Selfish Routing and the Price of Anarchy - Yale University ...

Selfish Routingand thePrice of AnarchyTim RoughgardenCornell University1

Algorithms forSelf-Interested AgentsOur focus: problems in which multipleagents (people, computers, etc.) interactMotivation: the Internet• decentralized operation and ownershipTraditional algorithmic approach:• agents classified as obedient or adversarial– examples: distributed algorithms, cryptography2

Algorithms and Game TheoryRecent trend: agents have own independentobjectives (and act accordingly)New goal: algorithms that account forstrategic behavior by self-interested agentsNatural tool: game theory• theory of “rational behavior” in competitive,collaborative settings– [von Neumann/Morgenstern 44]3

ObjectivesThis talk: understand consequences ofnoncooperative behavior• when is the cost of selfish behavior severe?–the “price of anarchy” [Koutsoupias/Papadimitriou 99]• what can we do about it?– design strategies, economic incentivesOur setting: routing in a congested network• will focus on [Roughgarden/Tardos FOCS ’00/JACM ’02]• and also [Roughgarden STOC ’02/JCSS to appear]4

Motivating ExampleExample: one unit of traffic wants to go froms to tl(x)=xdelay depends on congestionsl(x)=1tno congestion effectsQuestion: what will selfish network users do?• assume everyone wants smallest-possible delay5

Motivating ExampleClaim: all traffic will take the top link.l(x)=xFlow = 1-Єthis flowis envious!sl(x)=1tFlow = ЄReason:• Є > 0 ⇒ traffic on bottom is envious• Є = 0 ⇒ envy-free outcome– all traffic incurs one unit of delay6

Can We Do Better?Consider instead: traffic split equallyl(x)=xFlow = ½sImprovement:l(x)=1• half of traffic has delay 1 (same as before)• half of traffic has delay ½ (much improved!)tFlow = ½7

Braess’s ParadoxInitial Network:s½ ½x 1½ ½1xDelay = 1.5t8

Braess’s ParadoxInitial Network: Augmented Network:s½ ½x 1½ ½1xts½ ½x 1½0½1xtDelay = 1.5Now what?9

Braess’s ParadoxInitial Network: Augmented Network:s½ ½x 1½ ½1xtsx 101xtDelay = 1.5 Delay = 210

Braess’s ParadoxInitial Network: Augmented Network:s½ ½x 1½ ½1xtsx 101xtDelay = 1.5 Delay = 2All traffic incurs more delay! [Braess 68]• also has physical analogs [Cohen/Horowitz 91]11

The Mathematical Model• a directed graph G = (V,E)• k source-destination pairs (s 1 ,t 1 ), …, (s k ,t k )• a rate r i of traffic from s i to t i• for each edge e, a latency function l e (•)– assumed continuous and nondecreasingExample: (k,r=1)l(x)=xFlow = ½s 1 t 1l(x)=1Flow = ½12

Routings of TrafficTraffic and Flows:• f P = amount of traffic routed on s i -t i path P• flow vector f routing of trafficstSelfish routing: what flows arise as the routeschosen by many noncooperative agents?13

Nash FlowsSome assumptions:• agents small relative to network• want to minimize personal latencyDef: A flow is at Nash equilibrium (or is a Nash flow)if all flow is routed on min-latency paths[given current edge congestion]Example:sx1Flow = .5tFlow = .5sx1Flow = 1tFlow = 014

Some History• traffic model, definition of Nash flowsgiven by [Wardrop 52]– historically called user-optimal/user equilibrium• Nash flows exist, are (essentially) unique– due to [Beckmann et al. 56]• Nash flows also arise via distributedshortest-path protocols (e.g., OSPF, BGP)– as long as latency used for edge weights– convergence studied in [Tsitsiklis/Bertsekas 86]15

The Cost of a FlowDef: the cost C(f) of flow f = sum of alldelays incurred by traffic (aka total latency)sx1½½tCost = ½•½ +½•1 = ¾16

