On (Group) Strategy-proof Mechanisms without Payment for Facility Location Games Nguyen Kim Thang

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On (Group) Strategy-proof

Mechanisms without Payment for Facility Location Games

Nguyen Kim Thang

WINE’10

Tuesday, December 14, 2010

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Outline

Our main results Definitions Facility Games

Results

High level ideas.

Conclusion & Further directions

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Facility Games

A network is represented by a graph minimum-length path.

G(V, E) d(u, v) =

1

10

agents, agent has location ⁿ i x_i ∈ V

A det mechanism ^f ^: ^V ⁿ _→ ^V

x = !x₁, . . . , x_n" #→ P

f : V ⁿ → ∆(V )

A random mechanism Agent’s cost:

Agent’s cost: _E_F _∼_P [d(x_i, F )]

d(x_i, F )

x = !x₁, . . . , x_n" #→ F

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Facility Games

A network is represented by a graph minimum-length path.

G(V, E) d(u, v) =

1

10

agents, agent has location ⁿ i x_i ∈ V

A det mechanism ^f ^: ^V ⁿ _→ ^V

x = !x₁, . . . , x_n" #→ P

f : V ⁿ → ∆(V )

A random mechanism

Agent’s cost: d(x_i, F )

x = !x₁, . . . , x_n" #→ F

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Facility Games

1

10

A mechanism is group strategy-proof if for all , there existsS ⊂ N i ∈ S

cost(x_i, f (x^!_S, x₋_S)) > cost(x_i, f (x))

A mechanism is strategy-proof if

cost(x_i, f (x^!_i, x₋_i)) ≥ cost(x_i, f (x))

∀i

Social objective functions: ^!

i∈N

cost(x_i, F )

A mechanism is -approximation if ^α

cost(f ) ≤ α · OP T f

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Results

Line graphs General graphs Det

Ran

n GSP (n − 1) SP 1 GSP

1 GSP

(upper bound) (lower bound)

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Results

Line graphs General graphs Det

Ran

n GSP (n − 1) SP 1 GSP

1 GSP

(2 − 2/n) SP, n GSP (u.b)

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Results

Line graphs General graphs Det

Ran

n GSP (n − 1) SP

!

2 − 4 n^1/3

"

SP, Ω(n¹⁻^!) GSP

1 GSP

(2 − 2/n) SP, n GSP (u.b)

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Results

Designing (G)SP mechanism without money is expensive

Line graphs General graphs Det

Ran

n GSP (n − 1) SP

!

2 − 4 n^1/3

"

SP, Ω(n¹⁻^!) GSP

1 GSP

(2 − 2/n) SP, n GSP (u.b)

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Framework of lower bound

Instance 1 Instance 2 Instance m

Fix a graph

An instance differs from the previous one in some agents’ locations

Connect instances using strategy-proofness.

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GSP mechanisms

Dictatorship: open the facility at location of some fixed agent.

GSP and -approximation.ⁿ

Deterministic lower bound by theorem of

Schummer and Vohra. (Any SP mechanism on a cycle graph is dictatorship.)

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GSP mechanisms

Dictatorship: open the facility at location of some fixed agent.

GSP and -approximation.ⁿ

Deterministic lower bound by theorem of

Schummer and Vohra. (Any SP mechanism on a cycle graph is dictatorship.)

Thm: no randomized GSP mechanism is better than - approximation.

n¹⁻^3!/3

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Deterministic GSP mechanisms

u₁

u₂ u_n

v₁

v₂ 1

1 − ! v_n

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Deterministic GSP mechanisms

u₁

u₂ u_n

v₁

v₂ 1

1 − ! v_n

Case 1: facility is opened in U

All agents but the first one move to . The facility is not opened at

v₁

(n − 1)(1 − !)

2 − ! ≈ n − 1 2 v₁

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Deterministic GSP mechanisms

u₁

u₂ u_n

v₁

v₂ 1

1 − ! v_n

v₁

(n − 1)(1 − !)

2 − ! ≈ n − 1 2 v₁

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Deterministic GSP mechanisms

u₁

u₂ u_n

v₁

v₂ 1

1 − ! v_n

v₁

(n − 1)(1 − !)

2 − ! ≈ n − 1 2 v₁

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Deterministic GSP mechanisms

u₁

u₂ u_n

v₁

v₂ 1

1 − !

(n − 1)(1 − !)

2 − ! ≈ n − 1 2

Case 2: facility is opened in All agents but the last one move to .

V v_n

Again,

v_n

v₁

(n − 1)(1 − !)

2 − ! ≈ n − 1 2 v₁

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Deterministic GSP mechanisms

u₁

u₂ u_n

v₁

v₂ 1

1 − !

(n − 1)(1 − !) n − 1

Case 2: facility is opened in All agents but the last one move to .

