Resource-Aware Protocols for Network Cost-Sharing Games Moh a m a d L - - PowerPoint PPT Presentation
Resource-Aware Protocols for Network Cost-Sharing Games Moh a m a d L a ti f i a n, Sh a rif University of Technology George Christodoulou , University of Liverpool Vasilis Gkatzelis , Drexel University Alkmini Sgouritsa , University of Liverpool
George Christodoulou, University of Liverpool Vasilis Gkatzelis, Drexel University Alkmini Sgouritsa, University of Liverpool
Mohamad Latifian, Sharif University of Technology
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s v u t
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s v u t
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s v u t
u - t s - t
s - t
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u - t s - t
s - t
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u - t s - t
s - t
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u - t s - t
s - t
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s v u t
50 100 150 200 1 2 3 4 4 8 12 16 1 2 3 4 25 50 75 100 1 2 3 4 25 50 75 100 1 2 3 4 4 8 12 16 1 2 3 4
u - t s - t
s - t
Equal cost-sharing protocol
Equilibrium
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s v u t
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u - t s - t
s - t
Equal cost-sharing protocol
Equilibrium
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s v u t
50 100 150 200 1 2 3 4 4 8 12 16 1 2 3 4 25 50 75 100 1 2 3 4 25 50 75 100 1 2 3 4 4 8 12 16 1 2 3 4
u - t s - t
s - t
Equal cost-sharing protocol
Optimal Equilibrium
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s v u t
50 100 150 200 1 2 3 4 4 8 12 16 1 2 3 4 25 50 75 100 1 2 3 4 25 50 75 100 1 2 3 4 4 8 12 16 1 2 3 4
u - t s - t
s - t
Our goal is to design efgicient protocols.
Equal cost-sharing protocol
Optimal Equilibrium
Formal Definition
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and set of players
G N
Formal Definition
3
and set of players
G N
i si ti
Formal Definition
3
and set of players
G N
i si ti
ce(ℓ) ce(0) = 0
Formal Definition
3
and set of players
G N
i si ti
ce(ℓ) ce(0) = 0
Si si ti S = (S1, S2, …, Sn)
Formal Definition
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Formal Definition
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defines the cost share of in edge regarding the strategy profile
ξie(S) i e S
Formal Definition
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defines the cost share of in edge regarding the strategy profile
ξie(S) i e S
Formal Definition
4
defines the cost share of in edge regarding the strategy profile
ξie(S) i e S
Formal Definition
4
defines the cost share of in edge regarding the strategy profile
ξie(S) i e S
Formal Definition
4
defines the cost share of in edge regarding the strategy profile
ξie(S) i e S
Formal Definition
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ce(ℓ) ̂ ce(ℓ) ≥ ce(ℓ)
defines the cost share of in edge regarding the strategy profile
ξie(S) i e S
✓ Gives us more power to design efficient protocols
Formal Definition
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ce(ℓ) ̂ ce(ℓ) ≥ ce(ℓ)
defines the cost share of in edge regarding the strategy profile
ξie(S) i e S
✓ Gives us more power to design efficient protocols
Formal Definition
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ce(ℓ) ̂ ce(ℓ) ≥ ce(ℓ)
Evaluation of the protocol
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e∈E
ce(ℓe(S))
Evaluation of the protocol
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e∈E
ce(ℓe(S))
Γ
Evaluation of the protocol
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PoA(Γ) = sup
Γ∈Γ
maxS∈Eq(Γ) C(S) minS*∈F(Γ) C(S*)
e∈E
ce(ℓe(S))
Γ
Evaluation of the protocol
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PoA(Γ) = sup
Γ∈Γ
maxS∈Eq(Γ) C(S) minS*∈F(Γ) C(S*) PoA(Γ) = sup
Γ∈Γ
maxS∈Eq(Γ) ̂ C(S) minS*∈F(Γ) C(S*)
e∈E
ce(ℓe(S))
Γ
Evaluation of the protocol
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PoA(Γ) = sup
Γ∈Γ
maxS∈Eq(Γ) C(S) minS*∈F(Γ) C(S*) PoA(Γ) = sup
Γ∈Γ
maxS∈Eq(Γ) ̂ C(S) minS*∈F(Γ) C(S*)
e∈E
ce(ℓe(S))
Γ
Evaluation of the protocol
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PoA(Γ) = sup
Γ∈Γ
maxS∈Eq(Γ) C(S) minS*∈F(Γ) C(S*) PoA(Γ) = sup
Γ∈Γ
maxS∈Eq(Γ) ̂ C(S) minS*∈F(Γ) C(S*)
The goal is to design protocols with low Price of Anarchy.
Informational Power
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u - t s - t s - ts v u
50 100 150 200 1 2 3 4 4 8 12 16 1 2 3 4 25 50 75 100 1 2 3 4 25 50 75 100 1 2 3 4 4 8 12 16 1 2 3 4 (The cost share of player in edge )
ξie(S) i e
Informational Power
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u - t s - t s - ts v u
50 100 150 200 1 2 3 4 4 8 12 16 1 2 3 4 25 50 75 100 1 2 3 4 25 50 75 100 1 2 3 4 4 8 12 16 1 2 3 4 (The cost share of player in edge )
ξie(S) i e
Informational Power
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(The cost share of player in edge )
ξie(S) i e
Informational Power
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u - t s - t s - ts v
50 100 150 200 1 2 3 4 4 8 12 16 1 2 3 4 25 50 75 100 1 2 3 4 25 50 75 100 1 2 3 4 4 8 12 16 1 2 3 4 (The cost share of player in edge )
ξie(S) i e
and set of players using
G e
Informational Power
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s - t s - ts v
50 100 150 200 1 2 3 4 4 8 12 16 1 2 3 4 25 50 75 100 1 2 3 4 25 50 75 100 1 2 3 4 4 8 12 16 1 2 3 4 Classes of Games
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Classes of Games
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Classes of Games
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0s 1000s 2000s 3000s 4000s 0s 750s 1500s 2250s 3000s Classes of Games
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s t s1 t s2 s3 s4
Classes of Games
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s t s1 t s2 s3 s4
Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
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8
8
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Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
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Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
Theorem: This static-share leader-based cost-sharing protocol is stable, budget-balanced, resource aware, and it achieves PoA for SPGs with concave cost functions.
