ECO 426 (Market Design) - Lecture 5 · Ettore Damiano ECO 426 (Market Design) - Lecture 5. Kidney exchange - model Assumptions: Multi-way exchanges: No constraint on the number of

ECO 426 (Market Design) - Lecture 5

Ettore Damiano

October 19, 2015

Ettore Damiano ECO 426 (Market Design) - Lecture 5

Exchanging Kidneys

Two types of kidney exchanges


Exchanging Kidneys


Pairwise kidney exchange:


Exchanging Kidneys


Pairwise kidney exchange: exchange kidney with anotherpatient-donor pair


Exchanging Kidneys



t1

k1

t2

k2


Exchanging Kidneys



t1

k1

t2

k2


Exchanging Kidneys



t1

k1

t2

k2


Exchanging Kidneys



t1

k1

t2

k2

Exchange to list: donate kidney to patient on waiting list inexchange of a better spot on waiting list


Exchanging Kidneys



t1

k1

t2

k2


t1

k1

w1

w2

w3

w4....


Exchanging Kidneys



t1

k1

t2

k2


t1

k1

w1

w2

w3

w4....


Exchanging Kidneys



t1

k1

t2

k2


t1

k1

w1

w2

w3

w4....


Exchanging Kidneys



t1

k1

t2

k2


t1

k1

w1

w2

w3

w4....Looks similar to YRMH-IGYT


Kidney exchange problem

A Kidney exchange problem consists of:




A set of donor-patient pairs {(t1, k1), . . . , (tn, kn)}




A set of donor-patient pairs {(t1, k1), . . . , (tn, kn)}For each patient, ti , a set of compatible kidneysKi⊆K={k1,. . . ,kn}




A set of donor-patient pairs {(t1, k1), . . . , (tn, kn)}For each patient, ti , a set of compatible kidneysKi⊆K={k1,. . . ,kn}For each patient, ti , a (strict) preference ordering over the setof compatible kidneys Ki




A set of donor-patient pairs {(t1, k1), . . . , (tn, kn)}For each patient, ti , a set of compatible kidneysKi⊆K={k1,. . . ,kn}For each patient, ti , a (strict) preference ordering over the setof compatible kidneys Ki and the option of exchanging ownkidney, ki for priority w on the waiting list





Question: How do we organize a kidney exchange programsuch that






The outcome is Pareto efficient,






The outcome is Pareto efficient, it is not possible to improvefurther the welfare of all






The outcome is Pareto efficient, it is not possible to improvefurther the welfare of allFor each patient, the outcome is never worse than notparticipating in the mechanism,






The outcome is Pareto efficient, it is not possible to improvefurther the welfare of allFor each patient, the outcome is never worse than notparticipating in the mechanism, ensures broad participation, nodonor kidney is un-necessarily “wasted”






The outcome is Pareto efficient, it is not possible to improvefurther the welfare of allFor each patient, the outcome is never worse than notparticipating in the mechanism, ensures broad participation, nodonor kidney is un-necessarily “wasted”The mechanism is strategy proof,






The outcome is Pareto efficient, it is not possible to improvefurther the welfare of allFor each patient, the outcome is never worse than notparticipating in the mechanism, ensures broad participation, nodonor kidney is un-necessarily “wasted”The mechanism is strategy proof, patients have incentive todisclose their preferences honestly


Kidney exchange - model

Assumptions:



Assumptions:Multi-way exchanges:



Assumptions:Multi-way exchanges: No constraint on the number ofpatient-donor pairs that can participate in an exchange (i.e.multi-way exchanges are allowed)




t1

k1

t2k2

k3

t3k4t4




t1

k1

t2k2

k3

t3k4t4

List exchanges:




t1

k1

t2k2

k3

t3k4t4

List exchanges: Exchange to list are possible (i.e. exchanging akidney for a better spot on the wait-list)




t1

k1

t2k2

k3

t3k4t4

List exchanges: Exchange to list are possible (i.e. exchanging akidney for a better spot on the wait-list)Strict preferences:




