Stable Matching Oct. 11, 2018
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StableMatchingOct.11,2018
MaximumMatchingAmaximummatching isthelargestsubsetofagraph’sedgessuchthatnotwoedgesshareavertex.
Aretherededgesamaximummatching?
FindingMaximumMatchingsAnaugmentingpath startsandwithedgesdistinctnodesandedgesnotinthematching,thealternatesbetweenedgesinandnotinthematching.
Allnon-maximummatchingscanbeenlargedbyfindinganaugmentingpath.
IF
KidneyExchangeMechanism• AskeachPi Di pairfor
• BuildagraphG = (V,E) with
• ComputeandreturnamaximummatchingofG.
Ei = {PjDj | Dj can donate to Pi}<latexit sha1_base64="89JqWZT2FT6g/Zxbot72u119j40=">AAACIXicbZBNSwMxEIazflu/qh69BIvgSXZFsAcFQQWPFawtdMuSzU41mmSXZFYsS/+KF/+KFw+KeBP/jOnHQVtfGHh4Z4Zk3jiTwqLvf3lT0zOzc/MLi6Wl5ZXVtfL6xrVNc8OhzlOZmmbMLEihoY4CJTQzA0zFEhrx/Wm/33gAY0Wqr7CbQVuxGy06gjN0VlSunkeCHtOwqEV39MxVqEQyBIRHNKqgnGmapJohUExpj9bcRtiLyhV/zx+ITkIwggoZqRaVP8Mk5bkCjVwya1uBn2G7YAYFl9ArhbmFjPF7dgMth5opsO1icGGP7jgnoZ3UuNJIB+7vjYIpa7sqdpOK4a0d7/XN/3qtHDvVdiF0liNoPnyok8v+pf24aCIMcJRdB4wb4f5K+S0zjKMLteRCCMZPnoTr/b3A8eVB5eRoFMcC2SLbZJcE5JCckAtSI3XCyRN5IW/k3Xv2Xr0P73M4OuWNdjbJH3nfPz/kobg=</latexit><latexit sha1_base64="89JqWZT2FT6g/Zxbot72u119j40=">AAACIXicbZBNSwMxEIazflu/qh69BIvgSXZFsAcFQQWPFawtdMuSzU41mmSXZFYsS/+KF/+KFw+KeBP/jOnHQVtfGHh4Z4Zk3jiTwqLvf3lT0zOzc/MLi6Wl5ZXVtfL6xrVNc8OhzlOZmmbMLEihoY4CJTQzA0zFEhrx/Wm/33gAY0Wqr7CbQVuxGy06gjN0VlSunkeCHtOwqEV39MxVqEQyBIRHNKqgnGmapJohUExpj9bcRtiLyhV/zx+ITkIwggoZqRaVP8Mk5bkCjVwya1uBn2G7YAYFl9ArhbmFjPF7dgMth5opsO1icGGP7jgnoZ3UuNJIB+7vjYIpa7sqdpOK4a0d7/XN/3qtHDvVdiF0liNoPnyok8v+pf24aCIMcJRdB4wb4f5K+S0zjKMLteRCCMZPnoTr/b3A8eVB5eRoFMcC2SLbZJcE5JCckAtSI3XCyRN5IW/k3Xv2Xr0P73M4OuWNdjbJH3nfPz/kobg=</latexit><latexit sha1_base64="89JqWZT2FT6g/Zxbot72u119j40=">AAACIXicbZBNSwMxEIazflu/qh69BIvgSXZFsAcFQQWPFawtdMuSzU41mmSXZFYsS/+KF/+KFw+KeBP/jOnHQVtfGHh4Z4Zk3jiTwqLvf3lT0zOzc/MLi6Wl5ZXVtfL6xrVNc8OhzlOZmmbMLEihoY4CJTQzA0zFEhrx/Wm/33gAY0Wqr7CbQVuxGy06gjN0VlSunkeCHtOwqEV39MxVqEQyBIRHNKqgnGmapJohUExpj9bcRtiLyhV/zx+ITkIwggoZqRaVP8Mk5bkCjVwya1uBn2G7YAYFl9ArhbmFjPF7dgMth5opsO1icGGP7jgnoZ3UuNJIB+7vjYIpa7sqdpOK4a0d7/XN/3qtHDvVdiF0liNoPnyok8v+pf24aCIMcJRdB4wb4f5K+S0zjKMLteRCCMZPnoTr/b3A8eVB5eRoFMcC2SLbZJcE5JCckAtSI3XCyRN5IW/k3Xv2Xr0P73M4OuWNdjbJH3nfPz/kobg=</latexit><latexit sha1_base64="89JqWZT2FT6g/Zxbot72u119j40=">AAACIXicbZBNSwMxEIazflu/qh69BIvgSXZFsAcFQQWPFawtdMuSzU41mmSXZFYsS/+KF/+KFw+KeBP/jOnHQVtfGHh4Z4Zk3jiTwqLvf3lT0zOzc/MLi6Wl5ZXVtfL6xrVNc8OhzlOZmmbMLEihoY4CJTQzA0zFEhrx/Wm/33gAY0Wqr7CbQVuxGy06gjN0VlSunkeCHtOwqEV39MxVqEQyBIRHNKqgnGmapJohUExpj9bcRtiLyhV/zx+ITkIwggoZqRaVP8Mk5bkCjVwya1uBn2G7YAYFl9ArhbmFjPF7dgMth5opsO1icGGP7jgnoZ3UuNJIB+7vjYIpa7sqdpOK4a0d7/XN/3qtHDvVdiF0liNoPnyok8v+pf24aCIMcJRdB4wb4f5K+S0zjKMLteRCCMZPnoTr/b3A8eVB5eRoFMcC2SLbZJcE5JCckAtSI3XCyRN5IW/k3Xv2Xr0P73M4OuWNdjbJH3nfPz/kobg=</latexit>