= 1
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Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
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Preliminaries
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Preliminaries
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Preliminaries
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Preliminaries
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Preliminaries
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Lemma: In any symmetric instance where all the users need to connect to , there exists an
s t s t
Preliminaries
π
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Lemma: In any symmetric instance where all the users need to connect to , there exists an
s t s t
Preliminaries
π
he(S) e
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Lemma: In any symmetric instance where all the users need to connect to , there exists an
s t s t
Preliminaries
Never-Walk-Alone Protocol
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Red player’s path Green player’s path
Never-Walk-Alone Protocol
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Red player’s path Green player’s path
Never-Walk-Alone Protocol
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Red player’s path Green player’s path
Never-Walk-Alone Protocol
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Red player’s path Green player’s path The optimal path for two players
Never-Walk-Alone Protocol
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Red player’s path Green player’s path The optimal path for two players
players Never-Walk-Alone Protocol
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players Never-Walk-Alone Protocol
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players Never-Walk-Alone Protocol
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ζe(ℓ) = 2ce(ℓ) if e ∉ OPT(ℓ) or ℓ = 1
ce(ℓ) ℓ − 1
ξie(S) = { ζe(ℓe(S)) if i ≠ he(S) or ℓe(S) = 1 ϵe(ζe(ℓe(S))) otherwise,
Cost-share of players
players Never-Walk-Alone Protocol
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ζe(ℓ) = 2ce(ℓ) if e ∉ OPT(ℓ) or ℓ = 1
ce(ℓ) ℓ − 1
ξie(S) = { ζe(ℓe(S)) if i ≠ he(S) or ℓe(S) = 1 ϵe(ζe(ℓe(S))) otherwise,
Cost-share of players
players Never-Walk-Alone Protocol
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ζe(ℓ) = 2ce(ℓ) if e ∉ OPT(ℓ) or ℓ = 1
ce(ℓ) ℓ − 1
ξie(S) = { ζe(ℓe(S)) if i ≠ he(S) or ℓe(S) = 1 ϵe(ζe(ℓe(S))) otherwise,
Cost-share of players
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Efficiency of The Never-Walk-Alone Protocol
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Efficiency of The Never-Walk-Alone Protocol
Lemma: In the equilibrium all the users follow the optimal path from to .
s t
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Efficiency of The Never-Walk-Alone Protocol
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Efficiency of The Never-Walk-Alone Protocol
π
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Efficiency of The Never-Walk-Alone Protocol
π
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Efficiency of The Never-Walk-Alone Protocol
π
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Efficiency of The Never-Walk-Alone Protocol
π
Theorem: The PoA of the NWA protocol for DAGs with concave functions is at most , for an arbitrarily small value . If we assume , then PoA .
2 + ε ε > 0 n > 1 = 1 + ε
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Theorem: There is no resource-aware, budget-balanced protocol that can achieve a PoA better than for the case of multicast networks with constant cost functions.
n
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Theorem: There is no resource-aware, budget-balanced protocol that can achieve a PoA better than for the case of multicast networks with constant cost functions.
n
Theorem: There is no resource-aware protocol even with overcharging that can achieve a PoA better than for the case of multicast networks with constant cost functions.
n
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Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
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Theorem: The incremental cost-sharing protocol is stable, budget-balanced, oblivious, and it achieves PoA for SPGs with convex cost functions.
= 1
Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
14
Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
14
Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
Theorem: Any stable, budget-balanced, resource-aware cost-sharing protocol has PoA for directed acyclic graphs with convex cost functions.
= Ω(n)
14
Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
Theorem: There is no stable, resource-aware, cost-sharing mechanism with , for directed acyclic graphs with convex cost functions even with overcharging. PoA < 1.18
14
Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
Theorem: There is no resource-aware protocol even with overcharging that can achieve a PoA better than for the case of multicast networks with convex cost functions.
Ω( n)
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Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
15
Open Problems:
Concave Convex
SPG
(symmetric)
DAG
(symmetric)
Multicast
PoA = 2 + ε n > 1 → PoA = 1 + ε
PoA = 1
→ PoA = Ω(n)
→ PoA = Ω( n)
Budget Balance Overcharging
PoA = 1
→ PoA = Ω(n)
→ PoA > 1.18
Budget Balance Overcharging
Incremental protocol [Moulin ’99] Leader-based protocol for parallel links [Christodoulou et al. ’17] With overcharging
compositions of smaller SPGs, starting with copies of the basic SPG which is an edge.
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compositions of smaller SPGs, starting with copies of the basic SPG which is an edge.
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compositions of smaller SPGs, starting with copies of the basic SPG which is an edge.
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compositions of smaller SPGs, starting with copies of the basic SPG which is an edge.
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