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Selfish Routing and the
Price of Anarchy
Tim RoughgardenCornell University
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Algorithms for Self-Interested Agents
Our focus: problems in which multiple agents (people, computers, etc.) interact
Motivation: the Internet• decentralized operation and ownership
Traditional algorithmic approach:• agents classified as obedient or adversarial
– examples: distributed algorithms, cryptography
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Algorithms and Game Theory
Recent trend: agents have own independent objectives (and act accordingly)
New goal: algorithms that account for strategic behavior by self-interested agents
Natural tool: game theory• theory of “rational behavior” in competitive,
collaborative settings– [von Neumann/Morgenstern 44]
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Objectives
This 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 incentives
Our setting: routing in a congested network• will focus on [Roughgarden/Tardos FOCS ’00/JACM ’02]• and also [Roughgarden STOC ’02/JCSS to appear]
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Motivating Example
Example: one unit of traffic wants to go from s to t
Question: what will selfish network users do?• assume everyone wants smallest-possible delay
s t
ℓ(x)=x
ℓ(x)=1
delay depends on congestion
no congestion effects
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Motivating Example
Claim: all traffic will take the top link.
Reason:• Є > 0 ⇒ traffic on bottom is envious• Є = 0 ⇒ envy-free outcome
– all traffic incurs one unit of delay
s t
ℓ(x)=x
ℓ(x)=1
Flow = 1-Є
Flow = Єthis flow is envious!
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Can We Do Better?
Consider instead: traffic split equally
Improvement:• half of traffic has delay 1 (same as before)• half of traffic has delay ½ (much improved!)
s t
ℓ(x)=x
ℓ(x)=1
Flow = ½
Flow = ½
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Braess’s Paradox
Initial Network:
s tx 1
½
x1½
½
½
Delay = 1.5
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Braess’s Paradox
Initial Network: Augmented Network:
s tx 1
½
x1½
½
½
Delay = 1.5
s tx 1
½
x1½
½
½0
Now what?
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Braess’s Paradox
Initial Network: Augmented Network:
s tx 1
½
x1½
½
½
Delay = 1.5 Delay = 2
s tx 1
x10
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Braess’s Paradox
Initial Network: Augmented Network:
All traffic incurs more delay! [Braess 68]
• also has physical analogs [Cohen/Horowitz 91]
s tx 1
½
x1½
½
½
Delay = 1.5 Delay = 2
s tx 1
x10
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The Mathematical Model
• a directed graph G = (V,E)• k source-destination pairs (s1 ,t1), …, (sk ,tk)• a rate ri of traffic from si to ti• for each edge e, a latency function ℓe(•)
– assumed continuous and nondecreasing
s1 t1
ℓ(x)=x Flow = ½
Flow = ½ℓ(x)=1
Example: (k,r=1)
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Routings of Traffic
Traffic and Flows:• fP = amount of traffic routed on si-ti path P• flow vector f routing of traffic
Selfish routing: what flows arise as the routes chosen by many noncooperative agents?
s t
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Nash Flows
Some assumptions:• agents small relative to network• want to minimize personal latency
Def: A flow is at Nash equilibrium (or is a Nash flow) if all flow is routed on min-latency paths [given current edge congestion]
xs t
1Flow = .5
Flow = .5
s t1
Flow = 0
Flow = 1x
Example:
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Some History• traffic model, definition of Nash flows
given 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 distributed shortest-path protocols (e.g., OSPF, BGP)– as long as latency used for edge weights– convergence studied in [Tsitsiklis/Bertsekas 86]
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The Cost of a FlowDef: the cost C(f) of flow f = sum of all
delays incurred by traffic (aka total latency)
s t
x
1½½
Cost = ½•½ +½•1 = ¾
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The Cost of a FlowDef: the cost C(f) of flow f = sum of all
delays incurred by traffic (aka total latency)
Formally: if ℓP(f) = sum of latencies of edges of P (w.r.t. the flow f), then:
C(f) = ΣP fP • ℓP(f)
s ts t
x
1½½
Cost = ½•½ +½•1 = ¾
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Inefficiency of Nash FlowsNote: Nash flows do not minimize total latency • observed informally by [Pigou 1920]• lack of coordination leads to inefficiency
• Cost of Nash flow = 1•1 + 0•1 = 1• Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾
s t
x
10
1 ½
½
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How Bad Is Selfish Routing?
s t
x
10
1 ½
½
Pigou’s example is simple…
Central question: How inefficient are Nash flows in more realistic networks?
