ECO 426 (Market Design) - Lecture 4 · ECO 426 (Market Design) - Lecture 4 Ettore Damiano October 5, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 4. Matching when only one

ECO 426 (Market Design) - Lecture 4

Ettore Damiano

October 5, 2015

Ettore Damiano ECO 426 (Market Design) - Lecture 4

Matching when only one side has preferences

Some allocation problems can be modelled as two-sided matchingmarkets but with one side not having any preferences over thepossible allocations




Housing market




Housing market (allocating houses to individuals)





A collection of individuals, A (agents)





A collection of individuals, A (agents)each agent a ∈ A:






owns a “house,” ha,






owns a “house,” ha, (H is the set of all houses);






owns a “house,” ha, (H is the set of all houses);has (strict) preferences over the set of houses in the economy







the initial allocation might not be efficient (i.e. Paretoefficient)







the initial allocation might not be efficient (i.e. Paretoefficient)

mutually beneficial trades might be possible


Housing Market

Housing market vs. marriage market


Housing Market


one side of the market (houses) has no preferences overmatches;


Housing Market


one side of the market (houses) has no preferences overmatches;agents have an initial endowment (i.e. each agent owns ahouse)


Housing Market



the market starts from a default allocation where each agentis matched to his own house


Housing Market




Goal: find a matching that cannot be improved


Housing Market




Goal: find a matching that cannot be improved

it is not possible to reassign houses making some agent betteroff and making no agent worse off


Housing Market - CORE

An allocation is an assignment (matching) of agents to housessuch that




each agent is assigned exactly one house; and




each agent is assigned exactly one house; andeach house is assigned to exactly one agent.





An allocation in an housing market is described by a“bijection” μ : A → H.






In a housing market, each agent is endowed (owns) one house(e.g. a owns ha)






In a housing market, each agent is endowed (owns) one house(e.g. a owns ha)

What allocations would we expect to arise if agents can freelydispose of their endowment?



Agents in a group S ⊆ A own together (in a “coalition”) asubset of the houses in the market HS




The agents in a coalition S can “independently” distribute thehouses they own, HS , among themselves.





An assignment of the houses in HS to agents in S , is anallocation in the housing market where the set of agents is Sand the set of houses is HS





An assignment of the houses in HS to agents in S , is anallocation in the housing market where the set of agents is Sand the set of houses is HS

μS : S → HS .



Definition (Blocking) A coalition of agents S blocks anallocation μ, if there is an assignment μS of the houses ownedby the coalition to the members of the coalition S , such that:



Definition (Blocking) A coalition of agents S blocks anallocation μ, if there is an assignment μS of the houses ownedby the coalition to the members of the coalition S , such that:i) some member of S prefers μS to μ;



Definition (Blocking) A coalition of agents S blocks anallocation μ, if there is an assignment μS of the houses ownedby the coalition to the members of the coalition S , such that:i) some member of S prefers μS to μ; ii) no member of Sprefers μ to μS .




A blocking coalition can find a mutually beneficial trade (i.e.an exchange of houses among members of the coalition thatimproves all members’ welfare with respect to the allocation μ)





Definition (Core) An allocation is in the core of the housingmarket if it is not blocked by any coalition.

At a core allocation benefits from trade are exhausted





Definition (Core) An allocation is in the core of the housingmarket if it is not blocked by any coalition.

At a core allocation benefits from trade are exhaustedIn a marriage market, core matchings and stable matchingscoincide


Housing Market - TTC

Gale’s Top trading cycle algorithm




each agent points tohis/her preferred house





each house points to itsowner





each house points to itsowner

a2 -

h2

6

a1��

h1

6

h3

?a3��=

a4PPPP

PPPPi

h4

6

a5�� h5

6a6CCCCCO

h6

6



there is at least one cycle




a2 -

h2

6

a1��

h1

6

h3

?a3��=

a4PPP

PPPPPi

h4

6

a5�� h5

6a6CCCCCO

h6

6




a2 -

h2

6

a1��

h1

6

h3

?a3��=

a4PPP

PPPPPi

h4

6

a5�� h5

6a6CCCCCO

h6

6

remove all cycles assigning houses to agents




a2 -

h2

6

a1��

h1

6

h3

?a3��=

a4PPP

PPPPPi

h4

6

a5�� h5

6a6CCCCCO

h6

6


agents within a cycle exchange houses among each others



each remaining agentpoints to his/her preferredremaining house



each remaining agentpoints to his/her preferredremaining house a4PP

PPPP

PPi

?h4

6

a5�� h5

6a6CCCCCO

@@@Rh6

6




PPPP

PPi

?h4

6

a5�� h5

6a6CCCCCO

@@@Rh6

6





PPPP

PPi

?h4

6

a5�� h5

6a6CCCCCO

@@@Rh6

6


continue until no agent/house is left


TTC and core

Theorem The outcome of the TTC mechanism is the unique coreallocation of the housing market.


