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Intrinsic Robustness of the Price of

Anarchy

Tim RoughgardenStanford University

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The Mathematical Model

• a directed graph G = (V,E)

• k source-destination pairs (s1 ,t1), …, (sk ,tk)

• a rate (amount) ri of traffic from si to ti

• for each edge e, a cost function ce(•)– assumed nonnegative, continuous,

nondecreasing

s1 t1

c(x)=x Flow = ½

Flow = ½

c(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 are the equilibria?

s t

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Nash Flows

Some assumptions:• agents small relative to network (nonatomic

game)• want to minimize cost of their path

Def: A flow is at Nash equilibrium (or is a Nash flow) if all flow is routed on min-cost paths [given current edge congestion]

xs t

1Flow = .5

Flow = .5

s t1

Flow = 0

Flow = 1x

Example:

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History + Generalizations

• model, defn of Nash flows by [Wardrop 52]

• Nash flows exist, are (essentially) unique– due to [Beckmann et al. 56]– general nonatomic games: [Schmeidler 73]

• congestion game (payoffs fn of # of players)– defined for atomic games by [Rosenthal 73]– previous focus: Nash eq in pure strategies exist

• potential game (equilibria as optima)– defined by [Monderer/Shapley 96]

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The Cost of a Flow

Def: the cost C(f) of flow f = sum of all costs incurred by traffic (avg cost × traffic rate)

s t

x

1½½

Cost = ½•½ +½•1 = ¾

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The Cost of a Flow

Def: the cost C(f) of flow f = sum of all costs incurred by traffic (avg cost × traffic rate)

Formally: if cP(f) = sum of costs of edges of P (w.r.t. the flow f), then:

C(f) = P fP • cP(f)

s ts t

x

1½½

Cost = ½•½ +½•1 = ¾

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Inefficiency of Nash Flows

Note: Nash flows do not minimize the cost • observed informally by [Pigou 1920]

• Cost of Nash flow = 1•1 + 0•1 = 1• Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾• Price of anarchy := Nash/OPT ratio = 4/3

s t

x

10

1 ½

½

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Braess’s Paradox

Initial Network:

s tx 1

½

x1½

½

½

cost = 1.5

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Braess’s Paradox

Initial Network: Augmented Network:

s tx 1

½

x1½

½

½

cost = 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

cost = 1.5 cost = 2

s t

x 1

x10

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Braess’s Paradox

Initial Network: Augmented Network:

All traffic incurs more cost! [Braess 68]

• see also [Cohen/Horowitz 91], [Roughgarden 01]

s tx 1

½

x1½

½

½

cost = 1.5 cost = 2

s t

x 1

x10

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The Bad News

Bad 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 cost functions are polynomials

s t

xd

10

1 1-Є

Є

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Linear Cost Functions

First: focus on special case.

Def: linear cost fn is of form ce(x)=aex+be

Theorem: [Roughgarden/Tardos 00] for every network with linear cost fns:

≤ 4/3 ×

i.e., price of anarchy ≤ 4/3 in the linear case.

cost of Nash flow

cost of opt flow

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Sources of Inefficiency

Corollary of previous Theorem:• For linear cost 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 Networks

Theorem: [Roughgarden 02] fix any class of cost fns, and the worst Nash/OPT ratio occurs in a two-node, two-link network.

• under mild assumptions• inefficiency of Nash flows always has simple

explanation; simple networks are worst examples

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Simple Worst-Case Networks

Theorem: [Roughgarden 02] fix any class of cost fns, and the worst Nash/OPT ratio occurs in a two-node, two-link network.

• under mild assumptions• inefficiency of Nash flows always has simple

explanation; simple networks are worst examples

Proof Idea: Nash flows minimize potential function

• potential function “close” to total cost function

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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 POA ≈ d/log d

• quartic functions: worst-case POA ≈ 2

• 10% extra "capacity": worst-case POA ≈ 2

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But Are We at Equilibrium?

