Mix and Match

1. Mix and Match ItaiAshlagi, Felix Fischer, Ian Kash, Ariel Procaccia (Harvard SEAS)

2. Kidney Exchange • 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

3. The model (informally) • 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

4. A lower bound (to what?) Theorem: If there are at least two agents: No det. SP mechanism can give better than 2-approx to social welfare No rand. SP mechanism can give better than 4/3-approx to social welfare

5. A strategyproof mechanism • 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 between different agents on the same side of the partition • Among these return a matching with max cardinality (need tie breaking)

6. Example

7. Results • Theorem (main): Matchis SP for any number of agents 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 gives a 2-approx (!)

8. Discussion • Very attractive open problems! • Practical kidney exchange considerations • Evidence that hospitals are behaving strategically • Mix-and-Match gives ~ 90% efficiency

9. 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

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

11. 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

12. Questions?