Asymptotically Optimal Strategy-Proof Mechanismsfor Two-Facility Games

Pinyan Lu(Microsoft Research Asia)Xiaorui Sun(Shanghai Jiao Tong University)Yajun Wang(Microsoft Research Asia)Zeyuan Allen Zhu(Tsinghua University)

Where to build libraries

• people live in a city.• Goal: build new libraries and determine where

to place them– Each person wants a library to be as close to herself

as possible.• Design a mechanism to build the libraries– Players are located on a metric space .– Each player reports her location to the mechanism.– The mechanism decides locations to build the

facilities.

Requirements

• Social cost: the summation of the costs for each players.– Cost function: the distance to the closest facility.– Approximation ratio for mechanism :

• Strategy-proof mechanism does not encourage player to misreport its location.

Start from 1-facility

• Median function: – Strategy-proof

Mechanism

2-facility game

• Example: (

Mechanism

2-facility game

• Example: (– approximation ratio

Mechanism

-1 0 1

n-2

2-facility game

• Example: (– approximation ratio

• Good approximation mechanism?– If payment is allowed, the Vickrey-Clarke-Groves

mechanism gives both optimal and strategy-proof solution.

Randomized mechanism

• The mechanism selects facility locations according to some distribution.

• Each player’s cost function is the expected distance to the closest facility.

• Does randomness help approximation ratio?– Of course not for 1 facility game– What about 2 facility game?

2-facility result

Deterministic Randomized

Upper bound n-2[PT09]

Lower bound 1.5[PT09]

2-facility result

Deterministic Randomized

Upper bound n-2[PT09][LWZ09]

Lower bound 1.5[PT09]2[LWZ09] 1.045[LWZ09]

2-facility result

Deterministic Randomized

Upper bound n-2[PT09][LWZ09]

Lower bound 1.5[PT09]

2[LWZ09] 1.045[LWZ09]

2-facility result

Deterministic Randomized

Upper bound n-2[PT09][LWZ09]

4Lower bound 1.5[PT09]

2[LWZ09] 1.045[LWZ09]

(n-1)/2 lower bound

• For line metric space• Consider reported locations

• Lemma: Let be a deterministic strategy-proof mechanism for a line metric space with < Then there must be one facility at and one facility at for all .

( (

(n-1)/2 lower bound

• For line metric space• Consider reported locations

• Let – optimal social cost is at most – Mechanism gives social cost – approximation ratio– Contradiction

( (

Prove idea of the lemma

• Lemma: If < Then there must be one facility at and one facility at for all .

•

10−∞

(n-1)/21

Facility regions(n-1)/2

Prove idea of the lemma

• Fix , let varies in – Image set: All the possible facility locations in • Image set are several closed intervals.• If is in the image set, then there is a facility at if .

10−∞

(n-1)/21

(n-1)/2

𝑥

Prove idea of the lemma

• Fix , let varies in – Image set: All the possible facility locations in • Image set are several closed intervals.• If is in the image set, then there is a facility at if

10−∞

(n-1)/21

(n-1)/2

𝑥 𝑥 ′

Prove idea of the lemma

• Fix , let varies in – Image set: All the possible facility locations in • Image set are several closed intervals.• If is in the image set, then there is a facility at if

10−∞

(n-1)/21

(n-1)/2-11

𝑥 𝑥 ′

Prove idea of the lemma

• Fix , let varies in – Image set: All the possible facility locations in • Image set are several closed intervals.• If is in the image set, then there is a facility at if

10−∞

(n-1)/21

(n-1)/2-22

𝑥 𝑥 ′

Prove idea of the lemma

• Fix , let varies in – Image set: All the possible facility locations in • Image set are several closed intervals.• If is in the image set, then there is a facility at if

10−∞

(n-1)/21

(n-1)/2

𝑥 𝑥 ′

Prove idea of the lemma

• Fix , let varies in – Image set: All the possible facility locations in • Image set are several closed intervals.• If is in the image set, then there is a facility at if • If the image set is not , but contains

10

(n-1)/21

−∞−∞ 𝑦

(n-1)/2

𝑦+𝜀𝑧

Prove idea of the lemma

• Fix , let varies in – Consider

• There is still one facility at .• But at least (n-1)/2 approximation ratio.

10

(n-1)/21

−∞−∞ 𝑦

(n-1)/2

𝑦+𝜀

( (

2-facility result

Deterministic Randomized

Upper bound n-2[PT09][LWZ09]

4Lower bound 1.5[PT09]

2[LWZ09] 1.045[LWZ09]

Proportional Mechanism

• Given a profile x = () over general metric.• First facility : Uniformly choose The first

facility is placed at • Second facility : Let be the distance from

player to the first facility. Choose with probability The second facility is placed at

• Theorem: Proportional Mechanism is strategy- proof with approximation ratio 4.

Further work

• Upper bound for deterministic mechanism of 2-facility game over general metric is unbounded.

Deterministic Randomized

Upper bound N/A4

Lower bound

Further work

• Upper bound for deterministic mechanism of 2-facility game over general metric is unbounded.

• -facility game – Linear lower bound for deterministic mechanism

of -facility game.– Proportional mechanism does not apply for case.– Some tools(like image set) may be useful.

• Group strategy-proof mechanisms

Thank you!