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