1 intrinsic robustness of the price of anarchy tim roughgarden stanford university
TRANSCRIPT
1
Intrinsic Robustness of the Price of
Anarchy
Tim RoughgardenStanford University
2
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)
3
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
4
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:
5
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]
6
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
7
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
8
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
9
Braess’s Paradox
Initial Network:
s tx 1
½
x1½
½
½
cost = 1.5
10
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?
11
Braess’s Paradox
Initial Network: Augmented Network:
s tx 1
½
x1½
½
½
cost = 1.5 cost = 2
s t
x 1
x10
12
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
13
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
14
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
15
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
16
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
17
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
18
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
19
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.
.
20
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.
21
Weaker Equilibrium Concepts
pureNash
mixed Nash
correlated eq
no regret
22
Weaker Equilibrium Concepts
pureNash
mixed Nash
correlated eq
no regret
best-responsedynamics
23
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.
24
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
25
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]
26
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).
27
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!
28
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
29
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
30
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:
31
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.
32
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.
33