The Effectiveness of Stackelberg strategies and

Tolls for Network Congestion Games

Chaitanya SwamyUniversity of Waterloo

Network congestion gamesdirected graph G=(V,E) with source s, sink t latency functions le on edges: continuous, nondecreasing

le(x) = delay on edge e with x units of flow/trafficFlow has to be routed from s to t

Nonatomic game: infinite # of users controlling ε flow

Atomic splittable game: k users; user i controls Di flow that has to be routed (splittably) from s to t

Total volume of flow = 1 (so Σi Di = 1 in atomic case)

For a path P and flow f, lP(f) = ΣeP le(fe)

= latency of path P

lf(x) =

x

le(x) =

1s t

Price of Anarchy (PoA)Cost of a flow f, C(f) = Σe fe le(fe) = ΣP fP lP(f)

= total delay experienced by userso optimal flow C(o) = minfeasible flows f C(f) = OPT

Use Nash equilibrium to analyze selfish behaviorNash 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= maxNash flows fN

C(fN)/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 = li,e(x; e) = i.le(x) + e

i: user i’s sensitivity to delay

– i‘s flow routes selfishly wrt. latency f’ns li,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 latenciesthat 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:

G1 G1s1

t1 G2 G2

s2 t2

s t

G1 G1 G2

G2s t G1 G1

G1 G1

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

2x1

xx0.5

0.251

0.252x

1

xx

0.251

0.25

optimal flow o

LLF strategy g=0.5

Let g = LLF Stackelberg strategyh = induced Nash flow, i.e., h is Nash flow

wrt.

latency functions le’(x) = le(x+ge)

PoA of LLF for sepa graphsBasic property of Nash flow for nonatomic routing:f = (fP) is a Nash flow iff

fP>0 lP(f) lP(f’) for every s-t path P’

i.e., every flow-path used by Nash flow has minimum latency among all s-t pathsSepa 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.

fe>0 and f’e fe for all eP.

ii) Let P be any s-t path, f be any s-t flow. path P’ s.t.

fe>0 for all eP’, and P’ {eP: fe>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 eP, then lP(o) OPT/ .

(b) By part (i) of sepa lemma with f=o-g, f’=h, path P s.t. for all eP, he oe-ge, oe-ge>0.

So L* = Nash latency lP’(h) = lP(g+h) lP(o) OPT/ .ΣP’ hP’ lP’(g+h) (1-).L*.

(c) for an s-t path Q,

lQ(g+h) = ΣeQ: he=0 le(ge+he) + ΣeQ: he>0 le(ge+he) .

By part (ii) of sepa lemma (taking f=h), we get that path Q’ s.t. he>0 eQ’ and {eQ: he>0} Q’

ΣeQ: he>0 le(ge+he) ΣeQ’ le(ge+he) = lQ’(g+h) = L*

ΣQ gQ lQ(g+h) ΣQ gQ (lQ(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 li,e(x; te) = i.le(x) + e

on edge ei toll vs. time conversion factor for user i

•i routes her flow selfishly to minimize her disutility

•a flow profile (f1,…,fk) is an atomic Nash equilibrium, where fi is user i’s flow, if for each user i,

fi minimizes Σe fi,e li,e(fe; e) where f = Σi fi

Goal: find tolls that induce an optimal flow (if possible)

Heterogenous, multicommodity (or asymmetric) users: user i controls Di flow, has to be routed from si to ti

A convex program

Useful characterization of atomic Nash: given

flows (f1,…,fk), define Li,e(x; e) = i (le(x) + fi,ele’(x)) + e

Li,e measures the marginal cost of increasing user i’s flow on edge e

Then, (f1,…,fk) is an atomic Nash iff for each user i,

fi,P > 0 ΣeP Li,e(fe; e) ΣeP’ Li,e(fe; e) si-ti 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 Li,e(x; te) = i (le(x) + fi,ele’(x)) + e

(f1,…,fk) is an atomic Nash iff for each user i,

fi,P > 0 ΣeP Li,e(fe; te) ΣeP’ Li,e(fe; te) si-ti paths P’ (*)

Want Σi fi= H for some atomic Nash equilibrium (f1,…,fk)min A := Σi i (Σsi-ti paths P lP(H) + 0.5 Σe le’(He)fi,e

2)

s.t. Σi fi,e He edges e

Σsi-ti paths P fi,P = Di users i

fi,P 0 i, si-ti paths P

fi,e = Σ fi,Psi-ti paths Ps.t. eP

KKT conditions: (f1,…,fk) is an optimal solution iff e 0, zi :

– zi ΣeP e + ∂A/∂fi,P = ΣeP Li,e(fe; e) for every si-ti path P

Convex program (contd.)Define Li,e(x; te) = i (le(x) + fi,ele’(x)) + e

(f1,…,fk) is an atomic Nash iff for each user i,

fi,P > 0 ΣeP Li,e(fe; te) ΣeP’ Li,e(fe; te) si-ti paths P’ (*)

Want Σi fi= H for some atomic Nash equilibrium (f1,…,fk)min A := Σi i (Σsi-ti paths P lP(H) + 0.5 Σe le’(He)fi,e

2)

s.t. Σi fi,e He edges e

Σsi-ti paths P fi,P = Di users i

fi,P 0 i, si-ti paths P

fi,e = Σ fi,Psi-ti paths Ps.t. eP

Theorem: H is “induceable” iff optimal soln. (f1,…,fk) s.t. Σi

fi= H.

KKT conditions: (f1,…,fk) is an optimal solution iff e 0, zi :

– zi ΣeP e + ∂A/∂fi,P = ΣeP Li,e(fe; e) for every si-ti path P– fi,P > 0 zi = ΣeP e + ∂A/∂fi,P = ΣeP Li,e(fe; 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

Thank You