Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Local Smoothness and the
Price of Anarchy in
Atomic Splittable Congestion Games
Tim Roughgarden
Florian Schoppmann
January 23, 2011
1
Introduction and Model
Local Smoothness
The Price of Anarchy
I
Conclusion
(Papadimitriou & Koutsoupias 1999)
Loss due to lack of central control
PoA =
I
Tight Bounds Via Local Smoothness
SC(equilibrium)
equilibrium SC(optimum)
sup
Analogs:
I
I
Approximation Ratio (lack of
computing power)
Competitive Ratio (lack of
knowledge about future)
(Nash) Equilibrium:
I
No player can improve by unilateral change of strategy
2
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
How Efficient Is a System?
Autonomous decision makers:
I Local optimization,
distributed
Central Control:
I Global optimization
I
Not efficient
I
Efficient (if it works)
I
No regret
I
Unfair
I
Simple, “fault-tolerant”
I
Hard to implement
If PoA is low, no need to tell people what to do
I
Allows for a simple system design
If PoA is high, need to re-design system or give incentives
I
Mechanism Design, e.g., toll pricing
3
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Key Points
I
Main Definition: A general framework (“local
smoothness”) to bound the price of anarchy in games
with convex strategy sets
I
Theorem: Every PoA bound proved via local smoothness
holds even for correlated equilibria (i.e., with vanishing
per-step swap regret)
I
I
But does not apply to coarse-correlated equilibria
Theorem: Local smoothness provides optimal bounds for
atomic splittable congestion games
I
solves problem that had been open for several years
(Nisan et al. 2007, Algorithmic Game Theory )
4
Introduction and Model
Local Smoothness
Congestion Games
Tight Bounds Via Local Smoothness
(Rosenthal 1973)
Model: Finite set E of resources (e.g., network edges)
I
Finite set of players N
I
Player i’s strategy set = collection Pi of
subsets of E (e.g., paths)
I
cost `e (xe ) per resource, xe = load on e
I
Player i’s cost = sum of resource costs
I
Objective: Sum of player costs
E
Conclusion
5
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Atomic Splittable Congestion Games
Individ.
1
Each player i:
I
I
I
has weight wi
i
chooses ~x s.t.
0
P
i
p∈Pi xp
1
= wi
i
e∈E xe
has cost ci (~x ) =
· `e (xe )
where xei is i’s load on e
0
1
Coalition
1
0
x .5 2
.5
1
P
Conclusion
0
1
SC(�x ) = 3.25
Objective:
SC(~x ) :=
X
e∈E
xe · `e (xe ) =
X
i∈[n]
ci (~x )
6
Introduction and Model
Local Smoothness
Review: Smoothness
Tight Bounds Via Local Smoothness
(Roughgarden 2009)
Game (λ, µ)-smooth if ∀ outcomes ~x , ~y :
(*)
n
X
i=1
ci (~y i , ~x −i ) ≤ λ · SC(~y ) + µ · SC(~x )
Fact: The PoA of any (λ, µ)-smooth game is ≤
λ
1−µ
Conclusion
7
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Smoothness Is Not Strong Enough
For convex games, we derive better upper bounds by requiring
smoothness only for outcomes ~y “arbitrarily close to” ~x :
(*) ci (~y i , ~x −i ) − ci (~x ) ≥ ∇i ci (~x )T (~y i − ~x i )
~x is NE iff variational inequality holds:
∇i ci (~x )T (~y i − ~x i ) ≥ 0
Local Smoothness: Replace lhs of (*) by rhs
�x i �y i
8
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Local Smoothness
Key Definition: Game is (λ, µ)-locally smooth w.r.t. ~y if ∀~x ,
(*)
n
X
ci (~x ) + ∇i ci (~x )T (~y i − ~x i ) ≤ λ · SC(~y ) + µ · SC(~x )
i=1
Assuming (λ, µ)-local smooth w.r.t. ~y and ~x pure NE:
SC(~x ) =
X
≤
X
ci (~x )
[Def. of SC]
i
i
ci (~x ) + ∇i ci (~x )T (~y i − ~x i ) [Var. ineq.]
≤ λ · SC(~y ) + µ · SC(~x )
Hence: PoA of pure NE ≤
λ
1−µ
[(*)]
9
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Intermezzo: Equilibrium Concepts
I
Pure Nash = outcome ~x :
∀i, ~y i ∈ Si : ci (~x ) ≤ ci (~y i , ~x −i )
I
Conclusion
coarse correlated
correlated
mixed Nash
pure
Nash
Mixed Nash = product distribution P:
∀i, ~y i ∈ Si : E~x ∼P [ci (~x )] ≤ E~x ∼P [ci (~y i , ~x −i )]
I
Correlated = distribution P:
∀i, δi : Si → Si : E~x ∼P [ci (~x )] ≤ E~x ∼P [ci (δi (~x i ), ~x −i )]
I
Coarse-correlated = distribution P:
∀i, ~y i ∈ Si : E~x ∼P [ci (~x )] ≤ E~x ∼P [ci (~y i , ~x −i )]
10
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Why important?
