usage

hri(nStudents = ncol(s.prefs), nColleges = ncol(c.prefs), nSlots = rep(1,
nColleges), s.prefs = NULL, c.prefs = NULL, seed = NULL,
s.range = NULL, c.range = NULL, ...)

## Arguments

nStudents
integer indicating the number of students (in the college admissions problem) or men (in the stable marriage problem) in the market. Defaults to ncol(s.prefs).
nColleges
integer indicating the number of colleges (in the college admissions problem) or women (in the stable marriage problem) in the market. Defaults to ncol(c.prefs).
nSlots
vector of length nColleges indicating the number of places (i.e. quota) of each college. Defaults to rep(1,nColleges) for the marriage problem.
s.prefs
matrix of dimension nColleges x nStudents with the jth column containing student j's ranking over colleges in decreasing order of preference (i.e. most preferred first).
c.prefs
matrix of dimension nStudents x nColleges with the ith column containing college i's ranking over students in decreasing order of preference (i.e. most preferred first).
seed
integer setting the state for random number generation.
s.range
range of two intergers s.range = c(s.min, s.max), where s.min < s.max. Produces incomplete preference lists with the length of each student's list randomly sampled from the range [s.min, s.max]. Note: interval is only correct if either c.range or s.range is used.
c.range
range of two intergers c.range = c(c.min, c.max), where c.min < c.max. Produces incomplete preference lists with the length of each college's list randomly sampled from the range [c.min, c.max]. Note: interval is only correct if either c.range or s.range is used.
...
.

## Value

hri returns a list of the following elements.
s.prefs.smi
student-side preference matrix for the stable marriage problem with incomplete lists (SMI).

c.prefs.smi
college-side preference matrix for the stable marriage problem with incomplete lists (SMI).

s.prefs.hri
student-side preference matrix for the college admissions problem (a.k.a. hospital/residents problem) with incomplete lists (HRI).

c.prefs.hri
college-side preference matrix for the college admissions problem (a.k.a. hospital/residents problem) with incomplete lists (HRI).

matchings
edgelist of matched students and colleges, inculding the number of the match (matching) and two variables that indicate the student-optimal match (sOptimal) and college-optimal match (cOptimal)
.

## Description

Finds all stable matchings in either the hospital/residents problem (a.k.a. college admissions problem) or the related stable marriage problem. Dependent on the problem, the results comprise the student and college-optimal or the men and women-optimal matchings. The implementation allows for incomplete preference lists (some agents find certain agents unacceptable) and unbalanced instances (unequal number of agents on both sides). The function uses the Prosser (2014) constraint encoding based on either given or randomly generated preferences.

## Minimum required arguments

hri requires the following combination of arguments, subject to the matching problem.

nStudents, nColleges
Marriage problem with random preferences.

s.prefs, c.prefs
Marriage problem with given preferences.

nStudents, nSlots
College admissions problem with random preferences.

s.prefs, c.prefs, nSlots
College admissions problem with given preferences.

## References

Gale, D. and L.S. Shapley (1962). College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9--15.

Morizumi, Y., T. Hayashi and Y. Ishida (2011). A network visualization of stable matching in the stable marriage problem. Artificial Life Robotics, 16:40--43.

Prosser, P. (2014). Stable Roommates and Constraint Programming. Lecture Notes in Computer Science, CPAIOR 2014 Edition. Springer International Publishing, 8451: 15--28.

## Examples

## -----------------------
## --- Marriage problem

## 3 men, 2 women, random preferences:
hri(nStudents=7, nColleges=6, seed=4)
$s.prefs.smi [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 4 5 1 6 4 4 3 [2,] 1 6 5 4 5 2 4 [3,] 2 4 2 3 2 6 2 [4,] 5 1 4 5 3 5 1 [5,] 6 2 3 2 6 1 5 [6,] 3 3 6 1 1 3 6$c.prefs.smi
[,1] [,2] [,3] [,4] [,5] [,6]
[1,]    4    4    2    7    3    4
[2,]    2    3    7    2    4    2
[3,]    5    7    5    4    7    1
[4,]    1    5    3    1    2    7
[5,]    3    6    6    6    5    3
[6,]    6    2    1    3    6    6
[7,]    7    1    4    5    1    5

$matchings matching college slots student sOptimal cOptimal sRank cRank 1 1 1 1 3 1 0 1 5 2 1 2 2 5 1 0 3 4 3 1 3 3 7 1 0 1 2 4 1 4 4 1 1 0 1 4 5 1 5 5 2 1 0 1 4 6 1 6 6 4 1 0 1 1 7 2 1 1 1 0 1 2 4 8 2 2 2 5 0 1 3 4 9 2 3 3 7 0 1 1 2 10 2 4 4 2 0 1 3 2 11 2 5 5 3 0 1 2 1 12 2 6 6 4 0 1 1 1 attr(,"class") [1] "hri" ## 3 men, 2 women, given preferences: s.prefs <- matrix(c(1,2, 1,2, 1,2), 2,3) c.prefs <- matrix(c(1,2,3, 1,2,3), 3,2) hri(s.prefs=s.prefs, c.prefs=c.prefs)$s.prefs.smi
[,1] [,2] [,3]
[1,]    1    1    1
[2,]    2    2    2

$c.prefs.smi [,1] [,2] [1,] 1 1 [2,] 2 2 [3,] 3 3$matchings
matching college slots student sOptimal cOptimal sRank cRank
1        1       1     1       1        1        1     1     1
2        1       2     2       2        1        1     2     2

attr(,"class")
[1] "hri"

