Post on 19-Dec-2015
transcript
New Insights on Where to Locate a Library
Ariel D. Procaccia (Microsoft)
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Foreword Best advisor
award goes to... Thesis is about
computational social choice Approximation Learning Manipulation BEST
ADVISOR
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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
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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
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Take 2: leftmost location
B EC DA 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
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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 = n1 Optimal solution selects 1, social cost OPT = 1
Mechanism gives -approximation if for every instance, MECH/OPT
Leftmost location mechanism has ratio n1
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Take 3: the median Mechanism: select the median location The median is group strategyproof The median minimizes the social cost
EDBA DC D
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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
B
A
C
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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 E
C
B
A
G
F
D
F
C
B
A
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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 < n1 for SC
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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
(22/n)-approx mechanism for SC on any network
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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
A
B C
11 1
N
1/3
A
B C
1/31/3
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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
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Summary of social cost
NETWORK TOPOLOGY
general circle tree line Mechanism Target
LB (n) SP LB (n) SP UB 1 GSP UB 1 GSP det
SCUB 2 SPLB open
UB 2 GSPLB open UB 1 GSP UB 1 GSP rand
UB 2 GSPLB 2 SP
UB 2 GSPLB 2 SP
UB 2 GSPLB 2 SP
UB 2 GSPLB 2 SP
det
MCUB 2 GSPLB 2-o(1) SP
UB 3/2 GSPLB 3/2 GSP
UB 2 GSPLB 2-o(1) SP
UB 3/2 GSPLB 3/2 GSP
ran
?
?
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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
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
EDBA C
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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
EDA C
1/4 1/2 1/4
BB
1/4 1/2
d2d
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
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Minmax on general networks
Mechanism: choose A Gives a 2-approximation to the maximum
cost Lower bound of 2 still holds
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LRM on a circle Semicircle like an
interval on a line If all agents are on
one semicircle, can apply LRM
Meaningless otherwise
B
C
DE
1/4
F1/2
1/4
A
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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
3/8
B
A
CC
A
3/8
1/4
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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
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A randomized lower bound on trees
Theorem: there is no randomized strategyproof mechanism with approximation ratio better than 2o(1) for MC on trees
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Summary of maximum cost
NETWORK TOPOLOGY
general circle tree line Mechanism Target
LB (n) SP LB (n) SP UB 1 GSP UB 1 GSP det
SCUB 2 SPLB open
UB 2 GSPLB open UB 1 GSP UB 1 GSP rand
UB 2 GSPLB 2 SP
UB 2 GSPLB 2 SP
UB 2 GSPLB 2 SP
UB 2 GSPLB 2 SP
det
MCUB 2 GSPLB 2-o(1) SP
UB 3/2 SPLB 3/2 SP
UB 2 GSPLB 2-o(1) SP
UB 3/2 GSPLB 3/2 SP
ran?
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Bibliographic notes 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 networksWith Noga Alon, Michal Feldman, and Moshe Tennenholtz [under submission]
Locating a facility on a network Available from Google: Ariel Procaccia
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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
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Opt SP mech with
money + tractable
Class 1Opt SP mechanism with moneyProblem is intractable
Class 2No opt SP mech with money
Class 3No opt SP mech
w/o money
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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]
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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
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Thank Y u!
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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
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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
BA B
0 1
B
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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