How Bad is Selfish Routing Tim Roughgarden Eva Tardos presented by - PowerPoint PPT Presentation

How Bad is Selfish Routing Tim Roughgarden Eva Tardos presented by Yajun Wang (yalding@cs.ust.hk) for COMP670O Spring 2006, HKUST 1-1

Problem Formulation: Traffic Model • Given the rate of traffic between each pair of nodes in a network, find an assignment of traffic to minimize the total latency. • On each edge, the latency is load dependent • Each player controls a negligible fraction of the overall traffic. v v l ( x ) = 1 l ( x ) = x l ( x ) = 1 l ( x ) = x l ( x ) = 0 t s t s l ( x ) = 1 l ( x ) = x l ( x ) = 1 w l ( x ) = x w Braess’s Paradox 2-1

Formal Model • Graph G = ( V, E ) and k source-destination pairs { s i , t i } • P i denotes the set of (simple) s i − t i paths, and • P = ∪ i P i • A flow is a function: f : P → R + • A flow is feasible if : � P ∈P i f P = r i • Each edge has a nonnegative, differentiable, nondecreasing latency function l e ( · ) 3-1

Cost for Flows • Let ( G, r, l ) be an instance , and f is a flow. f e = � P : e ∈ P f P • Latency of a path P l P ( f ) = � e ∈ P l e ( f e ) • Cost of a flow f: C ( f ) = � P ∈P l P ( f ) f P = � e ∈ E l e ( f e ) f e • Players are small flows behave ”greedily” and ”selfishly” There are infinite number of players, each carry a negligible amout of flow. 4-1

Flows at Nash Equilibrium • Definition (Nash Equilibrium): A flow f is feasible for instance ( G, r, l ) is at Nash Equilibrium if for all i ∈ { 1 , . . . , k } , P 1 , P 2 ∈ P i , and δ ∈ [0 , f P 1 ] , we have l P 1 ( f ) ≤ l P 2 ( ˜ f ) , where  f P − δ if P = P 1  ˜ f P = f P + δ if P = P 2 f P if P / ∈ { P 1 , P 2 }  • Lemma: A flow f feasible for instance ( G, r, l ) is at Nash Equilibrium if and only if for all i ∈ { 1 , . . . , k } , P 1 , P 2 ∈ P i with f P 1 > 0 , l P 1 ( f ) ≤ l P 2 ( f ) . 5-1

Optimal Flows via Convex Programming • NonLinear Programming Formulation � Min c e ( f e ) e ∈ E subject to: � f P = r i ∀ i ∈ { 1 , . . . , k } P ∈P i � f e = f P ∀ e ∈ E P ∈P : e ∈ P f P ≥ 0 ∀ P ∈ P 6-1

Characteristic of Optimal Flows d Let c ′ e be the derivative dx c e ( x ) c ′ e ∈ P c ′ P ( f ) = � e ( f e ) • Lemma: A flow f is optimal for a convex program of the previous form if and only if for every i ∈ { 1 , . . . , k } and P 1 , P 2 ∈ P i with f P 1 > 0 , c ′ P 1 ( f ) ≤ c ′ P 2 ( f ) . 7-1

Characteristic of Optimal Flows d Let c ′ e be the derivative dx c e ( x ) c ′ e ∈ P c ′ P ( f ) = � e ( f e ) • Lemma: A flow f is optimal for a convex program of the previous form if and only if for every i ∈ { 1 , . . . , k } and P 1 , P 2 ∈ P i with f P 1 > 0 , c ′ P 1 ( f ) ≤ c ′ P 2 ( f ) . • Lemma: A flow f feasible for instance ( G, r, l ) is at Nash Equilibrium if and only if for all i ∈ { 1 , . . . , k } , P 1 , P 2 ∈ P i with f P 1 > 0 , l P 1 ( f ) ≤ l P 2 ( f ) . k � C ( f ) = L i ( f ) r i i =1 7-2

Nash Equilibrium and Optimal Flow Marginal cost function: e ( f e ) = ( l e ( f e ) f e ) ′ = l e ( f e ) + l ′ l ∗ e ( f e ) f e • Corollary: Let ( G, r, l ) be an instance in which x · l e ( x ) is a convex function for each edge e , with marginal cost functions l ∗ e . Then a flow f feasible for ( G, r, l ) is optimal if and only if it is at Nash equilibrium for the instance ( G, r, l ∗ ) 8-1

