Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Wardrop Equilibria and Price of Stability in Bottleneck Games With Splittable Traffic Vladimir Mazalov 1 Burkhard Monien 2 Florian Schoppmann 2 Karsten Tiemann 2 1 Karelian Research Center, Russian Academy of Sciences, Russia 2 University of Paderborn, Germany December 17, 2006 University of Paderborn Florian Schoppmann Dec. 17, 2006 1 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Motivation Communication, traffic, and logistics networks: ◮ Goal: high “network performance” ◮ Decentralized networks due to: ◮ Central coordination inherently impossible ◮ System of autonomous agents → tractable subproblems ◮ . . . Famous model due to Wardrop (1952): Selfish drivers minimize own travel time Computer scientist’s questions: ◮ Predictions → Game theory: Equilibria ◮ Quantify loss due to selfishness → “Prices of anarchy/stability” University of Paderborn Florian Schoppmann Dec. 17, 2006 2 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Motivation Communication, traffic, and logistics networks: ◮ Goal: high “network performance” ◮ Decentralized networks due to: ◮ Central coordination inherently impossible ◮ System of autonomous agents → tractable subproblems ◮ . . . Famous model due to Wardrop (1952): Selfish drivers minimize own travel time Computer scientist’s questions: ◮ Predictions → Game theory: Equilibria ◮ Quantify loss due to selfishness → “Prices of anarchy/stability” University of Paderborn Florian Schoppmann Dec. 17, 2006 2 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Single-Commodity Routing Games Definition (Bottleneck Game Γ ) a Γ = ( G , s , t , ( f e ) e ∈ E , r ) x 1 p 1 ◮ G = ( V , E ) multigraph s t 1 ◮ s , t ∈ V source/destination p 2 1 1 ◮ f e : R ≥ 0 → R ≥ 0 ∪ {∞} b nonnegative, continuous, and nondecreasing latency function ◮ V = { a , b , c , d } ◮ r ∈ R > 0 amount of s - t traffic ◮ E = { ( s , a ) , ( a , t ) , . . . } ◮ r = 1 ( G , s , t , ( f e ) e ∈ E ) is called a network. ◮ f ( s , a ) ( x ) = x , . . . P := { all simple paths from s to t } ◮ p 1 = { ( s , a ) , ( a , t ) } , . . . L := { λ ∈ R P ≥ 0 | � p ∈P λ p = r } set ◮ P = { p 1 , p 2 } of load vectors University of Paderborn Florian Schoppmann Dec. 17, 2006 3 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Motivation Revisited Classic Wardrop Model: a ◮ (Uncountably) infinitely many x 1 p 1 players, each having a negligible effect on the system s t 1 p 2 ◮ Mathematically: Nonatomic 1 1 anonymous games, zero-sets of b players do not matter ◮ Wardrop’s (1952) first principle (Wardrop equilibrium): The journey times on all the routes actually used are equal, and less than those which would be experienced by a single vehicle on any unused route. However, when assuming steady streams of flow, one might be interested in throughput, not individual travel time. University of Paderborn Florian Schoppmann Dec. 17, 2006 4 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Equilibria in Bottleneck Games ◮ For e ∈ E denote by l e ( λ ) := � p ∈P| e ∈ p λ p the edge load of e ◮ For p ∈ P denote by ℓ p ( λ ) := � ( f e ( l e ( λ ))) e ∈ p � ∞ the path latency of p . Definition (Wardrop Equilibrium) A load vector λ ∈ L is a Wardrop equilibrium (WE) iff for all p ∈ P with λ p > 0 it holds that ℓ p ( λ ) = min q ∈P { ℓ q ( λ ) } . What is different? Classic Wardrop Games: Bottleneck Games: ℓ p ( λ ) = � f e ( l e ( λ )) � 1 ℓ p ( λ ) = � f e ( l e ( λ )) � ∞ � = max e ∈ p { f e ( l e ( λ )) } = f e ( l e ( λ )) e ∈ p University of Paderborn Florian Schoppmann Dec. 17, 2006 5 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Measuring Social Utility Social cost defined as before ( → average path latency): � SC (Γ , λ ) := λ p · ℓ p ( λ ) p ∈P Optimal social cost: OPT (Γ) := λ ∈L (Γ) { SC (Γ , λ ) } min Worst-case ratios between stable states and the respective optima (Prices of anarchy/stability) for specific non-empty classes G of games: � SC (Γ , λ ) � PoA ( G ) := sup OPT (Γ) Γ ∈ G λ WE in Γ � � SC (Γ , λ ) �� PoS ( G ) := sup inf OPT (Γ) λ WE in Γ Γ ∈ G University of Paderborn Florian Schoppmann