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New Insights on Where to Locate a Library

New Insights on Where to Locate a Library. Ariel D. Procaccia ( Microsoft). Foreword. Best advisor award goes to... Thesis is about computational social choice Approximation Learning Manipulation. BEST ADVISOR. Where to locate a library on a street?.

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New Insights on Where to Locate a Library

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  1. New Insights on Where to Locate a Library Ariel D. Procaccia (Microsoft)

  2. Foreword • Best advisor award goes to... • Thesis is about computational social choice • Approximation • Learning • Manipulation BEST ADVISOR

  3. Where to locate a library on a street? • Want to locate a public facility (library, train station) on a street • n agents 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 an agent = its distance from the facility

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

  5. Take 2: leftmost location • Mechanism: select the leftmost reported location • Mechanism is strategyproof • A mechanism is group strategyproofif a coalition of agents cannot all gain by lying = the distance from at least one member does not decrease • Mechanism is group strategyproof A B B B C D E

  6. Social cost and approximation • Social cost (SC) of facility location = sum of distances to the agents • Leftmost location mechanism can be bad in terms of social cost • One agent at 0, n-1 agents at 1 • Mechanism selects 0, social cost MECH = n1 • Optimal solution selects 1, social cost OPT = 1 • Mechanism gives -approximationif for every instance, MECH/OPT   • Leftmost location mechanism has ratio  n1

  7. Take 3: the median • Mechanism: select the median location • The median is group strategyproof • The median minimizes the social cost B A C D D D E

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

  9. Median on trees • Suppose network is a tree • Mechanism: start from root, move towards majority of agents as long as possible • Mechanism minimizes social cost • Mechanism is (group) strategyproof A A B B D F F C C G E

  10. Strategyproof mechanisms in general networks • Schummer and Vohra [JET 2004] characterized the strategyproof mechanisms on general networks • Corollary: if network contains a cycle, there is no strategyproof mechanism with approx ratio < n1 for SC

  11. A randomized mechanism • A randomized mechanism randomly selects a location • Cost of agent = expected distance from the facility • Social cost = sum of costs = sum of expected distances • Random dictator mechanism: select an agent uniformly and return its location • Theorem: random dictator is a strategyproof (22/n)-approx mechanism for SC on any network

  12. Random dictator is not always group strategyproof • Consider a star with three arms of length one, with three agents at leaves • Cost of each agent = 4/3 • After moving to center, cost of each agent = 1 1/3 A A 1 N 1 1 1/3 1/3 B B C C

  13. Random dictator is sometimes group strategyproof • If the network is a line, random dictator is group strategyproof • Theorem: if the network is a circle, random dictator is group strategyproof

  14. Summary of social cost ? ?

  15. Minimizing the maximum cost • Maximum cost (MC) of facility location = max distance to the agents • Example: facility is a fire station • Optimal solution on a line = average of leftmost and rightmost locations, its max cost = d(A,E)/2 • Mechanism: select A • Mechanism is group strategyproof and gives a 2-approximation to MC • Theorem:There is no deterministic strategyproof mechanism with approx ratio smaller than 2 for MC on a line A C D E B

  16. 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 for MC is[½  (2  OPT) + ½  OPT] / OPT = 3/2 • LRM mechanism is strategyproof • Theorem:LRM Mechanism is group strategyproof • Theorem:There is no randomized strategyproof mechanism with approximation ratio better than 3/2 for MC on a line 2d d B B C A D E 1/2 1/4 1/2 1/4 1/4

  17. Minmax on general networks • Mechanism: choose A • Gives a 2-approximation to the maximum cost • Lower bound of 2 still holds

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

  19. Random Midpoint • Look at points antipodal to agents’ 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 agents are not on one semicircle, but  2 if they are B 1/4 A 3/8 C C A B 3/8

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

  21. A randomized lower bound on trees • Theorem:there is no randomized strategyproof mechanism with approximation ratio better than 2o(1) for MC on trees

  22. Summary of maximum cost ?

  23. Bibliographic notes • Approximate mechanism design without moneyWith Moshe Tennenholtz [EC’09] • Locating a facility on a line • Locating two facilities on a line • Locating one facility on a line when each player controls multiple locations • Strategyproof approximation mechanisms for location on networksWith NogaAlon, Michal Feldman, and Moshe Tennenholtz [under submission] • Locating a facility on a network • Available from Google: Ariel Procaccia

  24. A bit on algorithmic mechanism design • 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 • Money unavailable in many settings

  25. 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

  26. Approximate mechanism design without money • Can consider computationally tractable optimization problem • Approximation to obtain strategyproofness rather than circumvent computational complexity • Originates from work on incentive compatible regression learning and classification [Dekel+Fischer+P, SODA 08, Meir+P+Rosenschein, AAAI 08, IJCAI 09]

  27. Future work • I Promised “avalanche of challenging directions for future research” • I lied • Generally speaking: • Many technical open questions • Many extensions, can combine extensions • Completely different settings

  28. Thank Y u!

  29. Current work • Agents are vertices in directed graph, score is indegree • Must elect a subset of agents of size k • Objective function: sum of scores of elected agents • Strategy of an agent: outgoing edges • Utility of an agent: 1 if elected, 0 if not

  30. Lower bound of two • Theorem: there is no deterministic strategyproof mechanism with approx ratio smaller than 2 on a line • Suppose mechanism has ratio < 2 • Let A = 0, B = 1; OPT = ½ • Mechanism must locate facility at 0 < x < 1 • Let A = 0, B = x; OPT = x/2 • Mechanism must locate facility at 0 < y < x • B gains by reporting 1 A B B B 0 1

  31. Minmax on general networks • Mechanism: choose A • Gives a 2-approximation to the maximum cost • O = optimal location, X = some agent • d(A,X)  d(A,O) + d(O,X)  2  OPT • Lower bound of 2 still holds

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