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Smooth Games and Intrinsic Robustness

Christodoulou and Koutsoupias,Roughgarden

Slides stolen/modified from Tim Roughgarden

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Congestion Games

• Agent i has a set of strategies, Si, each strategy s in Si is a set of resources

• The cost to an agent is the sum of the costs of the resources r in s used by the agent when choosing s

• The cost of a resource is a function of the number of agents using the resource

fr(# agents)3

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Price of Anarchy

Price of anarchy: [Koutsoupias/Papadimitriou 99] quantify inefficiency w.r.t some objective function.– e.g., Nash equilibrium: an outcome such that

no player better off by switching strategies

Definition: price of anarchy (POA) of a game (w.r.t. some objective function):

optimal obj fn value

equilibrium objective fn value

the closer to 1 the better

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Network w/2 players:

s t

2x 12

5x50

Atomic identical flowis a Congestion game

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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

Costs: atomic and non-atomic flow

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

Linear costs

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Nash Equilibrium: To Minimize Cost:

Price of anarchy = 28/24 = 7/6.• if multiple equilibria exist, look at the worst

one

s t

2x 12

5x5

cost = 14+10 = 24

cost = 14+14 = 28

s t

2x 12

5x5

00

Atomic identical flowLinear costs

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Theorem: [Roughgarden/Tardos 00] for every non-atomic flow network with linear cost fns:

≤ 4/3 ×

i.e., price of anarchy non atomic flow ≤ 4/3 in the linear latency case.

cost of non-atomic Nash flow

cost of opt flow

POA non-atomic flow

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Abstract 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)

Key Definition: A game is (λ,μ)-smooth if, for every pair s,s* outcomes (λ > 0; μ < 1):

i Ci(s*i,s-i) ≤ λ●cost(s*) + μ●cost(s)

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Smooth => POA Bound

Next: “canonical” way to upper bound POA (via a smoothness argument).

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

Assuming (λ,μ)-smooth:

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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“Robust” POA

Best (λ,μ)-smoothness parameters:

cost(s) = i Ci(s)

≤ i Ci(s*i,s-i)

≤ λ●cost(s*) + μ●cost(s)Minimizing: λ/(1-μ).

Congestion games with affine cost functions are (5/3,1/3)-

smooth• Claim: For all non-negative integers y,

z :

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y(z + 1) ·53y2 +

13z2:

Thus,

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y(z + 1) ·53y2 +

13z2

) ay(z + 1) + by ·53(ay2 + by) +

13(az2 + bz)

Let s, s¤ be any two vectors of strategies in a congestion game,with loads x and x¤,in (s¤

i ;s¡ i ) the number of users of e is · xe + 1, we have

kX

i=1

Ci (s¤i ;s¡ i ) ·

X

e2E

(ae(xe + 1) + be)x¤e

·X

e2E

53(aex¤

e + be)x¤e +

X

e2E

13(aexe + be)xe

=53C(s¤) +

13C(s):

y = x¤e; z = xe

POA 5/ 2 Any congestion game(includes atomic unit

flow)

a,b ≥0

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Why Is Smoothness Stronger?

Key point: to derive POA bound, only needed

i Ci(s*i,s-i) ≤ λ●cost(s*) + μ●cost(s)

to hold in special case where s = a Nash eq and s* = optimal.

Smoothness: requires (*) for every pair s,s* outcomes.– even if s is not a pure Nash equilibrium

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The Need for Robustness

Meaning of a POA bound: if the game is at an equilibrium, then outcome is near-optimal.

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The Need for Robustness

Meaning of a POA bound: if the game is at an equilibrium, then outcome is near-optimal.

Problem: what if can’t reach equilibrium?• (pure) equilibrium might not exist• might be hard to compute, even

centrally– [Fabrikant/Papadimitriou/Talwar], [Daskalakis/

Goldberg/Papadimitriou], [Chen/Deng/Teng], etc.

• might be hard to learn in a distributed way

Worry: are POA bounds “meaningless”?

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Robust POA Bounds

High-Level Goal: worst-case bounds that apply even to non-equilibrium outcomes!

