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