TTC and core


The outcome of the TTC mechanism cannot be blocked


TTC and core



cannot make any agent matched in the first round better off


TTC and core



cannot make any agent matched in the first round better off(they are getting their favourite house)


TTC and core



cannot make any agent matched in the first round better off(they are getting their favourite house)cannot make any agent matched in the second round better offwithout making some of the agents matched in the first roundworse off


TTC and core



cannot make any agent matched in the first round better off(they are getting their favourite house)cannot make any agent matched in the second round better offwithout making some of the agents matched in the first roundworse offcannot make any agent matched in round n better off withoutmaking some agents matched in earlier rounds worse off.


TTC and strategic incentives

Theorem The TTC algorithm is a strategy proof mechanism.




an agent matched in round n cannot, by manipulating his/herpreferences, break any of the cycles that form before round n





preference manipulation cannot give the agent a house thatwas assigned earlier than round n.





preference manipulation cannot give the agent a house thatwas assigned earlier than round n.

getting an house that was assigned in a round later than ndoes not make the agent better off.


House allocation

An House allocation problem consists of


House allocation


A collection of N agents, A;


House allocation


A collection of N agents, A;A collection of N houses, H;


House allocation


A collection of N agents, A;A collection of N houses, H;For each agent (strict) a preference ordering over houses.

House allocation vs. housing market


House allocation




Same preference structure (i.e. only one side has preferences)


House allocation




Same preference structure (i.e. only one side has preferences)Same set of possible outcomes (i.e. assignment of a house toeach agent)


House allocation




Same preference structure (i.e. only one side has preferences)Same set of possible outcomes (i.e. assignment of a house toeach agent)Difference: agents do not “own” houses


House allocation




Same preference structure (i.e. only one side has preferences)Same set of possible outcomes (i.e. assignment of a house toeach agent)Difference: agents do not “own” houses (i.e. there is no initialallocation of houses to agents.)


House allocation





Goal: find an efficient assignment


House allocation





Goal: find an efficient assignment (i.e. one that cannot beimproved for all agents)


House allocation - mechanisms

Serial dictatorship



Serial dictatorship

Agents are given a priority ordering from 1 to N;



Serial dictatorship


Agents choose houses in order of their priority.



Serial dictatorship



Random serial dictatorship



Serial dictatorship




Same rules as in serial dictatorship;



Serial dictatorship





Priority ordering is chosen randomly from the set of all priorityorderings of agents.



Serial dictatorship






CORE from random assignment



Serial dictatorship







Randomly draw an initial assignment.



Serial dictatorship








Treat the initial assignment as an endowment.



Serial dictatorship








Treat the initial assignment as an endowment.

Use the TTC algorithm to find the unique core allocationgiven the initial assignment.


House allocation mechanisms - properties

Serial dictatorship, random serial dictatorship, and core fromrandom endowment are




Pareto efficient;




Pareto efficient; andStrategy proof.


House allocation with existing tenants

Housing market: each agent owns a house;




Housing allocation: no agents owns a house;





Housing allocation with existing tenants: some agents “own”houses others do not





Housing allocation with existing tenants: some agents “own”houses others do not

allocating student housing - with upper year students havingthe right to keep their current residence (i.e. “own” theirhouse)



Natural variation of the random serial dictatorship mechanism




Students without residence participate in a lottery determiningthe priority ordering;




Students without residence participate in a lottery determiningthe priority ordering;Students already in residence can:





keep their residence;





keep their residence; orgive up their residence and participate in the lottery.






Stock of available houses to choose from is formed by







empty houses;







empty houses; andhouses of students who choose to participate in the lottery.








