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Approximate Mechanism Design Without Money

Approximate Mechanism Design Without Money. Ariel Procaccia Center for Research on Computation and Society Harvard University. Algorithmic game theory. Computing solution concepts Quantifying the inefficiency of equilibria

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Approximate Mechanism Design Without Money

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  1. Approximate Mechanism Design Without Money Ariel Procaccia Center for Research on Computation and Society Harvard University

  2. Algorithmic game theory • Computing solution concepts • Quantifying the inefficiency of equilibria • Incentives in communication networks, P2P systems, and reputation systems • Computational mechanism design

  3. CS and GT: Ying and Yang

  4. Overview Facility location Approximate mechanism design w/o money Approximate mechanism design w/o money Kidney exchange Kidney exchange

  5. The model • Want to locate a public facility (library, train station, fire station) on a street • Players A, B, C,... report their ideal locations • A mechanism receives the reported locations as input, and returns the location of the facility • Given facility location, cost of a player = his distance from the facility

  6. Take 1: average • Suppose we have two players, A and B • Mechanism: take the average • A mechanism is strategyproof if players can never benefit from lying = the distance from their location cannot decrease by misreporting it • Problem: average is not strategyproof

  7. Take 2: leftmost location • Mechanism: select the leftmost reported location • Mechanism is strategyproof A B B C D E

  8. Minimizing the maximum cost • Maximum cost of facility location = max distance to the players • Example: facility is a fire station • Optimal solution = average of leftmost and rightmost locations; OPT = d(A,E)/2 • Mechanism gives -approximationif for every instance, MECH/OPT   • Leftmost location gives 2-approximation • Theorem:There is no deterministic strategyproof mechanism with approx ratio smaller than 2 A C D E B

  9. The Left-Right-Middle Mechanism • Left-Right-Middle (LRM) Mechanism: select leftmost location with prob. ¼, rightmost with prob. ¼, and average with prob. ½ • Approx ratio is[½  (2  OPT) + ½  OPT] / OPT = 3/2 • LRM mechanism is strategyproof • Theorem:There is no randomized strategyproof mechanism with approximation ratio better than 3/2 2  B B C A D E 1/2 1/4 1/2 1/4 1/4

  10. Facility location on a network • Players located on a network, represented as graph [SV04] • Examples: • Network of roads in a city • Telecommunications network: • Line • Hierarchical (tree) • Ring (circle) • Scheduling a daily task: circle A B C

  11. LRM on a circle • Semicircle like an interval on a line • If all players are on one semicircle, can apply LRM • Meaningless otherwise 1/4 A B C F D 1/2 E 1/4

  12. Random Midpoint • Look at points antipodal to players’ locations • Random Midpoint Mechanism: choose midpoint of arc between two antipodal points with prob. proportional to length • Theorem: mechanism is strategyproof • Approx ratio 3/2 if players are not on one semicircle, but  3 if they are B 1/4 A 3/8 C C A B 3/8

  13. A hybrid mechanism • Mechanism: • If players are on one semicircle, use LRM Mechanism • If players are not on one semicircle, use Random Midpoint Mechanism • Theorem: Mechanism is SP and gives 3/2-approximation when network is a circle • Lower bound of 3/2 holds on a circle

  14. Overview Facility location Facility location Approximate mechanism design w/o money Kidney exchange Kidney exchange

  15. Approx MD w/o money for CS • Algorithmic mechanism design (AMD) was introduced by Nisan and Ronen [STOC 1999] • The field deals with designing strategyproof (incentive compatible) approximation mechanisms for game-theoretic versions of optimization problems • All the work in the field considers mechanisms with payments • Keywords: without money

  16. Class 1 Opt SP mechanism with money Problem is intractable Opt SP mech with money + tractable Class 3 No opt SP mech w/o money Class 2 No opt SP mech with money

  17. Approx MD w/o money for econ • Work in econ on MD without money • Single peaked preferences • House allocation • Stable matching • No optimization target • Keyword: approximation

  18. In some settings, approximation facilitates truthfulness without money

  19. Work on approx MD w/o money • Facility location • [PT09,AFPT10,NST10,LWZ09,LSWZ10,Thang10,TAY10,FT10] • Machine learning • [DFP10,MPR08,MPR09,MPR10,MAMR11] • Approval voting • [AFPT10,CKM10] • Allocation of items • [GCR09,GC10] • Generalized assignment • [DG10] • Kidney exchange • [AFKP10]

  20. Overview Facility location Facility location Approximate mechanism design w/o money Approximate mechanism design w/o money Kidney exchange

  21. Motivation • Many types of kidney disease require transplantation • Potential donors may be incompatible with patient • Pairs of incompatible donor-patient pairs can sometimes exchange kidneys • Monetary transfers illegal • Previous work considered the donor/patient incentives • [Roth+Sonmez+Unver] Hospitals’ incentives may become a problem

  22. The model • Undirected graph • Vertices = donor-patient pairs • Edges = compatibility • Each player controls subset of vertices • Mechanism receives a graph and returns a matching • Utility of players = number of its matched vertices • Target: efficiency = # matched vertices • Strategy: subset of revealed vertices • But edges are public knowledge • Mechanism is strategyproof (SP) if it is a dominant strategy to reveal all vertices

  23. An example

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

  25. Another example

  26. Results • Theorem (main): Match is SP for any number of players and any partition  • For two players 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 (!) • In simulations gives 90% efficiency

  27. Bibliographic notes • Approximate mechanism design without moneyWith Moshe Tennenholtz (EC 2009) • Strategyproof approximation of the minimax on networksWith NogaAlon, Michal Feldman, and Moshe Tennenholtz (Math. of OR 2010) • Mix and MatchWith ItaiAshlagi, Felix Fischer, and Ian Kash (EC 2010) • Available from my website

  28. (Some of) my research interests • Economics: • Game theory and mechanism design • Social choice • Fair division • Computer science: • AI: multiagent systems, machine learning, decision making under uncertainty • Combinatorial optimization and approximation algorithms • Human computation

  29. Thank Y u!

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