The Cost of a FlowDef: the cost C(f) of flow f = sum of alldelays incurred by traffic (aka total latency)sx1½½tCost = ½•½ +½•1 = ¾Formally: if l P (f) = sum of latencies of edgesof P (w.r.t. the flow f), then:stC(f) = Σ P f P • l P (f)17

Inefficiency of Nash FlowsNote: Nash flows do not minimize total latency• observed informally by [Pigou 1920]• lack of coordination leads to inefficiencysx1 ½01½t• Cost of Nash flow = 1•1 + 0•1 = 1• Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾18

How Bad Is Selfish Routing?xPigou’s exampleis simple…s1 ½1t0½Central question: How inefficient are Nashflows in more realistic networks?Goal: prove that Nash flows are near-optimal• want laissez-faire approach to managing networks19

The Bad NewsBad Example:sx d1 1-Є01Є(r = 1, d large)tNash flow has cost 1, min cost ≈ 0⇒ Nash flow can cost arbitrarily more thanthe optimal (min-cost) flow– even if latency functions are polynomials20

Hardware Offsets SelfishnessApproach #1: use different type of guaranteeTheorem: [Roughgarden/Tardos 00] for everynetwork:Nash cost at rate r≤opt cost at rate 2r21

Hardware Offsets SelfishnessApproach #1: use different type of guaranteeTheorem: [Roughgarden/Tardos 00] for everynetwork:Nash cost at rate r≤opt cost at rate 2rAlso: M/M/1 fns (l(x)=1/(u-x), u = capacity) ⇒Nash w/capacities 2u≤opt w/capacities u22

Linear Latency FunctionsApproach #2: restrict class of allowablelatency functionsDef: linear latency fn is of form l e (x)=a e x+b eTheorem: [Roughgarden/Tardos 00] for everynetwork with linear latency fns:cost ofNash flow≤ 4/3 ×cost ofopt flow23

Sources of InefficiencyCorollary of previous Theorem:• For linear latency fns, worst Nash/OPTratio is realized in a two-link network!sx1 ½01½t• Cost of Nash = 1• Cost of OPT = ¾• simple explanation for worst inefficiency– confronted w/two routes, selfish usersovercongest one of them24

Simple Worst-Case NetworksTheorem: [Roughgarden 02] fix any class oflatency fns, and the worst Nash/OPT ratiooccurs in a two-node, two-link network.• under mild assumptions (convexity, richness)• inefficiency of Nash flows always has simpleexplanation; simple networks are worst examples25

Simple Worst-Case NetworksTheorem: [Roughgarden 02] fix any class oflatency fns, and the worst Nash/OPT ratiooccurs in a two-node, two-link network.• under mild assumptions (convexity, richness)• inefficiency of Nash flows always has simpleexplanation; simple networks are worst examplesProof Idea: Nash flows solve a certainminimization problem• not quite total latency, but close• electrical current is physical analog26

Computing the Price ofAnarchyApplication: worst-case examples simple ⇒worst-case ratio is easy to calculateExample: polynomials with degree ≤ d,nonnegative coeffs ⇒ price of anarchyΘ(d/log d)x ds1t27

Hardware Offsets SelfishnessTheorem: [Roughgarden/Tardos 00] for everynetwork:Nash cost at rate r≤opt cost at rate 2rCorollary: networks with M/M/1 delay fns ⇒Nash w/capacities 2u≤opt w/capacities u28

Key DifficultySuppose f a Nash flow, f * an opt flow at twicethe rate. Want to show that C(f * ) ≥ C(f).Note: cost of f can be written asC(f) = Σ e f e • l e (f e )Similarly: C(f * ) = Σ e f * • l e (f * )Problem: what is the relation between l e (f e )and l e (f * )?eee29

Key TrickIdea: lower bound cost of f * using adifferent set of latency fns c such that:• easy to lower bound cost of f * w.r.t. latency fns c• cost of f * w.r.t. fns c ≈ cost of f * w.r.t. fns l30

