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
Cost-Sharing Games v s t u 2
Cost-Sharing Games v s t u 2
Cost-Sharing Games s - t v s - t s t u - t u 2
Cost-Sharing Games s - t v s - t s t u - t u 2
Cost-Sharing Games s - t v 200 100 150 75 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s t 50 25 0 0 1 2 3 4 u - t 16 12 16 8 12 4 8 0 4 0 1 2 3 4 0 0 1 2 3 4 u 2
Cost-Sharing Games s - t v 200 100 150 75 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s t 50 25 0 0 1 2 3 4 u - t 16 12 16 8 12 4 8 0 4 0 1 2 3 4 0 0 1 2 3 4 u 2
Cost-Sharing Games s - t Equal cost-sharing protocol v 200 100 150 75 -12 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s t -23 50 25 0 0 1 2 3 4 u - t 16 12 16 8 12 -15 4 8 0 4 0 1 2 3 4 0 0 1 2 3 4 -50 u Equilibrium 2
Cost-Sharing Games s - t Equal cost-sharing protocol v 200 100 150 75 -12 -13 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s t -23 -13 50 25 0 0 1 2 3 4 u - t 16 12 16 8 12 -15 -5 4 8 0 4 0 1 2 3 4 0 0 1 2 3 4 -50 u Equilibrium 2
Cost-Sharing Games s - t Equal cost-sharing protocol v 200 100 150 75 -12 -13 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s t -23 -13 50 25 0 0 1 2 3 4 u - t 16 12 16 8 12 -15 -5 4 8 0 4 0 1 2 3 4 0 0 1 2 3 4 -50 -31 u Equilibrium Optimal 2
Cost-Sharing Games s - t Equal cost-sharing protocol v 200 100 150 75 -12 -13 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s t Our goal is to design e fg icient protocols. -23 -13 50 25 0 0 1 2 3 4 u - t 16 12 16 8 12 -15 -5 4 8 0 4 0 1 2 3 4 0 0 1 2 3 4 -50 -31 u Equilibrium Optimal 2
Cost-Sharing Games Form a l De f inition 3
Cost-Sharing Games Form a l De f inition • Graph G and set of players N 3
Cost-Sharing Games Form a l De f inition • Graph G and set of players N • Player needs to connect her source to her sink i s i t i 3
Cost-Sharing Games Form a l De f inition • Graph G and set of players N • Player needs to connect her source to her sink i s i t i • Edge has a cost function e c e ( ℓ ) - Non-decreasing - c e (0) = 0 3
Cost-Sharing Games Form a l De f inition • Graph G and set of players N • Player needs to connect her source to her sink i s i t i • Edge has a cost function e c e ( ℓ ) - Non-decreasing - c e (0) = 0 • Strategy is a path from to . This forms strategy pro fi le S i s i t i S = ( S 1 , S 2 , …, S n ) 3
Cost-Sharing Games Form a l De f inition 4
Cost-Sharing Games Form a l De f inition • Cost-sharing method ξ ie ( S ) de fi nes the cost share of in edge regarding the strategy i e pro fi le S 4
Cost-Sharing Games Form a l De f inition • Cost-sharing method ξ ie ( S ) de fi nes the cost share of in edge regarding the strategy i e pro fi le S - Stable 4
Cost-Sharing Games Form a l De f inition • Cost-sharing method ξ ie ( S ) de fi nes the cost share of in edge regarding the strategy i e pro fi le S - Stable - E ffi cient 4
Cost-Sharing Games Form a l De f inition • Cost-sharing method ξ ie ( S ) de fi nes the cost share of in edge regarding the strategy i e pro fi le S - Stable - E ffi cient - Budget-balance and overcharging 4
̂ Cost-Sharing Games Form a l De f inition • Cost-sharing method ξ ie ( S ) de fi nes the cost share of in edge regarding the strategy i e pro fi le S - Stable - E ffi cient - Budget-balance and overcharging c e ( ℓ ) ≥ c e ( ℓ ) c e ( ℓ ) 4
̂ Cost-Sharing Games Form a l De f inition • Cost-sharing method ξ ie ( S ) de fi nes the cost share of in edge regarding the strategy i e pro fi le S - Stable - E ffi cient - Budget-balance and overcharging c e ( ℓ ) ≥ c e ( ℓ ) c e ( ℓ ) ✓ Gives us more power to design e ffi cient protocols 4
̂ Cost-Sharing Games Form a l De f inition • Cost-sharing method ξ ie ( S ) de fi nes the cost share of in edge regarding the strategy i e pro fi le S - Stable - E ffi cient - Budget-balance and overcharging c e ( ℓ ) ≥ c e ( ℓ ) c e ( ℓ ) ✓ Gives us more power to design e ffi cient protocols - Increases the cost of the optimal solutions 4
Cost-Sharing Games Ev a lu a tion of the protocol 5