V v_n

v_n

v₁

(n − 1)(1 − !)

2 − ! ≈ n − 1 2 v₁

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Weakness

The argument does not carry.

u₁

u₂ u_n

v₁

v₂ 1

1 − ! v_n

An mechanism opens facility at with prob ¹ ₋ ^δ, ^δ

prevent agents

from collaborating.

2, . . . , n − 1 v_n, u₁

old cost of agent 2: 1 − !

new cost of agent 2: (1 − δ) · (1 − ") + δ · 1

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Weakness

u₁

u₂ u_n

v₁

v₂ 1

1 − ! v_n

prevent agents

from collaborating.

2, . . . , n − 1 v_n, u₁

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Weakness

u₁

u₂ u_n

v₁

v₂ 1

1 − ! v_n

prevent agents

from collaborating.

2, . . . , n − 1 v_n, u₁

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Solution

u₁ u₂

u_n

v₁

v₂ 1

v_n β₁

β₂ β_n₋₁ β₁

β₂

β_n₋₁

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Solution

u₁ u₂

u_n

v₁

v₂ 1

v_n β₁

β₂ β_n₋₁ β₁

β₂

β_n₋₁

Need: symmetry + asymmetry.

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Solution

u₁ u₂

u_n

v₁

v₂ 1

v_n β₁

β₂ β_n₋₁ β₁

β₂

β_n₋₁

Need: symmetry + asymmetry.

modify brown edges.

2n⁻^! > β₁ > β₂ > . . . > β_n₋₁ > n⁻^!

define the length in circular permutation

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Solution

u₁ u₂

u_n

v₁

v₂ 1

v_n β₁

β₂ β_n₋₁ β₁

β₂

β_n₋₁

harder to prevent agents from collaborating.

Why it works?

amplify the gap by recursive construction

Need: symmetry + asymmetry.

modify brown edges.

2n⁻^! > β₁ > β₂ > . . . > β_n₋₁ > n⁻^!

define the length in circular permutation

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Lower bound construction

u₁

u₂ u_n

v₁

v₂ 1

v_n

Lemma (informal):

P[f ((x)) = v_n] < 1 − !

! log n

log log n

"3!/2

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Lower bound construction

u₁

u₂ u_n

v₁

v₂ 1

v_n

Lemma (informal):

P[f ((x)) = v_n] < 1 − !

! log n

log log n

"3!/2

Recursive construction of multiple levels.

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Lower bound construction

u1

u2 u_n

v1

v₂ vn

u₁

u₂ un

v₂ vn

u₁

u₂ un

v₂ vn

Lemma (informal):

P[f ((x)) = v_n] < 1 − !

! log n

log log n

"3!/2

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Lower bound construction

u1

u2 u_n

v1

v₂ vn

u₁

u₂ un

v₁

v₂ vn

u₁

u₂ un

v₁

v₂ vn

Lemma (informal):

P[f ((x)) = v_n] < 1 − n^3!/2 P[f ((x)) = v_n] < 1 − !

! log n

log log n

"3!/2

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Randomized SP mechanisms

Mechanism (Random dictatorship): open the facility at with probability .

strategy-proof but not GSP 2-approximation

x_i 1/n

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Randomized SP mechanisms

x_i 1/n

Can we do better?

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Randomized SP mechanisms

x_i 1/n

Can we do better?

Thm:

no randomized SP mechanism is better than approximation.

!

2 − 4 n^1/3

"

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Randomized SP mechanism

s t

1 1

Idea 2!

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Randomized SP mechanism

s t

1 1

2!

Initially, “good” mechanism has prob. at 1/2 ^{s, t}

Idea

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Randomized SP mechanism

s t

There exists an agent, let say blue, with cost at least ^{1 +} ^!

1 1

2!

Idea

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Randomized SP mechanism

s t

1 1

2!

Idea

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Randomized SP mechanism

s t

1 1

2!

Idea

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Randomized SP mechanism

s t

Move appropriate blue agents to , by strategy- proof argue that .

s p_s ≈ p_t ≈ 1/2

1 1

2!

Idea

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Randomized SP mechanism

s t

s p_s ≈ p_t ≈ 1/2

1 1

2!

Idea

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Randomized SP mechanism

s t

s p_s ≈ p_t ≈ 1/2

1 1

2!

Idea

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Randomized SP mechanism

s t

s p_s ≈ p_t ≈ 1/2

1/2 · (1 + 2 + 2!) + 1/2 · (1 + 2!)

1 + 2! ≈ 2

Then:

1 + !

1 1

2!

Idea

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Conclusion

Open a constant facilities:

easy in term of optimization.

SP mechanism with bounded ratio?

Complete characterization of performance of randomized (G)SP mechanisms.

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On (Group) Strategy-proof Mechanisms without Payment for Facility Location Games Nguyen Kim Thang