Many US colleges use exactly this mechanism for assigningstudent residences









By participating in the lottery, a student in residence mightend up with a worse house










this might induce some students in residence not toparticipate in the lottery










this might induce some students in residence not toparticipate in the lotterythe outcome can be inefficient


YRMH-IGYT mechanism


YRMH-IGYT mechanism

Example: Existing tenants a1, a2, a3


YRMH-IGYT mechanism

Example: Existing tenants a1, a2, a3 with houses h1, h2, h3;


YRMH-IGYT mechanism

Example: Existing tenants a1, a2, a3 with houses h1, h2, h3;newcomers agents a4, a5, a6;


YRMH-IGYT mechanism

Example: Existing tenants a1, a2, a3 with houses h1, h2, h3;newcomers agents a4, a5, a6; empty houses hx , hy , hz ;


YRMH-IGYT mechanism

Example: Existing tenants a1, a2, a3 with houses h1, h2, h3;newcomers agents a4, a5, a6; empty houses hx , hy , hz ; randomlychosen priority ordering of agents {4,2,5,6,3,1}


YRMH-IGYT mechanism


assign agents their top choice inpriority order

a1

a2

a3

a4

a5

a6

h1

h2

h3

hx

hy

hz


YRMH-IGYT mechanism



a1

a2

a3

a5

a6

h1

h2

h3

hx

hz


YRMH-IGYT mechanism



a1

a3

a5

a6

h1

h2

h3

hz


YRMH-IGYT mechanism



until an agent requests anoccupied house

a1

a3

a6

h1

h3

hz


YRMH-IGYT mechanism




change the priority orderingplacing the existing tenantahead of requestor {4,2,5,3,6,1}

a1

a3

a6

h1

h3

hz


YRMH-IGYT mechanism





a1

a3

a6

h1

h3

hz


YRMH-IGYT mechanism





...repeat each time this happens{4,2,5,1,3,6}

a1

a3

a6

h1

h3

hz


YRMH-IGYT mechanism






a1

a3

a6

h1

h3

hz


YRMH-IGYT mechanism






a3

a6

h1

hz


YRMH-IGYT mechanism






a6 hz


YRMH-IGYT mechanism






YRMH-IGYT stands for: “You request my house - I get yourturn”


YRMH-IGYT properties

Theorem The YRMH-IGYT mechanism is Pareto efficient, strategyproof, and makes no existing tenant worse off.




Solve the participation problem





Relation to other mechanisms






Coincide with Serial dictatorship when no agent has a house






Coincide with Serial dictatorship when no agent has a houseCoincide with TTC when all agents are “tenants”






Coincide with Serial dictatorship when no agent has a houseCoincide with TTC when all agents are “tenants”IRMH-IGYT is a version of TTC with all unoccupied housespointing to the agent with the highest priority (among thoseremaining in the market)


Kidney Exchange


Kidney Exchange

Shortage of kidneys available for transplantation


Kidney Exchange


Yet, the “potential” supply (i.e. the number of not strictlyneeded functioning kidneys) vastly exceeds the demand


Kidney Exchange



How do we reduce the shortage?


Kidney Exchange




Increase the availability of cadaveric kidneys


Kidney Exchange




Increase the availability of cadaveric kidneysIncrease the supply of live donor kidneys


Kidney Exchange




Increase the availability of cadaveric kidneysIncrease the supply of live donor kidneys

cannot use any “price” mechanism because of legal (andmoral) constraints


Kidney donation

Live kidney donor must pass two compatibility tests before atransplantation is carried out


Kidney donation


Blood type match


Kidney donation


Blood type match

O-type patient can only accept O-type donorA-type patient can accept A and O-type donorB-type patient can accept B and O-type donorAB-type patient can accept all donors


Kidney donation


Blood type match


Tissue (HLA) compatibility


Kidney donation


Blood type match



Potential inefficiency: when a willing donor does not meet thecompatibility tests the kidney is “wasted”


Kidney donation


Blood type match




most donors are close friends/relatives, unwilling to donate toa stranger


Kidney donation


Blood type match




most donors are close friends/relatives, unwilling to donate toa stranger

Possible improvements: donor could be willing to donate tostranger if that improves the chances of their closefriend/relative receiving a kidney i.e a kidney exchange


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 at top of waitinglist in exchange of top 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 incetive todisclose their preferences honestly


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ECO 426 (Market Design) - Lecture 4 · ECO 426 (Market Design) - Lecture 4 Ettore Damiano October 5, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 4. Matching when only one