Ariel D. Procaccia (Microsoft)

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

 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 3

 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 4

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

 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  -approximation if for every instance, MECH/OPT    Leftmost location mechanism has ratio  n  1 6

 Mechanism: select the median location  The median is group strategyproof  The median minimizes the social cost E E D D B B A A 7 D D C C D D

 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 B B A A C C 8

 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 E E C C B B A A G G F F D D F F C C B B A A 9

 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 10

 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 11

 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 A A B B C C N N 1/3 A A B B C C 12

 If the network is a line, random dictator is group strategyproof  Theorem: if the network is a circle, random dictator is group strategyproof 13

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 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  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 E E D D B B A A C C 15

 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 E E D D A A C C 1/4 1/2 1/4 B B B B 1/2 d 2d  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 16

 Mechanism: choose A  Gives a 2-approximation to the maximum cost  Lower bound of 2 still holds 17

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

 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 B 3/8 B B A A C C C C A A 1/4 19

 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 20

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

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 Approximate mechanism design without money With 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 networks With Noga Alon, Michal Feldman, and Moshe Tennenholtz [under submission] Locating a facility on a network  Available from Google: Ariel Procaccia 23

 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 24

Opt SP mech with money + tractable Class 1 Opt SP mechanism with money Problem is intractable Class 2 No opt SP mech with money 25 Class 3 No opt SP mech w/o 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] 26

 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 27

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 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 29

 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 B B A A B B B B 30

 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 31