SLIDE 1
The Price of Anarchy in a Network Pricing Game. g
Sept 27, 2007 Allerton Conference
John Musacchio Assistant Professor Technology and Information Management University of California Santa Cruz University of California, Santa Cruz johnm@soe.ucsc.edu Joint work with: Shuang Wu University of California, Santa Cruz
SLIDE 2 Overview
Model
– Single source-destination pair
g p
– Competing providers – Non-atomic users – Traffic dependent latency – Elastic user demand
Model due to
– Acemoglu and Ozdaglar [1]
El ti d d t i
Elastic user demand extension:
– Hayrapetyan, Tardos, Wexler [3]
[1] D. Acemoglu and A. Ozdaglar, “Competition and Efficiency in Congested Markets,”
[3] A. Hayrapetyan, E. Tardos and T. Wexler, “A Network Pricing Game for Selfish Traffic,” Distributed Computing, March 2007.
SLIDE 3 Overview
Price of anarchy:
Social Welfare with Optimal Prices Social Welfare Nash with Nash prices = 3
2 Result due to Ozdaglar [2]
p
Result due to Ozdaglar [2]
We prove the same result a different way
– Ozdaglar proof: mathematical programming
O dag a p oo a e a ca p og a g argument
– Our proof: circuit analogy, linear algebra
[2] A. Ozdaglar, ``Price Competition with Elastic Traffic,'‘ to appear in Networks
SLIDE 4
Some Other Related Work
Roughgarden 02, 03
– Selfish routing games – Taxes to induce optimal routing
Johari and Tsitsiklis 05
– Cournot rate allocation mechanisms – Different situation – Very similar structure John Musacchio – Allerton 07
SLIDE 5
Organization
Overview Model Description Model Description Nash and Social Optimum Characterization Circuit Analogy Circuit Analogy PoA Proof Overview
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SLIDE 6
Wardrop Equilibrium for given prices
Path 1
Non-Atomic Users
l (f )
Path 2
p1 + l1(f1)
Delay+Price p
p2 + l2(f2)
Choose lowest “disutility”: pi+ l(fi) Delay+Price Delay
p2 Path 1 Path 2
Delay
p1
y Traffic Traffic 100% 100% 20% John Musacchio – Allerton 07 80%
SLIDE 7
Elastic Demand
?
Demand or “Disutility” Curve
Key assumption:
Disutil
User Surplus Concave Decreasing
ity
Total Flow (#of non-atomic users that connect)
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SLIDE 8
Social Optimum Pricing
p
Path 1 p1 Path 2
p2
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SLIDE 9
Network Pricing
p
Path 1 p1 Path 2
p2
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SLIDE 10 Wardrop Equilibrium for given (p1,p2,p3)
User Surplus For now consider linear latency case:
li(fi ) = aifi + bi
User Surplus
D
d
Provider 1 Profit
s
Disutility
p1 p2 p3 a1
L(f1)
f2 f3 f f1
b1
Figure from [3] Total Flow (#of non-atomic users that connect)
f
[3] A. Hayrapetyan, E. Tardos and T. Wexler, “A Network Pricing Game for Selfish Traffic,” Distributed Computing, March 2007.
SLIDE 11 Consequence of price change
Suppose: player 1 unilaterally
reduces price.
New Wardrop equilibrium disutility: d h
d
- New Wardrop equilibrium disutility: d-h.
Provider 1 Profit
Disutility
d
p3
d-h
p1
y
p3 p1 f2 f3 f1
f3 − h a3 h
Total Flow (#of non-atomic users that connect)
f
f + h/s
f2 − h a2
SLIDE 12
Convenient definition:
Nash Equilibrium Analysis
Convenient definition: New profit – old profit:
=0
Nash equilibrium condition:
=0 John Musacchio – Allerton 07
SLIDE 13
Social optimum pricing
Price so that users see the cost they impose on
society.