t1

k1

t2k2

k3

t3k4t4

List exchanges: Exchange to list are possible (i.e. exchanging akidney for a better spot on the wait-list)Strict preferences: No patient is indifferent between any two(compatible) alternatives




t1

k1

t2k2

k3

t3k4t4

List exchanges: Exchange to list are possible (i.e. exchanging akidney for a better spot on the wait-list)Strict preferences: No patient is indifferent between any two(compatible) alternatives (i.e. strict preferences over:compatible kidneys + trading donor for wait list + remainingwith own donor)




t1

k1

t2k2

k3

t3k4t4


Interpreting live donors’ kidneys as “owned” by theirrespective patients, the problem resembles one of houseallocation with existing tenants




t1

k1

t2k2

k3

t3k4t4


Interpreting live donors’ kidneys as “owned” by theirrespective patients, the problem resembles one of houseallocation with existing tenants

Maximizing “supply” of live donors as maximizing participation


TTC(and C)

TTC mechanism key properties


TTC(and C)

TTC mechanism key propertiesEach patient points to favorite kidney or the waiting list


TTC(and C)

TTC mechanism key propertiesEach patient points to favorite kidney or the waiting listEach kidney donor points to his/her patient


TTC(and C)


In a given round

t1k1

t5k5t3k3

t4 k4

t2 k2


TTC(and C)


In a given roundA cycle might form

t1k1

t5k5t3k3

t4 k4

t2 k2


TTC(and C)



each patient in thecycle receives thebest compatiblekidney available

t1k1

t5k5t3k3

t4 k4

t2 k2


TTC(and C)




A cycle might not form

t1k1

t5k5t3k3

t4 k4

t2 k2

t1k1 t2k2

t3k3

t4 k4. . .tn kn

w


TTC(and C)





some patient pointto wait-list

t1k1

t5k5t3k3

t4 k4

t2 k2

t1k1 t2k2

t3k3

t4 k4. . .tn kn

w


TTC(and C)





some patient pointto wait-list

If there is no cyclethere must be at leasta w -chain

t1k1

t5k5t3k3

t4 k4

t2 k2

t1k1 t2k2

t3k3

t4 k4. . .tn kn

w


TTC(and C)

Top Trading Cycles and Chains mechanism: key ideas


TTC(and C)


When a cycle form:


TTC(and C)


When a cycle form:

Carry out exchange


TTC(and C)


When a cycle form:

Carry out exchangeRemove kidneys and patients in cycle and restart


TTC(and C)


When a cycle form:


When no cycle form


TTC(and C)


When a cycle form:


When no cycle form

There can be more than one chain


TTC(and C)


When a cycle form:


When no cycle form


w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2


TTC(and C)


When a cycle form:


When no cycle form


w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2

Multiple chains can be in “competition” with each other


TTC(and C)


When a cycle form:


When no cycle form


w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2

Multiple chains can be in “competition” with each other

Need a chain selection rule


chain selection

Examples of chain selection rules


chain selection


Choose based on length

w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2


chain selection



longest chain

w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2


chain selection



longest chainminimal chain

w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2


chain selection




Choose based ondonor-patient priority

w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2


chain selection




Choose based ondonor-patient priority

Choose chain with thehighest prioritydonor-patient pair(e.g. t3, k3)

w t1 k1

t5 k5

t3 k3

t4 k4

t2 k2


“tail kidney”

The “tail kidney” in a chain (i.e. the kidney of the lastpatient receiving a transplant in the kidney exchange) is notstrictly needed for the exchange


“tail kidney”


w t1 k1

t5 k5

t3 k3

t4 k4t2 k2


“tail kidney”


w t1 k1

t5 k5

t3 k3

t4 k4t2 k2

The tail kidney can be


“tail kidney”


w t1 k1

t5 k5

t3 k3

t4 k4t2 k2


Assigned to some compatible patient on wait list (i.e. listexchange)


“tail kidney”


w t1 k1

t5 k5

t3 k3

t4 k4t2 k2


Assigned to some compatible patient on wait list (i.e. listexchange) (might have welfare consequences, Pareto efficiencyis not guaranteed)


