Asymptotically Optimal Strategy Proof Mechanisms for TwoFacility 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 •

Requirements •

Start from 1 -facility • Mechanism

2 -facility game • Mechanism

2 -facility game • Mechanism n-2 -1 0 1

2 -facility game •

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 Upper bound n-2[PT 09] Lower bound 1. 5[PT 09] Randomized

2 -facility result Deterministic Upper bound n-2[PT 09] Lower bound 1. 5[PT 09] 2[LWZ 09] Randomized 1. 045[LWZ 09]

2 -facility result Deterministic Upper bound Lower bound Randomized n-2[PT 09] 1. 045[LWZ 09]

2 -facility result Deterministic Upper bound Lower bound Randomized n-2[PT 09] 1. 045[LWZ 09]

(n-1)/2 lower bound •

(n-1)/2 lower bound •

Prove idea of the lemma • (n-1)/2 Facility regions (n-1)/2 1 0 1

Prove idea of the lemma • (n-1)/2 1 0 1

Prove idea of the lemma • (n-1)/2 1 0 1

Prove idea of the lemma • 1 (n-1)/2 -1 (n-1)/2 1 0 1

Prove idea of the lemma • 2 (n-1)/2 -2 (n-1)/2 1 0 1

Prove idea of the lemma • (n-1)/2 1 0 1

Prove idea of the lemma • (n-1)/2 1 0 1

Prove idea of the lemma • (n-1)/2 1 0 1

2 -facility result Deterministic Upper bound Lower bound Randomized n-2[PT 09] 1. 045[LWZ 09]

Proportional Mechanism •

Further work • Upper bound for deterministic mechanism of 2 -facility game over general metric is unbounded. Deterministic Upper bound Lower bound N/A Randomized 4

Further work •

Thank you!