Check if a two-sided matching is stable

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Description

This function checks if a given matching is stable for a particular set of preferences. This stability check can be applied to both the stable marriage problem and the college admission problem. The function requires preferences to be specified in cardinal form. If necessary, the function rankIndex can be used to turn ordinal preferences into cardinal utilities.

Usage

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galeShapley.checkStability(proposerUtils, reviewerUtils, proposals, engagements)

Arguments

proposerUtils

is a matrix with cardinal utilities of the proposing side of the market. If there are n proposers and m reviewers, then this matrix will be of dimension m by n. The i,jth element refers to the payoff that proposer j receives from being matched to reviewer i.

reviewerUtils

is a matrix with cardinal utilities of the courted side of the market. If there are n proposers and m reviewers, then this matrix will be of dimension n by m. The i,jth element refers to the payoff that reviewer j receives from being matched to proposer i.

proposals

is a matrix that contains the number of the reviewer that a given proposer is matched to: the first row contains the reviewer that is matched to the first proposer, the second row contains the reviewer that is matched to the second proposer, etc. The column dimension accommodates proposers with multiple slots.

engagements

is a matrix that contains the number of the proposer that a given reviewer is matched to. The column dimension accommodates reviewers with multiple slots.

Value

true if the matching is stable, false otherwise

Examples

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# define cardinal utilities
uM = matrix(c(0.52, 0.85,
              0.96, 0.63,
              0.82, 0.08,
              0.55, 0.34), nrow = 4, byrow = TRUE)
uW = matrix(c(0.76, 0.88, 0.74, 0.02,
              0.32, 0.21, 0.02, 0.79), ncol = 4, byrow = TRUE)
# define matching
results = list(
     proposals = matrix(c(2, 1), ncol = 1),
     engagements = matrix(c(2, 1, NA, NA), ncol = 1))
# check stability
galeShapley.checkStability(uM, uW, results$proposals, results$engagements)

# if preferences are in ordinal form, we can use galeShapley.validate
# to transform them into cardinal form and then use checkStability()
prefM = matrix(c(2, 1,
                 3, 2,
                 4, 4,
                 1, 3), nrow = 4, byrow = TRUE)
prefW = matrix(c(1, 1, 1, 2,
                 2, 2, 2, 1), ncol = 4, byrow = TRUE)
# define matching
results = list(proposals = matrix(c(2, 1), ncol = 1),
               engagements = matrix(c(2, 1, NA, NA), ncol = 1))
# check stability
pref.validated = galeShapley.validate(proposerPref = prefM,
                                      reviewerPref = prefW)
galeShapley.checkStability(pref.validated$proposerUtils,
                           pref.validated$reviewerUtils,
                           results$proposals,
                           results$engagements)

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