## --------------------------------
## --- College admission problem

## 7 students, 2 colleges with 3 slots each, random preferences:
hri(nStudents=7, nSlots=c(3,3), seed=21)
$s.prefs.smi [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 4 4 4 1 4 4 1 [2,] 5 5 5 2 5 5 2 [3,] 6 6 6 3 6 6 3 [4,] 1 1 1 4 1 1 4 [5,] 2 2 2 5 2 2 5 [6,] 3 3 3 6 3 3 6$c.prefs.smi
[,1] [,2] [,3] [,4] [,5] [,6]
[1,]    2    2    2    5    5    5
[2,]    1    1    1    7    7    7
[3,]    3    3    3    6    6    6
[4,]    7    7    7    2    2    2
[5,]    4    4    4    3    3    3
[6,]    5    5    5    4    4    4
[7,]    6    6    6    1    1    1

$s.prefs.hri [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 2 2 2 1 2 2 1 [2,] 1 1 1 2 1 1 2$c.prefs.hri
[,1] [,2]
[1,]    2    5
[2,]    1    7
[3,]    3    6
[4,]    7    2
[5,]    4    3
[6,]    5    4
[7,]    6    1

$matchings matching college slots student sOptimal cOptimal sRank cRank 1 1 1 1 1 1 0 4 2 2 1 1 2 3 1 0 5 3 3 1 1 3 7 1 0 3 4 4 1 2 6 2 1 0 3 4 5 1 2 4 5 1 0 1 1 6 1 2 5 6 1 0 2 3 7 2 1 2 1 0 1 5 2 8 2 1 1 2 0 1 4 1 9 2 1 3 3 0 1 6 3 10 2 2 4 5 0 1 1 1 11 2 2 6 6 0 1 3 3 12 2 2 5 7 0 1 5 2 attr(,"class") [1] "hri" ## 7 students, 2 colleges with 3 slots each, given preferences: s.prefs <- matrix(c(1,2, 1,2, 1,NA, 1,2, 1,2, 1,2, 1,2), 2,7) c.prefs <- matrix(c(1,2,3,4,5,6,7, 1,2,3,4,5,NA,NA), 7,2) hri(s.prefs=s.prefs, c.prefs=c.prefs, nSlots=c(3,3))$s.prefs.smi
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    1    1    1    1    1    1    1
[2,]    2    2    2    2    2    2    2
[3,]    3    3    3    3    3    3    3
[4,]    4    4   NA    4    4   NA   NA
[5,]    5    5   NA    5    5   NA   NA
[6,]    6    6   NA    6    6   NA   NA

$c.prefs.smi [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 1 1 1 1 1 [2,] 2 2 2 2 2 2 [3,] 3 3 3 4 4 4 [4,] 4 4 4 5 5 5 [5,] 5 5 5 NA NA NA [6,] 6 6 6 NA NA NA [7,] 7 7 7 NA NA NA$s.prefs.hri
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    1    1    1    1    1    1    1
[2,]    2    2   NA    2    2   NA   NA

$c.prefs.hri [,1] [,2] [1,] 1 1 [2,] 2 2 [3,] 3 4 [4,] 4 5 [5,] 5 NA [6,] 6 NA [7,] 7 NA$matchings
matching college slots student sOptimal cOptimal sRank cRank
1        1       1     1       1        1        1     1     1
2        1       1     2       2        1        1     2     2
3        1       1     3       3        1        1     3     3
4        1       2     4       4        1        1     4     3
5        1       2     5       5        1        1     5     4

attr(,"class")
[1] "hri"

## 7 students, 3 colleges with 3 slots each, incomplete preferences:
hri(nStudents=7, nSlots=c(3,3,3), seed=21, s.range=c(1,3))
$s.prefs.smi [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 7 1 1 7 1 1 4 [2,] 8 2 2 8 2 2 5 [3,] 9 3 3 9 3 3 6 [4,] 1 NA NA 4 NA 4 7 [5,] 2 NA NA 5 NA 5 8 [6,] 3 NA NA 6 NA 6 9 [7,] NA NA NA NA NA 7 NA [8,] NA NA NA NA NA 8 NA [9,] NA NA NA NA NA 9 NA$c.prefs.smi
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,]    5    5    5    7    7    7    7    7    7
[2,]    6    6    6    6    6    6    4    4    4
[3,]    2    2    2    4    4    4    6    6    6
[4,]    3    3    3   NA   NA   NA    1    1    1
[5,]    1    1    1   NA   NA   NA   NA   NA   NA

$s.prefs.hri [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 3 1 1 3 1 1 2 [2,] 1 NA NA 2 NA 2 3 [3,] NA NA NA NA NA 3 NA$c.prefs.hri
[,1] [,2] [,3]
[1,]    5    7    7
[2,]    6    6    4
[3,]    2    4    6
[4,]    3   NA    1
[5,]    1   NA   NA

\$matchings
matching college slots student sOptimal cOptimal sRank cRank
1        1       1     3       2        1        1     3     3
2        1       1     1       5        1        1     1     1
3        1       1     2       6        1        1     2     2
4        1       2     4       7        1        1     1     1
5        1       3     8       1        1        1     2     4
6        1       3     7       4        1        1     1     2

attr(,"class")
[1] "hri"

## --------------------
## --- Summary plots

## 200 students, 200 colleges with 1 slot each
res <- hri(nStudents=200, nColleges=200, seed=12)
plot(res)

plot(res, energy=TRUE)