Nash Equilibrium and Optimal Flow (cont’) • Lemma: An instance ( G, r, l ) with continuous, nonde- creasing latency functions admits a feasible flow at Nash equilibrium. Moreover, if f, ˜ f are flows at Nash equilib- rium, then C ( f ) = C ( ˜ f ) . � x Proof: Set h e ( x ) = 0 l e ( t ) dt � Min h e ( f e ) � f P = r i ∀ i ∈ { 1 , . . . , k } e ∈ E P ∈P i � f e = f P ∀ e ∈ E P ∈P : e ∈ P Note, h ′ e ( x ) = l e ( x ) f P ≥ 0 ∀ P ∈ P 9-1

”Unique” Nash Equilibrium • Lemma: An instance ( G, r, l ) with continuous, nonde- creasing latency functions admits a feasible flow at Nash equilibrium. Moreover, if f, ˜ f are flows at Nash equilib- rium, then C ( f ) = C ( ˜ f ) . � x Set h e ( x ) = 0 l e ( t ) dt Proof (cont’) : � ˜ Min h e ( f e ) If f e � = f e , the function e ∈ E h e ( x ) must be linear and l e is a constant function This implies l e ( f e ) = l e ( ˜ f e ) . C ( f ) = � k i =1 L i ( f ) r i = C ( ˜ f ) . 10-1

Nontrivial Upper Bound for Price of Anarchy For instance ( G, r, l ) , let f ∗ be an optimal flow and f be a flow at Nash equilibrium. C ( f ) ρ = ρ ( G, r, l ) = C ( f ∗ ) Corollary: Suppose the instance ( G, r, l ) and the constant α ≥ 1 satisfy: � x x · l e ( x ) ≤ α · 0 l e ( t ) dt ρ ( G, r, l ) ≤ α 11-1

Nontrivial Upper Bound for Price of Anarchy (cont’) Corollary: Suppose the instance ( G, r, l ) and the constant α ≥ 1 satisfy: � x x · l e ( x ) ≤ α · 0 l e ( t ) dt ρ ( G, r, l ) ≤ α Proof: � C ( f ) = l e ( f e ) f e e ∈ E � f e � ≤ α l e ( t ) dt 0 e ∈ E � f ∗ N.E optimizes this ob- e � ≤ α l e ( t ) dt jective function. 0 e ∈ E � l e ( f ∗ e ) f ∗ ≤ α e e ∈ E α · C ( f ∗ ) = 12-1

Upper Bound for Polynomial Latency Function Corollary: Suppose the instance ( G, r, l ) has the latency func- tions: l e ( x ) = � p i =0 a e,i x i a e,i ≥ 0 ρ ( G, r, l ) ≤ p + 1 Remarks: It is not tight. l e ( x ) = a e x + b e for a e , b e ≥ 0 ρ ≤ 2 Tight Bound: ρ ≤ 4 / 3 For higher degree polynomial latency functions: ρ = O ( p ln p ) 13-1

A Bicriteria Result for General Latency Functions Negative Result: : l ( x ) = 1 s ρ = 4 / 3 t l ( x ) = x Optimal flows assgins ( p + 1) − 1 /p on the If l ( x ) = x p : lower link, which has a total latency: 1 − p ( p + 1) − ( p +1) /p → 0 ρ → ∞ 14-1

Augment Analysis for General Latency Function • Theorem: If f is a flow at Nash equilibrium for ( G, r, l ) and f ∗ is feasible for ( G, 2 r, l ) , then C ( f ) ≤ C ( f ∗ ) Let ¯ e (¯ � � l e ( f ∗ e ) f ∗ e − C ( f ∗ ) f ∗ l e ( f ∗ e ) − l e ( f ∗ = e )) � l e ( f e ) if x ≤ f e ¯ e e ∈ E l e ( x ) = l e ( x ) if x ≥ f e � ≤ l e ( f e ) f e e ∈ E = C ( f ) ¯ l P ( f ∗ ) ≥ ¯ ¯ � � � l P ( f ∗ ) f ∗ L i ( f ) f ∗ ≥ l P ( f 0 ) ≥ L i ( f ) P P e i P ∈P i � = 2 L i ( f ) r i i = 2 C ( f ) 15-1