Key TrickIdea: lower bound cost of f * using adifferent set of latency fns c such that:• easy to lower bound cost of f * w.r.t. latency fns c• cost of f * w.r.t. fns c ≈ cost of f * w.r.t. fns lThe construction:graph of lgraph of cl e (f e )0l e (f e )00 f e0 f e31

Lower Bounding OPTAssume for simplicity: only one commodity.• all traffic in Nash flow has same latency, say L• cost of Nash flow easy to compute: C(f) = rL32

Lower Bounding OPTAssume for simplicity: only one commodity.• all traffic in Nash flow has same latency, say L• cost of Nash flow easy to compute: C(f) = rLKey observation: latency of path P w.r.t.latency fns c with no congestion is l P (f)l e (f e )00 f epath latencyin Nash flow33

Lower Bounding OPTAssume for simplicity: only one commodity.• all traffic in Nash flow has same latency, say L• cost of Nash flow easy to compute: C(f) = rLKey observation: latency of path P w.r.t.latency fns c with no congestion is l P (f)l e (f e )00 f epath latencyin Nash flow⇒ cost of f * w.r.t. c is at least 2rL = 2C(f)34

Bounding the OverestimateSo far: cost of f * w.r.t. c is ≥ 2C(f).Claim: (will finish proof of Thm)[cost of f * w.r.t. c] -C(f * ) ≤ C(f).35

Bounding the OverestimateSo far: cost of f * w.r.t. c is ≥ 2C(f).Claim: (will finish proof of Thm)[cost of f * w.r.t. c] -C(f * ) ≤ C(f).Reason: difference in costs on edge e isl e (f e )0typical value ofc e (f* e )f* e - l e (f* e )f*e0 f*e f e36

Bounding the OverestimateSo far: cost of f * w.r.t. c is ≥ 2C(f).Claim: (will finish proof of Thm)[cost of f * w.r.t. c] -C(f * ) ≤ C(f).Reason: difference in costs on edge e isl e (f e )0typical value ofc e (f* e )f* e - l e (f* e )f*e0 f*e f e⇒ c e (f* e )f * e - l e (f*e )f * ≤ l e e (f e )f esum over edgesto get Claim37

SummaryGoal: prove that loss in network performancedue to selfish routing is not too large.Problem: a Nash flow can costfar more than an optimal flow.Solutions:• compare Nash to opt flow with extra traffic• restrict class of allowable edge latencyfunctions, obtain bounded price of anarchysx d1 1-Є01Єt38

Coping with SelfishnessGoal: design/manage networks so thatselfish routing “not too bad”⇒ adds algorithmic dimensionApproach #1: Network design• want to avoid Braess’s ParadoxstResults: [Roughgarden FOCS ‘01]• Braess’s Paradox can be arbitrarily severe inlarger networks, hard to efficiently detect• also [Lin/Roughgarden/Tardos, in prep]39

Coping with SelfishnessApproach #2: Stackelberg routing• some traffic routed centrally, selfish users reactto congestion accordingly• [Roughgarden STOC ‘01]: Stackelberg routing candramatically improve over the Nash flowApproach #3: Edge pricing• use economic incentives (taxes) to influenceselfish behavior• [Cole/Dodis/Roughgarden EC ‘03 + STOC ‘03]:explore this idea for selfish routing40

Future Research• Explore other game-theoretic environmentsusing an approximation framework– [Czumaj/Krysta/Voecking STOC ‘02], [Vetta FOCS ‘02], etc.• Approximation algorithms for network design– also interesting without game-theoretic constraints– [Kumar/Gupta/Roughgarden FOCS ‘02]– [Gupta/Kumar/Roughgarden STOC ‘03]• Algorithms for key game-theoretic concepts– Nash/market equilibria (e.g., [Devanur et al FOCS ‘02])41

ExtensionsFact: positive results continue to hold for:• approximate Nash flows [RT00]– users route on approximately min-latency paths• finitely many agents, splittable flow [RT00]– weakens assumption that agents are small• “nonatomic congestion games”, gameswithout combinatorial structure of anetwork [RT02]42