“tail kidney”


w t1 k1

t5 k5

t3 k3

t4 k4t2 k2


Assigned to some compatible patient on wait list (i.e. listexchange) (might have welfare consequences, Pareto efficiencyis not guaranteed)Remain available to remaining patients on the kidney exchangeprogram


“tail kidney”


w t1 k1

t5 k5

t3 k3

t4 k4t2 k2


Assigned to some compatible patient on wait list (i.e. listexchange) (might have welfare consequences, Pareto efficiencyis not guaranteed)Remain available to remaining patients on the kidney exchangeprogram (guarantees Pareto efficiency)


chain selection

Combining chain selection and tail kidney options


chain selection


1 Choose longest chain and remove tail kidneys


chain selection


1 Choose longest chain and remove tail kidneys (not strategyproof,


chain selection


1 Choose longest chain and remove tail kidneys (not strategyproof, not Pareto efficient)


chain selection



2 Choose longest chain and keep tail kidney


chain selection



2 Choose longest chain and keep tail kidney (not strategy proof,


chain selection



2 Choose longest chain and keep tail kidney (not strategy proof,Pareto efficient)


chain selection




3 Choose minimal chain and keep tail kidney


chain selection




3 Choose minimal chain and keep tail kidney (strategy proof


chain selection




3 Choose minimal chain and keep tail kidney (strategy proof ,Pareto efficient)


chain selection





4 Choose chain starting with highest priority patient-donor pairand remove tail kidney


chain selection





4 Choose chain starting with highest priority patient-donor pairand remove tail kidney (Strategy proof,


chain selection





4 Choose chain starting with highest priority patient-donor pairand remove tail kidney (Strategy proof, not Pareto efficient)


chain selection






5 Choose chain starting with highest priority patient-donor pairand keep tail kidney


chain selection






5 Choose chain starting with highest priority patient-donor pairand keep tail kidney (Strategy proof,


chain selection






5 Choose chain starting with highest priority patient-donor pairand keep tail kidney (Strategy proof, Pareto efficient)


chain selection






5 Choose chain starting with highest priority patient-donor pairand keep tail kidney (Strategy proof, Pareto efficient) -equivalent to YRMH-IGYT


chain selection







Key properties


chain selection







Key properties

Minimal chains for strategy proofness


chain selection







Key properties

Minimal chains for strategy proofnessKeep kidney for Pareto efficiency


Constrained kidney exchange

Practical shortcomings:




Multi-way exchanges can be difficult to implement




Multi-way exchanges can be difficult to implementBeing illegal to enter a contractual agreement for a “kidneyexchange” all surgeries must be performed simultaneously toensure compliance with the agreed exchange





Pairwise kidney exchange requires four “simultaneous”surgeries





Pairwise kidney exchange requires four “simultaneous”surgeries (two nephrectomies two kidney transplants)





Pairwise kidney exchange requires four “simultaneous”surgeries (two nephrectomies two kidney transplants)Trilateral exchange requires six “simultaneous” surgeries etc.






Preferences are “in practice” not strict







Compatibility is treated as a binary variable (0-1)








List exchanges pose a “selection” problem









Most common blood type is O-type









Most common blood type is O-typeMost likely donor kidney exchanged to wait-list will be O-typeincompatible (otherwise the donating patient would have it)









Most common blood type is O-typeMost likely donor kidney exchanged to wait-list will be O-typeincompatible (otherwise the donating patient would have it)List exchanges may harm O-type patient on wait-list


Pairwise Kidney exchange with binary preferences

A constrained bilateral Kidney exchange problem with binary(compatibility based) preferences consists of:




A set of donor-patient pairs {(t1, k1), . . . , (tn, kn)}









The set of agents N and a compatibility matrix, R , suffice todescribe the problem






R is an N × N matrix






R is an N × N matrix with

ri,j =

{1 if i and j are compatible

0 otherwise






R is an N × N matrix with

ri,j =

{1 if i and j are compatible

0 otherwise

Objective: Find a collection of bilateral kidney exchangeamong mutually compatible donor-patient pairs