Worst-Case Ratio with Linear Latency Fuctions l e = a e x + b e with a e , b e ≥ 0 l ∗ e = 2 a e x + b e • Lemma: If ( G, r, l ) be an instance with edge latency func- tions l e ( x ) = a e x + b e for each edge e ∈ E . Then (a) a flow f is at Nash equilibrium in G if and only if for P, P ′ ∈ P i with f P > 0 , � e ∈ P a e f e + b e ≤ � e ∈ P ′ a e f e + b e (b) a flow f ∗ is (globally) Optimal in G if and only if for P, P ′ ∈ P i with f ∗ P > 0 , e ∈ P 2 a e f ∗ e ∈ P ′ 2 a e f ∗ � e + b e ≤ � e + b e 16-1

Worst-Case Ratio with Linear Latency Fuctions (cont’) • Lemma: Suppose ( G, r, l ) has linear latency functions and f is a flow at Nash equilibrium. Then (a) The flow f/ 2 is optimal for ( G, r/ 2 , l ) (b) the marginal cost of increasing the flow on a path P for f/ 2 equals the latency of P for f l ∗ P ( f/ 2) = l P ( f ) Creating optimal flow in two steps: ( f is at Nash equilibrium) (1) Send a flow optimal for instance ( G, r/ 2 , l ) . C ( f ) / 4 (2) Augment to one optimal for instance ( G, r, l ) . C ( f ) / 2 17-1

Augment Cost for Linear Latency Functions • Lemma: ( G, r, l ) has linear latency functions and f ∗ is an optimal flow. Let L ∗ i ( f ∗ ) be the minimum marginal cost for s i − t i paths. For any δ > 0 , a feasible flow f for ( G, (1 + δ ) r, l ) : C ( f ) ≥ C ( f ∗ ) + δ � k i =1 L ∗ i ( f ∗ ) r i x · l e ( x ) = a e x 2 + b e is convex. e ) f ∗ + ( f e − f ∗ ) l ∗ l e ( f e ) f e ≥ l e ( f ∗ e ( f ∗ e ) 18-1

Augment Cost for Linear Latency Functions • Proof: � C ( f ) = l e ( f e ) f e e ∈ E � � l e ( f ∗ e ) f ∗ ( f e − f ∗ e ) l ∗ e ( f ∗ ≥ e + e ) e ∈ E e ∈ E k � � C ( f ∗ ) + l ∗ P ( f ∗ )( f P − f ∗ = P ) i =1 P ∈P i k � � C ( f ∗ ) + L ∗ i ( f ∗ ) ( f P − f ∗ ≥ P ) i =1 P ∈P i k � C ( f ∗ ) + δ L ∗ i ( f ∗ ) r i = i =1 19-1

Worst-Case Ratio with Linear Latency Fuctions (cont’) • Lemma: If ( G, r, l ) has linear latency functions, then ρ ( G, r, l ) ≤ 4 / 3 Proof: Let f be a flow at N.E. f/ 2 is optimal for ( G, r/ 2 , l ) . Moreover, L ∗ i ( f/ 2) = L i ( f ) . 1 e + 1 k 4 a e f 2 i ( f/ 2) r i C ( f/ 2) = 2 b e f e � C ( f ∗ ) L ∗ ≥ C ( f/ 2) + 2 1 i =1 � ( a e f 2 ≥ e + b e f e ) k 4 C ( f/ 2) + 1 e � = L i ( f ) r i 1 2 = 4 C ( f ) i =1 C ( f/ 2) + 1 = 2 C ( f ) 3 ≥ 4 C ( f ) 20-1

Extensions: • Approximate Nash Equilibrium: If f is at ǫ N.E, and f ∗ is feasible for ( G, 2 r, l ) , then C ( f ) ≤ 1+ ǫ 1 − ǫ C ( f ∗ ) . • Finite Agents: Splittable Flow C ( f ) ≤ C ( f ∗ ) . • Finite Agents: Unsplittable Flow If for some α < 2 , l e ( x + r i ) ≤ α · l e ( x ) , x ∈ [0 , � j � = i r j ] α 2 − α C ( f ∗ ) . C ( f ) ≤ 21-1