Goal: prove that Nash flows are near-optimal • want laissez-faire approach to managing networks
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The Bad NewsBad Example: (r = 1, d large)
Nash flow has cost 1, min cost ≈ 0
⇒ Nash flow can cost arbitrarily more than the optimal (min-cost) flow– even if latency functions are polynomials
s t
xd
10
1 1-Є
Є
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Hardware Offsets Selfishness
Approach #1: use different type of guarantee
Theorem: [Roughgarden/Tardos 00] for every network:
≤Nash cost at rate r opt cost at rate 2r
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Hardware Offsets Selfishness
Approach #1: use different type of guarantee
Theorem: [Roughgarden/Tardos 00] for every network:
≤
Also: M/M/1 fns (ℓ(x)=1/(u-x), u = capacity) ⇒
≤
Nash cost at rate r opt cost at rate 2r
Nash w/capacities 2u opt w/capacities u
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Linear Latency Functions
Approach #2: restrict class of allowable latency functions
Def: linear latency fn is of form ℓe(x)=aex+be
Theorem: [Roughgarden/Tardos 00] for every network with linear latency fns:
≤ 4/3 × cost of Nash flow
cost of opt flow
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Sources of Inefficiency
Corollary of previous Theorem:• For linear latency fns, worst Nash/OPT
ratio is realized in a two-link network!
• simple explanation for worst inefficiency– confronted w/two routes, selfish users
overcongest one of them
s t
x
10
1 ½
½
• Cost of Nash = 1
• Cost of OPT = ¾
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Simple Worst-Case NetworksTheorem: [Roughgarden 02] fix any class of
latency fns, and the worst Nash/OPT ratio occurs in a two-node, two-link network.
• under mild assumptions (convexity, richness)• inefficiency of Nash flows always has simple
explanation; simple networks are worst examples
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Simple Worst-Case NetworksTheorem: [Roughgarden 02] fix any class of
latency fns, and the worst Nash/OPT ratio occurs in a two-node, two-link network.
• under mild assumptions (convexity, richness)• inefficiency of Nash flows always has simple
explanation; simple networks are worst examples
Proof Idea: Nash flows solve a certain minimization problem
• not quite total latency, but close• electrical current is physical analog
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Computing the Price of Anarchy
Application: worst-case examples simple ⇒worst-case ratio is easy to calculate
Example: polynomials with degree ≤ d, nonnegative coeffs ⇒ price of anarchy Θ(d/log d)
s t
xd
1
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Hardware Offsets Selfishness
Theorem: [Roughgarden/Tardos 00] for every network:
≤
Corollary: networks with M/M/1 delay fns ⇒
≤
Nash cost at rate r opt cost at rate 2r
Nash w/capacities 2u opt w/capacities u
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Key Difficulty
Suppose f a Nash flow, f* an opt flow at twice the rate. Want to show that C(f*) ≥ C(f).
Note: cost of f can be written as
C(f) = Σe fe• ℓe(fe)
Similarly: C(f*) = Σe f*• ℓe(f*)
Problem: what is the relation between ℓe(fe)and ℓe(f*)?
e
e
e
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Key Trick
Idea: lower bound cost of f* using a different 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 ℓ
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Key Trick
Idea: lower bound cost of f* using a different 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 ℓ
The construction:
ℓe(fe)
00 fe
graph of ℓ
ℓe(fe)
00 fe
graph of c
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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) = rL
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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) = rL
Key observation: latency of path P w.r.t. latency fns c with no congestion is ℓP(f)
ℓe(fe)
00 fe
path latency in Nash flow
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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) = rL
Key observation: latency of path P w.r.t. latency fns c with no congestion is ℓP(f)
⇒ cost of f* w.r.t. c is at least 2rL = 2C(f)
ℓe(fe)
00 fe
path latency in Nash flow
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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).
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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 is
ℓe(fe)
00 fe
typical value of ce(fe)fe - ℓe(fe)fe
* * * *
fe*
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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 is
⇒ ce(fe)fe - ℓe(fe)fe ≤ ℓe(fe)fe
ℓe(fe)
00 fe
typical value of ce(fe)fe - ℓe(fe)fe
* * * *
fe* sum over edges
to get Claim* ** *
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SummaryGoal: prove that loss in network performance
due to selfish routing is not too large.
Problem: a Nash flow can cost far more than an optimal flow.
Solutions: • compare Nash to opt flow with extra traffic• restrict class of allowable edge latency
functions, obtain bounded price of anarchy
s t
xd
10
1 1-Є
Є
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Coping with Selfishness
Goal: design/manage networks so that selfish routing “not too bad”
⇒ adds algorithmic dimension
Approach #1: Network design• want to avoid Braess’s Paradox
Results: [Roughgarden FOCS ‘01]• Braess’s Paradox can be arbitrarily severe in
larger networks, hard to efficiently detect• also [Lin/Roughgarden/Tardos, in prep]
s t
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Coping with SelfishnessApproach #2: Stackelberg routing• some traffic routed centrally, selfish users react
to congestion accordingly• [Roughgarden STOC ‘01]: Stackelberg routing can
dramatically improve over the Nash flow
Approach #3: Edge pricing• use economic incentives (taxes) to influence
selfish behavior• [Cole/Dodis/Roughgarden EC ‘03 + STOC ‘03]:
explore this idea for selfish routing
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Future Research• Explore other game-theoretic environments
using 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])
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Extensions
Fact: 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”, games without combinatorial structure of a network [RT02]