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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 AnarchyPrice 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 strategiesDefinition: price of anarchy (POA) of a
game (w.r.t. some objective function):
optimal obj fn valueequilibrium objective fn value
the closer to 1 the better
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Network w/2 players:
s t2x 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 tx
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 t2x 12
5x5cost = 14+10 = 24
cost = 14+14 = 28
s t2x 12
5x500
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 BoundNext: “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” POABest (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 + 1
3z2:
Thus,
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y(z + 1) · 53y2 + 1
3z2
) ay(z + 1) + by · 53(ay2 + by) + 1
3(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 havekX
i=1Ci (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¤) + 1
3C(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 RobustnessMeaning of a POA bound: if the game is at
an equilibrium, then outcome is near-optimal.
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The Need for RobustnessMeaning 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 BoundsHigh-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 Nashcorrelated 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"
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pureNash
mixed Nashcorrelated 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 RobustnessTheorem: [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 RobustnessTheorem: [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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