2. MIX AND MATCH Itai Ashlagi, Felix Fischer, Ian Kash, Ariel Procaccia (Harvard SEAS)

3. Kidney Exchange 2  Many types of kidney disease require transplantation  Potential donors sometimes incompatible with patient  Pairs of incompatible donor-patient pairs can sometimes exchange kidneys  Previous work considered the donor/patient incentives  [Roth+Sonmez+Unver] Hospitals’ incentives may become a problem

4. The model (informally) 3  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 strategy to reveal all vertices

5. 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?) 4

6. A strategyproof mechanism 5  Let  = (  1,  2 ) be a bipartition of the agents  The M ATCH  mechanism:  Consider matchings that maximize the number of “internal edges” and do not have any edges between different agents on the same side of the partition  Among these return a matching with max cardinality (need tie breaking)

7. Example 6

8. Results 7  Theorem (main): M ATCH  is SP for any number of agents and any partition   For two agents M ATCH {1},{2} gives a 2-approx  For more gives no approximation  The M IX - AND -M ATCH mechanism:  Mix: choose a random partition   Match: Execute M ATCH   Theorem: M IX - AND -M ATCH is universally SP and gives a 2-approx (!)

9. Discussion 8  Very attractive open problems!  Practical kidney exchange considerations  Evidence that hospitals are behaving strategically  M IX - AND -M ATCH gives ~ 90% efficiency

10. Approximate MD Without Money  [Procaccia and Tennenholtz. Approximate mechanism design without money. In EC’09]  Session: Approximate mechanism design without money  Algorithmic mechanism design was introduced by Nisan and Ronen [STOC’99]  The field deals with designing truthful approximation mechanisms for game-theoretic versions of optimization problems  All the work in the field considers mechanisms with payments  Money unavailable in many settings 9

11. Opt SP mech with money + tractable Class 1 Opt SP mechanism with money Problem intractable Class 2 No opt SP mech with money Class 3 No opt SP mech w/o money 10 Some cool animations

12. Variety of domains  Kidney exchange  Ashlagi+Kash+Fischer+P [EC’10]  Regression learning and classification  Dekel+Fischer+P [SODA’08  JCSS]  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 11