1 / 27

Asymptotically Optimal Strategy-Proof Mechanisms for Two-Facility Games

Asymptotically Optimal Strategy-Proof Mechanisms for 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.

jackson-daugherty
Télécharger la présentation

Asymptotically Optimal Strategy-Proof Mechanisms for Two-Facility Games

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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)

  2. 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.

  3. Requirements • Social cost: the summation of the costs for each players. • Cost function: the distance to the closest facility. • Approximation ratio for mechanism : • Strategy-proofmechanism does not encourage player to misreport its location.

  4. Start from 1-facility • Median function: • Strategy-proof Mechanism

  5. 2-facility game • Example: ( Mechanism

  6. 2-facility game • Example: ( • approximation ratio Mechanism n-2 1 0 -1

  7. 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.

  8. 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?

  9. 2-facility result

  10. 2-facility result

  11. 2-facility result

  12. 2-facility result

  13. (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 . ( (

  14. (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 ( (

  15. Prove idea of the lemma • Lemma: If < Then there must be one facility at and one facility at for all . Facility regions (n-1)/2 (n-1)/2 1 0 1

  16. 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 . (n-1)/2 (n-1)/2 1 0 1

  17. 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 (n-1)/2 (n-1)/2 1 0 1

  18. 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 (n-1)/2 (n-1)/2-1 1 1 0 1

  19. 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 (n-1)/2 (n-1)/2-2 2 1 0 1

  20. 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 (n-1)/2 (n-1)/2 1 0 1

  21. 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 (n-1)/2 (n-1)/2 1 0 1

  22. Prove idea of the lemma • Fix , let varies in • Consider • There is still one facility at . • But at least (n-1)/2 approximation ratio. ( ( (n-1)/2 (n-1)/2 1 0 1

  23. 2-facility result

  24. 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. Choosewith probability The second facility is placed at • Theorem: Proportional Mechanism is strategy- proof with approximation ratio 4.

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

  26. 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

  27. Thank you!

More Related