Tight connection to learning:
I
Correlated equilibria ∼ empirical distribution of sequences
with vanishing per-step swap regret
I
Coarse-correlated equilibria ∼ empirical distribution of
sequences with vanishing per-step regret
Reduced assumptions:
I
Bound applies to repeated-play sequences that do not
converge in any sense
I
Nash equilibria in general hard to compute
I
Simple algorithms for regret-minimization exist
11
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Robustness of (Local) Smoothness Bounds
I
Roughgarden (2009):
Smoothness bounds all
coarse-correlated equilibria
I
Theorem: Local smoothness
bounds all correlated equilibria
I
Theorem: Local smoothness
does not bound
coarse-correlated equilibria
coarse correlated
correlated
mixed Nash
pure
Nash
12
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
13
Local Smoothness Bounds All Correlated Equilibria
Key Claim:
(*)
E~x ∼P ∇i ci (~x )T (~y i − ~x i ) ≥ 0
Proof of Key Claim (by contradiction, assume lhs of (*) < 0):
I
Define ~x := ((1 − ) · ~x i + · ~y i , ~x −i )
I
By dominated convergence theorem:
Z
Z
ci (~x ) − ci (~x )
lim
dP(~x ) = ∇i ci (~x )T (~y i − ~x i ) dP(~x )
&0
= E~x ∼P [∇i ci (~x )T (~y i − ~x i )] < 0
I
For small enough , E~x ∼P [ci (~x )] < E~x ∼P [ci (~x )]
E
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Local Smoothness Does Not Bound All CCEs
Example:
I
I
n = 2, strat. sets = [0, 1], c1 (~x ) = c2 (~x ) = (x1 − x2 )2 + P(0, α) = P(1, 1 − α) =
1
2
defines a CCE (for α ≤ 14 )
1
1–α
α
0
0
1
14
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Tight Bounds for Atomic Splittable CGs
Let γ(L) := smallest upper bound on PoA provable via local
smoothness, for set of cost functions L
Theorem: The PoA in every atomic splittable congestion game
is at most γ(L). This bound is tight.
Previous work:
I
Cominetti et al. (2006): Special case of our local
smoothness framework (λ fixed at 1)
I
Harks (2007): Tight upper bounds (tight by our results)
15
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Core of upper-bound proof
Find minimal
(*)
λ
1−µ
y · `(x) +
s.t. ∀` ∈ L, x ≥ 0, y > 0 with `(y ) > 0:
y2 0
· ` (x) ≤ λ · y · `(y ) + µ · x · `(x)
4
We define:
2
y · `(x) + y4 · `0 (x) − µ · x · `(x)
g`,x,y (µ) :=
y · `(y ) · (1 − µ)
Then (*) is equivalent to g`,x,y (µ) ≤
λ
.
1−µ
Best upper bound provable using local smoothness:
γ(L) = inf
µ∈[0,1)
sup
`∈L
x≥0,y >0,`(y )>0
g`,x,y (µ)
16
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Plots of Functions g`,x,y (µ) When ` Fixed
λ
1−µ
γ({�})
x · �(x)
2
= y · �(x) + y4 · �� (x)
µ
0
0.5
1
17
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Lower-Bound Construction for Special Case (I)
Here: Special case `, x, y with x · `(x) = y · `(x) +
y2
4
· `0 (x)
For concreteness: `(z) = z 3 , x = 32 , y = 1, q := 2x − 1 ∈ N
Let λ, µ arbitrary s.t. g`,x,y (µ) =
λ
.
1−µ
Family of instances:
I
n players with weight ≈ 1
I
n resources
I
Each player: 1 “optimal”,
n−1
“non-opt.” strategies
q
I
In NE ~x , players put α on
opt, β on non-opt.
“non-optimal”
strategies
“optimal”
strategy
18
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Lower-Bound Construction for Special Case (II)
For Nash equilibrium ~x , we need:
I
The load on each resource is exactly x:
α + (n − 1) · β = x
I
Marginal costs are equal on used strategies:
`(x) + α · `0 (x) = q · (`(x) + β · `0 (x))
I
On each resource, one player puts load 12 ± o(1) and all
others put load o(1). Solving the above equations gives:
α=
n − 1 q · x n→∞ y
+
−−−→
2n
n
2
and β =
x − α n→∞
−−−→ 0
n−1
19
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Lower-Bound Construction for Special Case (III)
For the optimum ~y , we need:
I
If every player only uses its “optimal” strategy, each
resource has load y + o(1). The weight of any player is:
n−1
x − α n→∞
−−−→ 1
α+
·β =α+
q
2x − 1
h
i
n→∞ 1
x−α
q = 2x − 1, β = n−1 , α −−−→ 2
In NE ~x , social cost contributed by each resource is
x · `(x) = λ · y · `(y ) + µ · x · `(x),
i.e.,
λ
1−µ
times as in optimum ~y . =⇒ PoA = ( 32 )4
[x · `(x) = y · `(x) +
y2
4
· `0 (x), g`,x,y (µ) =
λ
]
1−µ
20
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Comparison (Polynomials)
Degree
Atomic
splittable
1
2
3
4
5
6
7
8
d
(Roughg. 2003, Aland et al. 2006)
Atomic
unsplittable
(weighted)
1.500
2.618
2.549
9.909
5.063
47.82
11.09
277.0
26.32
1,858
66.88
14,099
180.3
118,926
512.0
1,101,126
√
( 1+
d+1 d+1
)
2
Conclusion
Θ( logd d )d+1
Nonatomic
1.333
1.626
1.896
2.151
2.394
2.630
2.858
3.081
Θ( logd d )
21
Introduction and Model
Local Smoothness
Tight Bounds Via Local Smoothness
Conclusion
Take-Home Points
I
Market power can be harmful (“price of collusion”)
I
Essentially all PoA results for congestion games conform
to the (local) smoothness framework
I
Local smoothness can also be used to reprove the PoA for
resource allocation games (Johari & Tsitsiklis 2004)
Remaining open problems:
I
PoA for coarse-correlated equilibria
I
PoA in symmetric games (different player weights)
22