Intrinsic Robustness of the Price of Anarchy

1
Intrinsic Robustness of the Price of Anarchy

• Tim Roughgarden
• Stanford 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

Example (k,r1)
c(x)x
Flow ½
s1
t1
c(x)1
Flow ½
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?

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

Example
Flow 1
Flow .5
x
x
s
t
s
t
1
1
Flow .5
Flow 0
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)

x
½
s
t
½
1
Cost ½½ ½1 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)

x
½
s
t
½
1
Cost ½½ ½1 ¾
8
Inefficiency of Nash Flows

• Note Nash flows do not minimize the cost
• observed informally by Pigou 1920
• Cost of Nash flow 11 01 1
• Cost of optimal (min-cost) flow ½½ ½1 ¾
• Price of anarchy Nash/OPT ratio 4/3

x
1
½
s
t
1
0
½
9
Braesss Paradox

• Initial Network

cost 1.5
10
Braesss Paradox

• Initial Network Augmented Network

½
½
x
1
0
s
t
½
½
x
1
cost 1.5
Now what?
11
Braesss Paradox

• Initial Network Augmented Network

x
1
0
s
t
x
1
cost 1.5
cost 2
12
Braesss Paradox

• Initial Network Augmented Network
• All traffic incurs more cost! Braess 68
• see also Cohen/Horowitz 91, Roughgarden 01

x
1
0
s
t
x
1
cost 1.5
cost 2
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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

14
Linear Cost Functions

• First focus on special case.
• Def linear cost fn is of form ce(x)aexbe
• 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

• 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

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

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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

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
no regret
correlated eq
mixed Nash
pure Nash
22
Weaker Equilibrium Concepts
no regret
correlated eq
mixed Nash
pure Nash
best- response dynamics
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.

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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

25
An Upper Bound Template

• Suppose we have
• cost(s) ?i Ci(s) defn of
  cost
• ?i Ci(si,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(si,s-i)
  s a Nash eq
• ??cost(s) µ?cost(s)
  ()
• Then POA (of pure Nash eq) ?/(1-µ).

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An Upper Bound Template

• Suppose we have
• cost(s) ?i Ci(s) defn of
  cost
• ?i Ci(si,s-i)
  s a Nash eq
• ??cost(s) µ?cost(s)
  ()
• Then POA (of pure Nash eq) ?/(1-µ).
• Definition A game is (?,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 (?,µ)-smooth game, expected cost
  of each outcomes in the 5 sets above is at most
  ?/(1-µ).
• such a POA bound automatically far more general

30
Illustration
So in every (?,µ)-smooth game with a sum
objective, inefficiency of outcomes in the 5
sets looks like
worst correlated equilibium
worst pure Nash
worst mixed Nash
worst no regret sequence
optimal outcome
1
?/(1-µ)
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
pure Nash
no regret sequence
optimal outcome
mixed Nash
1
?/(1-µ)
for tightest choice of ?,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 (?,µ)-smooth game, average cost of
  every no-regret sequence at most ?/(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.