V = {PiDi pairs}<latexit sha1_base64="yjejohkouEt2O/Vspp+xVDxdPKM=">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</latexit><latexit sha1_base64="yjejohkouEt2O/Vspp+xVDxdPKM=">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</latexit><latexit sha1_base64="yjejohkouEt2O/Vspp+xVDxdPKM=">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</latexit><latexit sha1_base64="yjejohkouEt2O/Vspp+xVDxdPKM=">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</latexit>
E = {(i, j) | PjDj 2 Ei ^ PiDi 2 Ej}<latexit sha1_base64="+YAOoz1LvsamGbOzPOhaur6HXLM=">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</latexit><latexit sha1_base64="+YAOoz1LvsamGbOzPOhaur6HXLM=">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</latexit><latexit sha1_base64="+YAOoz1LvsamGbOzPOhaur6HXLM=">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</latexit><latexit sha1_base64="+YAOoz1LvsamGbOzPOhaur6HXLM=">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</latexit>
IsthismechanismDSIC?ShouldagentsreportFi �Ei ?
ShouldagentsreportFi �Ei ?
Depends on the alg
tf
PriorityMatchingSupposepatientsarerankedbypriority:• Howlongthey’vebeenwaitingforatransplant.• Howbadlytheyneedatransplant.
Howcanwemodifythemechanismtoaccountforpriorities?
IncentivesforHospitals• Hospitalswanttomatchtheirownpatients.• Theycanidentifymatchesbetweentheirownpatients.
Whatarethemaximummatchings?
FEI
EE l
01111 I
C
TheoremTherecannotexistamechanismthatmaximizespatientwelfare(findsamaximummatching)andisDSICforthehospitals(ensurestheyreporttruthfullyaboutalltheirpatients).
Proof:thecounterexamplewejustsaw.
ResidencyMatchingEveryyear,thousandsofdoctorsfinishmedicalschoolandapplytoresidencyprograms.
Doctorsarematchedtohospitalsbyacentralizedmechanism,whichwewillmodel.
D = setofdoctorsH = setofhospitals
| D | = | H |
Ps3 lol IH
Doctorsreport:
Hospitalsreport:
Giventhesereports,themechanismcomputesastablematching.
preference ordering of all hospitals
D Hz Hit H Hz Hs
preference ordering of all doctors
Hi Ds Dz Dz D Da
A matching is stable if it hasNo blocking pairsA pair d h is blocking if
each prefers the other over the mechanism
Deferred Acceptance Algorithmwhile 7 unmatched d e D
d applies to favorite heftthat hasn't rejected them
III
isi tchn a
d't h's current matchif h prefers dy dh permanently rejects dtentatively natch d h
elsepermanently rejects d
finalize matches
Runningtimeworst case lol I Hl
WhatsthIs it a matching
Is it stableDoctors go down
their list
Hospitals move up by re matching
d h
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