Tight Bounds for Cost-Sharing in Weighted Congestion Games Martin Gairing University of Liverpool Joint work with: Konstantinos Kollias (Stanford University) Grammateia Kotsialou (University of Liverpool) Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 1 / 18
Weighted Congestion Games Setting: Players with asymmetric demands (weights) compete for resources Each one selects which resources she will use A joint cost C e ( x ) = x · c e ( x ) is induced on resource e (x = total users weight) The joint cost is paid for by the users of e Players try to minimize their individual costs Applications: Scheduling Network design Selfish routing Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 2 / 18
Example: A Routing Game x 3 → 64 1 s t 3 16 · x Players: The two s - t flows with weights 1 and 3 Resources: The two edges with cost functions x 3 and 16 · x Joint cost on top edge is 64. How is it shared between red and blue? Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 3 / 18
Cost-Sharing and Equilibria Cost-Sharing Method: Input: Weights of edge users and a cost function Output: Cost shares for the users Interpretation in Delay Settings: Flows of packets with Poisson rates equal to players’ weights Cost functions are M/M/1 aggregate queueing delays Messing around with packets in the queue changes the aggregate delays of individual players but not the overall aggregate delay Pure Nash Equilibrium (PNE): Focus on each i , fix the paths of others We have a PNE if every i is already using her best path Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 4 / 18
Proportional Sharing Informal Definition (Proportional Sharing) α fraction of the total weight → α fraction of the cost x 3 → 16 + 48 1 s t 3 16 · x This is a PNE Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 5 / 18
Proportional Sharing Informal Definition (Proportional Sharing) α fraction of the total weight → α fraction of the cost x 3 1 s t 3 16 · x → 16 + 48 This is not a PNE, both want to deviate Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 6 / 18
Proportional Sharing Informal Definition (Proportional Sharing) α fraction of the total weight → α fraction of the cost x 3 → 1 1 s t 3 16 · x → 48 This is a PNE Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 7 / 18
Proportional Sharing Informal Definition (Proportional Sharing) α fraction of the total weight → α fraction of the cost x 3 → 27 1 s t 3 16 · x → 16 Optimal paths (also a PNE) Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 8 / 18
Shapley Value Informal Definition (Shapley Value) The cost share of a player is the average joint cost jump she causes over all possible orderings. c 0 ( x ) c 0 ( x ) x 3 → 64 Order: Order: 1 3 3 1 37 63 27 1 1 s t 3 1 2 3 4 1 2 3 4 16 · x x x Resulting cost shares for x 3 are 19 and 45 Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 9 / 18
Weighted Shapley Value Informal Definition (Weighted Shapley Value) Shapley Value with distribution over orderings. Players have sampling weights and the last one is repeatedly drawn from the remaining set. c 0 ( x ) c 0 ( x ) x 3 → 64 Order: Order: 1 3 3 1 37 63 27 1 1 s t 3 1 2 3 4 1 2 3 4 16 · x x x Suppose sampling weights = weights. Cost shares become 7.5 and 56.5 Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 10 / 18
Price of Anarchy and Price of Stability Price of Anarchy (PoA) PoA( Ξ , C ) = worst case total cost in equilibrium optimal total cost Price of Stability (PoS) PoS( Ξ , C ) = best case total cost in equilibrium optimal total cost with Ξ = cost-sharing method, C = set of allowable cost functions Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 11 / 18
Price of Anarchy and Price of Stability Price of Anarchy (PoA) PoA( Ξ , C ) = worst case total cost in equilibrium optimal total cost Price of Stability (PoS) PoS( Ξ , C ) = best case total cost in equilibrium optimal total cost with Ξ = cost-sharing method, C = set of allowable cost functions Thought process: 1 The application gives us C 2 We decide on a cost-sharing method Ξ 3 How bad is the worst cast and best case equilibrium outcome? I for the worst-case network (given Ξ , C ) Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 11 / 18