Cost-Sharing Games Ev a lu a tion of the protocol • Total Cost C ( S ) = ∑ c e ( ℓ e ( S )) e ∈ E 5
Cost-Sharing Games Ev a lu a tion of the protocol • Total Cost C ( S ) = ∑ c e ( ℓ e ( S )) e ∈ E • Price of Anarchy (PoA) over class of games Γ max S ∈ Eq ( Γ ) C ( S ) PoA ( Γ ) = sup min S * ∈ F ( Γ ) C ( S *) Γ∈ Γ 5
̂ Cost-Sharing Games Ev a lu a tion of the protocol • Total Cost C ( S ) = ∑ c e ( ℓ e ( S )) e ∈ E • Price of Anarchy (PoA) over class of games Γ max S ∈ Eq ( Γ ) C ( S ) PoA ( Γ ) = sup min S * ∈ F ( Γ ) C ( S *) Γ∈ Γ - With overcharging max S ∈ Eq ( Γ ) C ( S ) PoA ( Γ ) = sup min S * ∈ F ( Γ ) C ( S *) Γ∈ Γ 5
̂ Cost-Sharing Games Ev a lu a tion of the protocol • Total Cost C ( S ) = ∑ c e ( ℓ e ( S )) e ∈ E • Price of Anarchy (PoA) over class of games Γ max S ∈ Eq ( Γ ) C ( S ) PoA ( Γ ) = sup min S * ∈ F ( Γ ) C ( S *) Γ∈ Γ - With overcharging max S ∈ Eq ( Γ ) C ( S ) PoA ( Γ ) = sup min S * ∈ F ( Γ ) C ( S *) Γ∈ Γ 5
̂ Cost-Sharing Games Ev a lu a tion of the protocol • Total Cost C ( S ) = ∑ c e ( ℓ e ( S )) e ∈ E • Price of Anarchy (PoA) over class of games Γ The goal is to design protocols with low Price of Anarchy. max S ∈ Eq ( Γ ) C ( S ) PoA ( Γ ) = sup min S * ∈ F ( Γ ) C ( S *) Γ∈ Γ - With overcharging max S ∈ Eq ( Γ ) C ( S ) PoA ( Γ ) = sup min S * ∈ F ( Γ ) C ( S *) Γ∈ Γ 5
Cost-Sharing Games Inform a tion a l Power v s - t 200 100 150 75 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s 50 25 0 0 1 2 3 4 u - t 16 16 12 12 8 8 4 4 0 0 1 2 3 4 0 0 1 2 3 4 u 6
Cost-Sharing Games Inform a tion a l Power • What does the protocol know in de fi ning ξ ie ( S ) i e (The cost share of player in edge ) v s - t 200 100 150 75 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s 50 25 0 0 1 2 3 4 u - t 16 16 12 12 8 8 4 4 0 0 1 2 3 4 0 0 1 2 3 4 u 6
Cost-Sharing Games Inform a tion a l Power • What does the protocol know in de fi ning ξ ie ( S ) i e (The cost share of player in edge ) - Oblivious : Set of players using e 6
Cost-Sharing Games Inform a tion a l Power • What does the protocol know in de fi ning ξ ie ( S ) i e (The cost share of player in edge ) - Oblivious : Set of players using e - Omniscient : Everything about the game v s - t 200 100 150 75 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s 50 25 0 0 1 2 3 4 u - t 16 16 12 12 8 8 4 4 0 0 1 2 3 4 0 0 1 2 3 4 6
Cost-Sharing Games Inform a tion a l Power • What does the protocol know in de fi ning ξ ie ( S ) i e (The cost share of player in edge ) - Oblivious : Set of players using e - Omniscient : Everything about the game - Resource-aware : Everything about G and set of players using e v s - t 200 100 150 75 100 50 50 25 0 0 0 1 2 3 4 0 1 2 3 4 s - t 100 75 s 50 25 0 0 1 2 3 4 16 16 12 12 8 8 4 4 0 0 1 2 3 4 0 0 1 2 3 4 6
Cost-Sharing Games Cl a sses of G a mes 7
Cost-Sharing Games Cl a sses of G a mes • Classes of graphs - Directed Acyclic Graphs (DAGs) - Series Parallel Graph 7
Cost-Sharing Games Cl a sses of G a mes 4000s • Classes of graphs 3000s - Directed Acyclic Graphs (DAGs) 2000s - Series Parallel Graph 1000s • Classes of cost functions 3000s 0s - Convex 2250s - Concave 1500s 750s 0s 7
Cost-Sharing Games Cl a sses of G a mes • Classes of graphs - Directed Acyclic Graphs (DAGs) s t - Series Parallel Graph • Classes of cost functions s 2 - Convex s 3 - Concave s 1 t • Symmetric and multicast s 4 7
Cost-Sharing Games Cl a sses of G a mes • Classes of graphs - Directed Acyclic Graphs (DAGs) Concave Convex s t - Series Parallel Graph PoA = 1 PoA = 1 SPG Leader-based protocol for parallel links (symmetric) • Classes of cost functions Incremental protocol [Moulin ’99] [Christodoulou et al. ’17] PoA = 2 + ε → PoA = Ω ( n ) Budget Balance s 2 - Convex DAG n > 1 → PoA = 1 + ε (symmetric) → PoA > 1.18 Overcharging s 3 With overcharging - Concave s 1 t → PoA = Ω ( n ) Budget Balance • Symmetric and multicast Multicast Overcharging → PoA = Ω ( n ) s 4 7
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