intrinsic robustness of the price of anarchy
DESCRIPTION
Intrinsic Robustness of the Price of Anarchy. Tim Roughgarden Stanford University. The Mathematical Model. a directed graph G = (V,E) k source-destination pairs (s 1 ,t 1 ), …, (s k ,t k ) a rate (amount) r i of traffic from s i to t i for each edge e , a cost function c e (•) - PowerPoint PPT PresentationTRANSCRIPT
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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
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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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