What We Know Proportional Sharing: PoA I polynomial costs: PoA = Θ ( d/ ln d ) d +1 [ Awerbuch et al.’05 , Christodoulou & Koutsoupias’05 , Aland et al.’06 ] I Recipe for general C . [ Roughgarden’09, Bhawalkar et al.’10 ] unweighted: d ≤ PoS ≤ d + 1 [ Christodoulou & Koutsoupias’05, Caragiannis et al.’06, Christodoulou & Gairing’13 ] There are games with no PNE [ Libman & Orda’01 , Fotakis et al.’04 , Goemans et al.’05 ] Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 12 / 18
What We Know Proportional Sharing: PoA I polynomial costs: PoA = Θ ( d/ ln d ) d +1 [ Awerbuch et al.’05 , Christodoulou & Koutsoupias’05 , Aland et al.’06 ] I Recipe for general C . [ Roughgarden’09, Bhawalkar et al.’10 ] unweighted: d ≤ PoS ≤ d + 1 [ Christodoulou & Koutsoupias’05, Caragiannis et al.’06, Christodoulou & Gairing’13 ] There are games with no PNE [ Libman & Orda’01 , Fotakis et al.’04 , Goemans et al.’05 ] Weighted Shapley Values: PoA [ Kollias & Roughgarden’11, Gkatzelis et al.’14 ] Existence of PNE only guaranteed by weighted Shapley values [ Gopalakrishnan et al.’13 ] Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 12 / 18
Our Results: PoA Assumptions: 1 Convexity: Every cost function c ∈ C is continuous, nondecreasing, and convex. 2 Closure under dilation: If c ( x ) ∈ C → c ( ax ) ∈ C . 3 Consistency: Cost shares depend only on how players contribute to the joined cost. 4 Fairness: The cost share of a player on a resource is a convex function of her weight. Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 13 / 18
Our Results: PoA Assumptions: 1 Convexity: Every cost function c ∈ C is continuous, nondecreasing, and convex. 2 Closure under dilation: If c ( x ) ∈ C → c ( ax ) ∈ C . 3 Consistency: Cost shares depend only on how players contribute to the joined cost. 4 Fairness: The cost share of a player on a resource is a convex function of her weight. Main result 1: General PoA bounds A recipe for computing the PoA given: I the set of allowable cost functions, and I the cost sharing method. Matching upper and lower bounds. Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 13 / 18
Recipe (sketch) for PoA P is NE; P ∗ is optimum Smoothness argument: X X C ( P ) = ξ c e ( i, S e ( P )) i ∈ N e ∈ P i Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18
Recipe (sketch) for PoA P is NE; P ∗ is optimum Smoothness argument: X X X X C ( P ) = ξ c e ( i, S e ( P )) ≤ ξ c e ( i, S e ( P ) ∪ { i } ) e ∈ P ⇤ i ∈ N e ∈ P i i ∈ N i Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18
Recipe (sketch) for PoA P is NE; P ∗ is optimum Smoothness argument: X X X X C ( P ) = ξ c e ( i, S e ( P )) ≤ ξ c e ( i, S e ( P ) ∪ { i } ) e ∈ P ⇤ i ∈ N e ∈ P i i ∈ N i X X = ξ c e ( i, S e ( P ) ∪ { i } ) e ∈ E i ∈ S e ( P ⇤ ) Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18
Recipe (sketch) for PoA P is NE; P ∗ is optimum Smoothness argument: X X X X C ( P ) = ξ c e ( i, S e ( P )) ≤ ξ c e ( i, S e ( P ) ∪ { i } ) e ∈ P ⇤ i ∈ N e ∈ P i i ∈ N i X X = ξ c e ( i, S e ( P ) ∪ { i } ) ≤ λ · C ( P ∗ ) + µ · C ( P ) e ∈ E i ∈ S e ( P ⇤ ) Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18
Recipe (sketch) for PoA P is NE; P ∗ is optimum Smoothness argument: X X X X C ( P ) = ξ c e ( i, S e ( P )) ≤ ξ c e ( i, S e ( P ) ∪ { i } ) e ∈ P ⇤ i ∈ N e ∈ P i i ∈ N i X X = ξ c e ( i, S e ( P ) ∪ { i } ) ≤ λ · C ( P ∗ ) + µ · C ( P ) e ∈ E i ∈ S e ( P ⇤ ) Convex Program λ min 1 − µ s.t. µ ≤ 1 X ξ c ( i, T ∪ { i } ) ≤ λ · w T ⇤ · c ( w T ⇤ ) + µ · w T · c ( w T ) , ∀ c, T, T ∗ i ∈ T ⇤ Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18
Recipe (sketch) for PoA P is NE; P ∗ is optimum Smoothness argument: X X X X C ( P ) = ξ c e ( i, S e ( P )) ≤ ξ c e ( i, S e ( P ) ∪ { i } ) e ∈ P ⇤ i ∈ N e ∈ P i i ∈ N i X X = ξ c e ( i, S e ( P ) ∪ { i } ) ≤ λ · C ( P ∗ ) + µ · C ( P ) e ∈ E i ∈ S e ( P ⇤ ) Convex Program λ min 1 − µ s.t. µ ≤ 1 X ξ c ( i, T ∪ { i } ) ≤ λ · w T ⇤ · c ( w T ⇤ ) + µ · w T · c ( w T ) , ∀ c, T, T ∗ i ∈ T ⇤ Primal solution ⇒ upper bound Lagrangian Dual solution ⇒ construction of matching lower bound Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18
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