Asymptotically Optimal Strategy-Proof Mechanisms for Two-Facility Games

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!