Dec. 17, 2006 6 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Measuring Social Utility (II) Define for a set of latency functions F : ◮ G ( F ) := class of bottleneck games with latency functions drawn from F ◮ P ( F ) := class of all games in G ( F ) whose graph only consists of parallel edges First objective: Similar result as Roughgarden/Tardos (2002) for bottleneck games? For instance, let P 1 denote set of affine latency functions (with positive coefficients). PoA ( G ( P 1 )) = ? University of Paderborn Florian Schoppmann Dec. 17, 2006 7 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Reusable Results? The classic model has been studied extensively—which results carry over, which not? ◮ Trivially, no difference on parallel edges ◮ It makes a difference whether non-simple paths are allowed. For e � = ( s , t ) : 3 1 1 s t p 1 f e ( x ) := 4 − x , f ( s , t ) ( x ) := 3 − x p 2 Equilibrium: λ = ( 1 , 2 ) a ◮ Wardrop equilibria always exist (Schmeidler, 1973) University of Paderborn Florian Schoppmann Dec. 17, 2006 8 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Reusable Results? The classic model has been studied extensively—which results carry over, which not? ◮ Trivially, no difference on parallel edges ◮ It makes a difference whether non-simple paths are allowed. For e � = ( s , t ) : 3 1 1 s t p 1 f e ( x ) := 4 − x , f ( s , t ) ( x ) := 3 − x p 3 p 2 Equilibria: λ = ( 1 , 2 , 0 , . . . ) λ ′ = ( 0 , 1 , 2 , 0 , . . . ) a ◮ Wardrop equilibria always exist (Schmeidler, 1973) University of Paderborn Florian Schoppmann Dec. 17, 2006 8 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Series Parallel Graphs Recursively defined: ◮ Base case: s t ◮ An arbitrary multigraph G is series parallel iff it can be constructed from two series parallel graphs. If s 1 G 1 t 1 and s 2 G 2 t 2 are series parallel then G 1 s 1 G 1 G 2 t 2 s t and G 2 are series parallel. University of Paderborn Florian Schoppmann Dec. 17, 2006 9 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Strong Cuts—From Equilibria to Maximum Flows Definition (Strong Cut) Let Γ = ( G , s , t , ( f e ) e ∈ E , r ) be a bottleneck game and λ ∈ L a Wardrop equilibrium. A cut S � V is called strong with respect to Γ and λ iff f e ( l e ( λ )) ≥ SC (Γ , λ ) for all edges leaving S and r l e ( λ ) = 0 for all edges e ∈ E going into S . Note: SC (Γ , λ ) = min p ∈P { ℓ p ( λ ) } is unique path latency for all used r paths in λ 5 5 a 1 1 6 − x 6 − x 6 s t 1 1 2 − x 6 − x b 1 1 University of Paderborn Florian Schoppmann Dec. 17, 2006 10 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Existence of Strong Cuts Lemma Let G be a series parallel graph, Γ = ( G , s , t , ( f e ) e ∈ E , r ) , and λ WE of Γ . Then a strong cut S � V with respect to Γ , λ exists. Proof (By structural induction). Induction hypothesis: Every series parallel graph G fulfills: If Γ is a bottleneck game on G and λ is WE for Γ , then a strong cut w.r.t. Γ , λ exists. ◮ Base Case: Trivial ◮ Induction Step, Parallel Connection: For i ∈ { 1 , 2 } : Let r i be traffic through G i G 1 Then strong cuts S i w.r.t. s t ( G i , s , t , ( f e ) e ∈ E i , r i ) , λ exist. G 2 Combine ( S = S 1 ∪ S 2 ) to get strong cut for G w.r.t. Γ , λ University of Paderborn Florian Schoppmann Dec. 17, 2006 11 / 27 · ·
Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion Existence of Strong Cuts Lemma Let G be a series parallel graph, Γ = ( G , s , t , ( f e ) e ∈ E , r ) , and λ WE of Γ . Then a strong cut S � V with respect to Γ , λ exists. Proof (By structural induction). Induction hypothesis: Every series parallel graph G fulfills: If Γ is a bottleneck game on G and λ is WE for Γ , then a strong cut w.r.t. Γ , λ exists. ◮ Base Case: Trivial ◮ Induction Step, Parallel Connection: For i ∈ { 1 , 2 } : Let r i be traffic through G i G 1 Then strong cuts S i w.r.t. s t ( G i , s , t , ( f e ) e ∈ E i , r i ) , λ exist. G 2 Combine ( S = S 1 ∪ S 2 ) to get strong cut for G w.r.t. Γ , λ University of Paderborn Florian Schoppmann Dec. 17, 2006 11 / 27 · ·
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