• best-response dynamics, pre-convergence– [Mirrokni/Vetta 04], [Goemans/Mirrokni/Vetta 05],

[Awerbuch/Azar/Epstein/Mirrokni/Skopalik 08]

• correlated equilibria– [Christodoulou/Koutsoupias 05]

• coarse correlated equilibria aka “price of total anarchy” aka “no-regret players”– [Blum/Even-Dar/Ligett 06],

[Blum/Hajiaghayi/Ligett/Roth 08]

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Lots of previous work uses smoothness Bounds

• atomic (unweighted) selfish routing [Awerbuch/Azar/Epstein 05], [Christodoulou/Koutsoupias 05], [Aland/Dumrauf/Gairing/Monien/Schoppmann 06], [Roughgarden 09]

• nonatomic selfish routing [Roughgarden/Tardos 00],[Perakis 04] [Correa/Schulz/Stier Moses 05]

• weighted congestion games [Aland/Dumrauf/Gairing/Monien/Schoppmann 06],

[Bhawalkar/Gairing/Roughgarden 10]

• submodular maximization games [Vetta 02], [Marden/Roughgarden 10]

• coordination mechanisms [Cole/Gkatzelis/Mirrokni 10]

Beyond Pure Nash Equilibria (Static)

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pureNash

mixed Nash

correlated eq

CCE

For all s;s0i : Es» ¾[Ci (s)] · E s¡ i » ¾¡ i [Ci (s0

i ;s¡ i )]¾= ¾1 £ ¾2 £ ¢¢¢£ ¾k

For all s;s0i : Es» ¾[Ci (s)] · E s» ¾[Ci (s0

i ;s¡ i )]

For all s;s0i : E s» ¾[Ci (s)jsi ] · E s» ¾[Ci (s0

i ;s¡ i )jsi ]

Mixed:

Correlated:

Coarse Correlated:

¾6= ¾1 £ ¾2 £ ¢¢¢£ ¾k

Beyond Nash Equilibria (non-Static)

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

– if every player uses e.g. “multiplicative weights” then get o(1) regret in poly-time

– empirical distribution = "coarse correlated eq" 21

pureNash

mixed Nash

correlated eq

no-regret

An Out-of-Equilibrium Bound

Theorem: [Roughgarden STOC 09] in a (λ,μ)-smooth game, average cost of every no-regret sequence at most

[λ/(1-μ)] x cost of optimal outcome.

(the same bound we proved for pure Nash equilibria)

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Smooth => No-Regret Bound

• notation: s1,s2,...,sT = no regret; s* = optimal

Assuming (λ,μ)-smooth:

t cost(st) = t i Ci(st) [defn of cost]

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Smooth => No-Regret Bound

• notation: s1,s2,...,sT = no regret; s* = optimal

Assuming (λ,μ)-smooth:

t cost(st) = t i Ci(st) [defn of cost]

= t i [Ci(s*i,st

-i) + ∆i,t] [∆i,t:= Ci(st)- Ci(s*i,st

-

i)]

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Smooth => No-Regret Bound

• notation: s1,s2,...,sT = no regret; s* = optimal

Assuming (λ,μ)-smooth:

t cost(st) = t i Ci(st) [defn of cost]

= t i [Ci(s*i,st

-i) + ∆i,t] [∆i,t:= Ci(st)- Ci(s*i,st

-

i)]

≤ t [λ●cost(s*) + μ●cost(st)] + i t ∆i,t [(*)]

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Smooth => No-Regret Bound

• notation: s1,s2,...,sT = no regret; s* = optimal

Assuming (λ,μ)-smooth:

t cost(st) = t i Ci(st) [defn of cost]

= t i [Ci(s*i,st

-i) + ∆i,t] [∆i,t:= Ci(st)- Ci(s*i,st

-

i)]

≤ t [λ●cost(s*) + μ●cost(st)] + i t ∆i,t [(*)]

No regret: t ∆i,t ≤ 0 for each i.

To finish proof: divide through by T.

Intrinsic Robustness

Theorem: [Roughgarden STOC 09] for every set C, unweighted congestion games with cost functions restricted to C are tight:

maximum [pure POA] = minimum [λ/(1-μ)]congestion games

w/cost functions in C(λ ,μ): all such gamesare (λ ,μ)-smooth

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Intrinsic Robustness

Theorem: [Roughgarden STOC 09] for every set C, unweighted congestion games with cost functions restricted to C are tight:

maximum [pure POA] = minimum [λ/(1-μ)]

• weighted congestion games [Bhawalkar/ Gairing/Roughgarden ESA 10] and submodular maximization games [Marden/Roughgarden CDC 10] are also tight in this sense

congestion gamesw/cost functions in C

(λ ,μ): all such gamesare (λ ,μ)-smooth

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