The Effectiveness of Stackelberg strategies and Tolls for Network Congestion Games Chaitanya Swamy University of Waterloo

Network congestion games directed graph G=(V,E) with source s, sink t latency functions l e on edges: continuous, nondecreasing l e (x) = delay on edge e with x units of flow/traffic Flow has to be routed from s to t Nonatomic game: infinite # of users controlling ε flow Atomic splittable game: k users; user i controls D i flow that has to be routed (splittably) from s to t Total volume of flow = 1 (so Σ i D i = 1 in atomic case) For a path P and flow f, l P (f)= Σ e  P l e (f e ) = latency of path P l f (x) = x l e (x) = 1 st

Price of Anarchy (PoA) Cost of a flow f, C(f)= Σ e f e l e (f e ) = Σ P f P l P (f) = total delay experienced by users o  optimal flowC(o)= min feasible flows f C(f) = OPT Use Nash equilibrium to analyze selfish behavior Nash equilibrium  combination of players’ strategies where no user has incentive to deviate unilaterally Price of anarchy (PoA) of network game = ratio of cost of worst Nash flow to OPT = max Nash flows f N C(f N )/C(o) PoA is unbounded for both nonatomic and atomic congestion games (as k  ) even when G has only 2 parallel links; Roughgarden & Tardos, Roughgarden

Two ways of reducing the PoA a) Stackelberg strategies – central authority controls some  -fraction of flow and routes it in any desired way – remaining (1-  )-fraction is routed selfishly – simple, no communication needed b/w system and selfish users, no notion of currency required – Korilis, Lazar & Orda (KLO97): first considered Stackelberg strategies to improve system performance  motivation came from virtual-private-network design, where system must allocate bandwidth on preassigned virtual paths

Two ways of reducing PoA b) Network tolls – impose tolls  e on edges: net disutility incurred by user i on edge e = l i,e (x;  e ) =  i. l e (x) +  e  i : user i’s sensitivity to delay – i‘s flow routes selfishly wrt. latency f’ns l i,e (x;  e ) – classic means of congestion control: proposed by Pigou way back in 1920 (P20). – known to be quite effective for nonatomic routing: optimal flow can be induced via tolls

Related Work Stackelberg strategies KLO97: for parallel-link graphs + MM1 latency f’ns. gave conditions under which  Stackelberg strategy that induces an optimal flow Roughgarden (R05): for parallel-link graphs  strategy that reduces PoA to 1/  for arbitrary latencies that reduces PoA to 4/(3+  ) for linear latencies Kumar & Marathe: PTAS for finding best strategy recent work: Kaporis & Spirakis, Sharma & Williamson, Karakostas & Kolliopoulos (KK06), Correa & Stier-Moses (CS06)

Related Work (contd.) Network tolls P20, Beckman, McGuire & Winston: marginal- cost tolls induce OPT for homogenous users Cole, Dodis & Roughgarden: tolls inducing optimal flow exist for heterogenous users Fleischer, Jain & Mahdian; Karakostas & Kolliopoulos; Yang & Huang: can find “optimal tolls” for heterogenous, multicommodity users Not much known in atomic case. Hayrapetyan, Tardos & Wexler; Cominetti, Correa & Stier- Moses: show C(atomic Nash)  C(nonatomic Nash) in some cases  optimal tolls exist in these cases Nonatomic routing

Our Results Stackelberg strategies: obtain first results for graphs more general than parallel-link graphs – series-parallel graphs: show that PoA is at most 1/  +1 for arbitrary latencies – general graphs: obtain latency-class specific bounds quantifying trade-off b/w price of anarchy and  PoA = 1;if  =1 PoA for latency-class;if  =0 (with no flow control) (Independently KK06 have obtained such results for linear latencies; CS06 have also obtained some results.) – parallel-link graphs: PoA is at most  + (1-  )(PoA without any flow control)  PoA always improves by controlling flow

Our Results Stackelberg strategies: obtain first results for graphs more general than parallel-link graphs – series-parallel graphs: PoA  1/  +1 for arbitrary latencies – general graphs: obtain latency-class specific bounds quantifying trade-off b/w price of anarchy and  – parallel-link graphs: PoA   + (1-  )(PoA with no control) Network tolls: optimal tolls exist for atomic splittable users, even heterogenous, multicommodity users – tolls can be computed by solving a convex program – results extend to general atomic splittable congestion games – completely characterize flows “induceable” via tolls

Series-Parallel (sepa) Graphs sepa graphs with ends s, t are defined inductively: is a sepa graph Given two sepa graphs:, Series construction: G 1 s1s1 t1t1 G 2 s2s2 t2t2 st G 1 G 2 s t G 1 s t Example: Parallel construction:

Largest-Latency-First (LLF) Compute an optimal flow o Saturate paths of o starting from largest latency path until  units are routed Generalization of the LLF strategy introduced by R05 for parallell-link graphs 2x 1 x x x 1 x x optimal flow o LLF strategy g  =0.5 Let g = LLF Stackelberg strategy h = induced Nash flow, i.e., h is Nash flow wrt. latency functions l e ’(x) = l e (x+g e )

PoA of LLF for sepa graphs Basic property of Nash flow for nonatomic routing: f = (f P ) is a Nash flow iff f P >0  l P (f)  l P (f’) for every s-t path P’ i.e., every flow-path used by Nash flow has minimum latency among all s-t paths Sepa lemma: Given a sepa graph with ends s, t: i) If f, f’ are two s-t flows where f routes more flow than f’, then there exists a path P s.t. f e >0 and f’ e  f e for all e  P. ii) Let P be any s-t path, f be any s-t flow.  path P’ s.t. f e >0 for all e  P’, and P’  {e  P: f e >0}.

