MIX AND MATCH
Itai Ashlagi, Felix Fischer, Ian Kash,
Ariel Procaccia (Harvard SEAS)
Kidney Exchange
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Many types of kidney disease requiretransplantation
Potential donors sometimes incompatible withpatient
Pairs of incompatible donor-patient pairs cansometimes exchange kidneys
Previous work considered the donor/patientincentives
[Roth+Sonmez+Unver] Hospitals’ incentives maybecome a problem
The model (informally)
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Set of agents (hospitals)
Undirected graph
Vertices = donor-patient pairs
Edges = compatibility
Each agent controls subset of vertices
Mechanism receives a graph and returns a matching
No payments!
Utility of agent = number of its matched vertices
Target: # matched vertices = social welfare
Agents can hide vertices and match them later
But graph is public knowledge
Mechanism is strategyproof (SP) if it is a dominant strategyto reveal all vertices
Theorem: If there are at least two agents:
1.No det. SP mechanism can give better than 2-approx to social welfare
2.No rand. SP mechanism can give better than 4/3-approx to social welfare
A lower bound (to what?)
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![C:\Documents and Settings\Aprocaccia\Local Settings\Temporary Internet Files\Content.IE5\V4AWB8CZ\MC900434713[1].wmf](data/images/img7.png){kind=small}
![C:\Documents and Settings\Aprocaccia\Local Settings\Temporary Internet Files\Content.IE5\V4AWB8CZ\MC900434713[1].wmf](data/images/img8.png){kind=small}
A strategyproof mechanism
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Let = (1,2) be a bipartition of the agents
The MATCH mechanism:
Consider matchings that maximize the number of“internal edges” and do not have any edges betweendifferent agents on the same side of the partition
Among these return a matching with max cardinality(need tie breaking)
Example
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Results
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Theorem (main): MATCH is SP for any number ofagents and any partition
For two agents MATCH{1},{2} gives a 2-approx
For more gives no approximation
The MIX-AND-MATCH mechanism:
Mix: choose a random partition
Match: Execute MATCH
Theorem: MIX-AND-MATCH is universally SP and givesa 2-approx (!)
Discussion
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Very attractive open problems!
Practical kidney exchange considerations
Evidence that hospitals are behaving strategically
MIX-AND-MATCH gives ~ 90% efficiency
Approximate MD Without Money
[Procaccia and Tennenholtz.Approximate mechanism designwithout money. In EC’09]
Session: Approximate mechanismdesign without money
Algorithmic mechanism designwas introduced by Nisan andRonen [STOC’99]
The field deals with designingtruthful approximationmechanisms for game-theoreticversions of optimizationproblems
All the work in the field considersmechanisms with payments
Money unavailable in manysettings
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Opt SPmech withmoney +tractable
Class 1
Opt SP mechanism withmoney
Problem intractable
Class 2
No opt SP mechwith money
Class 3
No opt SP mech
w/o money
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Some cool animations
Variety of domains
Kidney exchange
Ashlagi+Kash+Fischer+P [EC’10]
Regression learning and classification
Dekel+Fischer+P [SODA’08JCSS]
Meir+P+Rosenschein [AAAI’08, IJCAI’09, AAMAS’10]
Facility location
P+Tennenholtz [EC’09], Alon+Feldman+P+Tennenholtz [MOR],Nissim+Smorodinsky+Tennenholtz
Lu+Wang+Zhou [WINE’09], Lu+Sun+Wang+Zhu [EC’10]
Allocation of items
Guo+Conitzer [AAMAS’10]
Generalized assignment
Dughmi+Ghosh [EC’10]
Approval
Alon+Fischer+P+Tennenholtz
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