Priority Mechanism


Priority Mechanism

Order donor-patient pairs according to priorities


Priority Mechanism

Order donor-patient pairs according to prioritiesExample: Nine patient-donor pairs {t1, t2, . . . , t9}


Priority Mechanism

Order donor-patient pairs according to prioritiesExample: Nine patient-donor pairs {t1, t2, . . . , t9} priorityordering {1, 8, 4, 2, 6, 3, 7, 9, 5}


Priority Mechanism


medical priority and/or random


Priority Mechanism



t1 t2

t5 t4

t6

t3 t8

t7 t9


Priority Mechanism



t1 t2

t5 t4

t6

t3 t8

t7 t9


Priority Mechanism



match top priority patient if possible(i.e. if there is a patient-donor pairmutually compatible with the priority1 patient-donor pair), else skip

t1 t2

t5 t4

t6

t3 t8

t7 t9


Priority Mechanism




match priority 2 patient, if possible, inconjunction with priority 1 agent elseskip priority 2 agent

t1 t2

t5 t4

t6

t3 t8

t7 t9


Priority Mechanism





. . .

match priority n patient, if possible, inconjunction with all earlier priorities,else skip

t1 t2

t5 t4

t6

t3 t8

t7 t9


Priority Mechanism





. . .

match priority n patient, if possible, inconjunction with all earlier priorities,else skip

t1 t2

t5 t4

t6

t3 t8

t7 t9


Priority Mechanism

The priority mechanism is


Priority Mechanism


Pareto efficient


Priority Mechanism


Pareto efficientStrategy proof


Priority Mechanism



Limits:


Priority Mechanism



Limits:

Allowing tri-lateral exchanges can make many more transplantspossible


Priority Mechanism



Limits:

Allowing tri-lateral exchanges can make many more transplantspossibleAdditional benefits from more complex multi-lateral exchangesdecline rapidly


Priority Mechanism



Limits:


Example: Blood incompatible pairs (O-B,O-A,A-B,A-B,B-A);


Priority Mechanism



Limits:


Example: Blood incompatible pairs (O-B,O-A,A-B,A-B,B-A);HLA incompatible pairs (A-A,A-A,A-A,B-O)


Priority Mechanism



Limits:



Only bilateral exchanges:


Priority Mechanism



Limits:



Only bilateral exchanges: (A-B,B-A) (A-A,A-A) (O-B,B-O)


Priority Mechanism



Limits:



Only bilateral exchanges: (A-B,B-A) (A-A,A-A) (O-B,B-O)Bilateral and trilateral:


Priority Mechanism



Limits:



Only bilateral exchanges: (A-B,B-A) (A-A,A-A) (O-B,B-O)Bilateral and trilateral: (A-B,B-A) (A-A,A-A,A-A)(B-O,O-A,A-B)


Kidney exchange programs

New England Program for Kidney Exchange (2004)




Priority mechanism




Priority mechanismup to 4-lateral exchanges




Priority mechanismup to 4-lateral exchangeslist exchanges allowed




Priority mechanismup to 4-lateral exchangeslist exchanges allowedaltruistic donor exchanges




Priority mechanismup to 4-lateral exchangeslist exchanges allowedaltruistic donor exchanges (i.e. chains starting from analtruistic live donor, rather than a list exchange)





Ohio - Living Kidney Donor Program






Performed a Six-way paired kidney exchange (September 2011)






Performed a Six-way paired kidney exchange (September 2011)

National program under construction


NEAD

NEAD: Never Ending Altruistic Donor Chain


NEAD


Alliance for Paired Donation


NEAD


Alliance for Paired Donation - 10 kidney transplant chain


NEAD



National Kidney Registry


NEAD



National Kidney Registry - 30 kidney transplant chain


Documents

ECO 426 (Market Design) - Lecture 5 · Ettore Damiano ECO 426 (Market Design) - Lecture 5. Kidney exchange - model Assumptions: Multi-way exchanges: No constraint on the number of