Theorem: PoA of LLF is at most 1/  +1 on sepa graphs. Proof: (a) Due to LLF strategy, for any path P, if (o-g) e >0 for all e  P, then l P (o)  OPT/ . (b) By part (i) of sepa lemma with f=o-g, f’=h,  path P s.t. for all e  P, h e  o e -g e, o e -g e >0. So L * = Nash latency  l P ’(h) = l P (g+h)  l P (o)  OPT/ .  Σ P’ h P’ l P’ (g+h)  (1-  ). L *. (c) for an s-t path Q, l Q (g+h) = Σ e  Q: h e =0 l e (g e +h e ) + Σ e  Q: h e >0 l e (g e +h e ). By part (ii) of sepa lemma (taking f=h), we get that  path Q’ s.t. h e >0  e  Q’ and {e  Q: h e >0}  Q’  Σ e  Q: h e >0 l e (g e +h e )  Σ e  Q’ l e (g e +h e ) = l Q’ (g+h) = L *  Σ Q g Q l Q (g+h)  Σ Q g Q ( l Q (g)+L * )  C(g)+ . L *  C(g+h)  C(g)+L *  OPT. (1/  +1)

Tolls for atomic splittable users Given tolls   = {  e }: user i experiences net disutility l i,e (x; t e ) =  i. l e (x) +  e on edge e  i  toll vs. time conversion factor for user i i routes her flow selfishly to minimize her disutility a flow profile (f 1,…,f k ) is an atomic Nash equilibrium, where f i is user i’s flow, if for each user i, f i minimizes Σ e f i,e l i,e (f e ;  e ) where f = Σ i f i Goal: find tolls that induce an optimal flow (if possible) Heterogenous, multicommodity (or asymmetric) users: user i controls D i flow, has to be routed from s i to t i

A convex program Useful characterization of atomic Nash: given flows (f 1,…,f k ), define L i,e (x;  e ) =  i ( l e (x) + f i,e l e ’(x) ) +  e L i,e measures the marginal cost of increasing user i’s flow on edge e Then, (f 1,…,f k ) is an atomic Nash iff for each user i, f i,P > 0  Σ e  P L i,e (f e ;  e )  Σ e  P’ L i,e (f e ;  e )  s i -t i paths P’ (*) Key idea: (*) can be interpreted as the Kuhn-Karusch- Tucker (KKT) conditions of a suitable convex program derivative wrt. x

Convex program (contd.) Define L i,e (x; t e ) =  i ( l e (x) + f i,e l e ’(x) ) +  e (f 1,…,f k ) is an atomic Nash iff for each user i, f i,P > 0  Σ e  P L i,e (f e ; t e )  Σ e  P’ L i,e (f e ; t e )  s i -t i paths P’ (*) Want Σ i f i = H for some atomic Nash equilibrium (f 1,…,f k ) min A := Σ i  i ( Σ s i -t i paths P l P (H) Σ e l e ’(H e )f i,e 2 ) s.t. Σ i f i,e  H e  edges e Σ s i -t i paths P f i,P = D i  users i f i,P  0  i, s i -t i paths P f i,e = Σ f i,P s i -t i paths P s.t. e  P KKT conditions: (f 1,…,f k ) is an optimal solution iff  e  0, z i : – z i  Σ e  P  e + ∂A/∂f i,P = Σ e  P L i,e (f e ;  e ) for every s i -t i path P

Convex program (contd.) Define L i,e (x; t e ) =  i ( l e (x) + f i,e l e ’(x) ) +  e (f 1,…,f k ) is an atomic Nash iff for each user i, f i,P > 0  Σ e  P L i,e (f e ; t e )  Σ e  P’ L i,e (f e ; t e )  s i -t i paths P’ (*) Want Σ i f i = H for some atomic Nash equilibrium (f 1,…,f k ) min A := Σ i  i ( Σ s i -t i paths P l P (H) Σ e l e ’(H e )f i,e 2 ) s.t. Σ i f i,e  H e  edges e Σ s i -t i paths P f i,P = D i  users i f i,P  0  i, s i -t i paths P f i,e = Σ f i,P s i -t i paths P s.t. e  P Theorem: H is “induceable” iff  optimal soln. (f 1,…,f k ) s.t. Σ i f i = H. KKT conditions: (f 1,…,f k ) is an optimal solution iff  e  0, z i : – z i  Σ e  P  e + ∂A/∂f i,P = Σ e  P L i,e (f e ;  e ) for every s i -t i path P – f i,P > 0  z i = Σ e  P  e + ∂A/∂f i,P = Σ e  P L i,e (f e ;  e )

Open Questions Stackelberg routing on general graphs: bounded PoA for arbitrary latencies? What about multicommodity networks? Stackelberg strategies for other objectives? Understanding of atomic splittable Nash equilibria

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