Since 2002: price of anarchy (i.e., worst Nash/OPT ratio) analyzed in many models.

Critique: Usual interpretation of a POA bound presumes players reach equilibrium.

.

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But Are We at Equilibrium?

Since 2002: price of anarchy (i.e., worst Nash/OPT ratio) analyzed in many models.

Critique: Usual interpretation of a POA bound presumes players reach equilibrium.

Soln #1: Justify via convergence theorems.

Soln #2: [taken here] Prove bounds for much bigger sets than just Nash equilibria.

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Weaker Equilibrium Concepts

pureNash

mixed Nash

correlated eq

no regret

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Weaker Equilibrium Concepts

pureNash

mixed Nash

correlated eq

no regret

best-responsedynamics

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Main Result (Informal)

Informal Theorem: [Roughgarden 09] under “surprisingly general” conditions, a bound on the price of anarchy (for pure Nash) extends automatically to all 5 bigger sets.

Example Application: selfish routing games (nonatomic or atomic) with cost functions in an arbitrary fixed set.

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The Setup

• n players, each picks a strategy si

• player i incurs a cost Ci(s)

Important Assumption: objective function is cost(s) := i Ci(s)

Next: generic template for upper bounding price of anarchy of pure Nash equilibria.

• notation: s = a Nash eq; s* = an optimal

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An Upper Bound Template

Suppose we have:

cost(s) = i Ci(s) [defn of cost]

≤ i Ci(s*i,s-i) [s a Nash

eq]

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An Upper Bound Template

Suppose we have:

cost(s) = i Ci(s) [defn of cost]

≤ i Ci(s*i,s-i) [s a Nash

eq] ≤ λ●cost(s*) + μ●cost(s) [(*)]

Then: POA (of pure Nash eq) ≤ λ/(1-μ).

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An Upper Bound Template

Suppose we have:

cost(s) = i Ci(s) [defn of cost]

≤ i Ci(s*i,s-i) [s a Nash eq]

≤ λ●cost(s*) + μ●cost(s) [(*)]

Then: POA (of pure Nash eq) ≤ λ/(1-μ).

Definition: A game is (λ,μ)-smooth if (*) holds for every pair s,s* outcomes.

• not only when s is a pure Nash eq!

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Main Result #1

Examples: selfish routing, linear cost fns.• every nonatomic game is (1,1/4)-smooth• every atomic game is (5/3,1/3)-smooth

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Main Result #1

Examples: selfish routing, linear cost fns.• every nonatomic game is (1,1/4)-smooth• every atomic game is (5/3,1/3)-smooth

Theorem 1: in a (λ,μ)-smooth game, expected cost of each outcomes in the 5 sets above is at most λ/(1-μ).– such a POA bound “automatically” far more

general

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Illustration

worstcorrelated equilibium

worstno regretsequence

1

optimaloutcome

worstpureNash

worstmixedNash

λ/(1-μ)

So: in every (λ,μ)-smooth game with a sum objective, inefficiency of outcomes in the 5 sets looks like:

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Main Result #2

Theorem 2 (informal): in sufficiently rich classes of games, smoothness arguments suffice for a tight worst-case bound (even for pure Nash equilibria).

correlated equilibium

no regretsequence

1

optimaloutcome

pureNash

mixedNash

λ/(1-μ)for tightestchoice of λ,μ

Special Case of Result #1

Definition: a sequence s1,s2,...,sT of outcomes is no-regret if:

• for each player i, each fixed action qi:– average cost player i incurs over sequence no

worse than playing action qi every time

– simple hedging strategies can be used by players to enforce this (for suff large T)

Result: in a (λ,μ)-smooth game, average cost of every no-regret sequence at most λ/(1-μ) cost of optimal outcome.

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Take-Home Points

• guarantees on equilibrium quality possible in interesting problem domains

• the most common way of proving such bounds automatically yields a much more robust guarantee– and this technique often gives tight bounds

Future research agenda: broader understanding of performance guarantees for adaptive systems.

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