Latency cost on link i: (aifi * + bi)fi * Marginal cost:
2aifi
* + bi
Latency seen by user:
aifi
* + bi
Difference:
aifi
* * *
Conclusion: pi
* = aifi * achieves social optimum
John Musacchio – Allerton 07
SLIDE 14 Circuit analogy
Nash Eq ilibri m Social Optimum:
Voltage
d
Nash Equilibrium:
Voltag
d*
Social Optimum:
d
Current e
f f d*
Current e
d f * f δ1 a p1 a δ2 a δ3
Power Provider P
a1 a2 a3 a1 f a2 a3
= Profit
a1 f1
+
a2
+
a3
+
a1 a2 a3
+
b1
+
+
+
+
+
+
SLIDE 15 Nash vs. Social Opt. – Original Game
Nash Equilibrium: Social Optimum: Nash Equilibrium: Social Optimum:
d*
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SLIDE 16 Nash vs. Social Opt. – Modification 1
N h E ilib i S i l O ti Nash Equilibrium: Social Optimum:
d*
Fl & S i l W lf Fl & S i l W lf Flow & Social Welfare Unchanged Flow & Social Welfare Not Reduced Price of Anarchy Not Reduced
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SLIDE 17 Nash vs. Social Opt. – Modification 2
Nash Equilibrium: Social Optimum: V V Nash Equilibrium: Social Optimum:
d* Fl U h d
- Flow Unchanged
- Social Welfare reduced
by light green area
- Flow Unchanged
- Social Welfare reduced
by light green area
Price of Anarchy Not Reduced
y g g
SLIDE 18 Circuit analogy
Nash Equilibrium: Social Optimum: Nash Equilibrium: Social Optimum: V
s
V
s δ1 δ2 δ3 a a a
Power = Provider Power = Provider
a a1 a a2 a a3 a1 a2 a3
Surplus Surplus
b1 a1
+
a2
+
a3
+
a1
+
a2
+
a3
+
SLIDE 19 Matrix – Vector Notation
F = ⎡ ⎢ ⎣ f1 f2 ⎤ ⎥ ⎦ F ∗ = ⎡ ⎢ ⎣ f ∗
1
f ∗
2
⎤ ⎥ ⎦ M = ⎡ ⎢ ⎣ 1 1 ... 1 1 ... ⎤ ⎥ ⎦ ⎣. . .fn ⎦ ⎣. . .f ∗
n
⎦ ⎣. . . . . . ... ⎦ ⎡a1 ... ⎤ ⎡δ1 ... δ ⎤ A = ⎡ ⎢ ⎣ 0 a2 ... . . . . . . ... ⎤ ⎥ ⎦ ∆ = ⎡ ⎢ ⎣ 0 δ2 ... . . . . . . ... ⎤ ⎥ ⎦
Providers used in Nash equilibrium can become Providers used in Nash equilibrium can become “undercut” in social optimum
w l o g Providers: 1 m not undercut A = · ¯ A, ¸ ∆ = · ¯ ∆, ¸ F = · ¯ F ¸ w.l.o.g. Providers: 1,…, m not undercut m+1,…,n undercut A = · 0, A ¸ ∆ = · 0, ∆ ¸ F = · F ¸
John Musacchio – Allerton 07
SLIDE 20 Relations between flow vectors
V V
δ1 δ2 δ3
V
s a a a
V
s
Nash:
b a1 a1 b a2 a2 b a3 a3 b a1 a1 b a2 a2 b a3 a3 b1
+
+
+
+
+
SLIDE 21 Social Welfare
V V
δ1 δ2 δ3
V
s a a a
V
s
Nash:
a1
1
a1 a2 a2 a3 a3 b a1 a1 b a2 a2 b a3 a3 b1
+
+
+
+
+
SLIDE 22
Social Welfare Comparison Metric
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SLIDE 23 Useful Algebraic Identities
Define: Then: Proof:
- relation of δi’s and A; matrix inversion lemma
i
;
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SLIDE 24
Use Identities
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SLIDE 25 Change Coordinates
(i) P b l i Z l (ii) Thi i ht b ti
1
(i) (i) Parabola in Z always ≥ 0. (ii) This might be negative
1 2β ||Q||2
1
Z Case 1: (ii) Positive done. C 2 U i t f “ ll” t h th t t t l “ d t
¯
Case 2: Use existence of a “small” to show that total “undercut flow” Z is small. Tedious algebra |(i)| > |(ii)|
aj
SLIDE 26
Worst Case
Nash: Social Optimum: D 1 1 Disutility Flow Flow Price = 1 Flow = 1 Price = 0 Flow = 2 1 2 2 Flow 1 Social Welfare =1 Flow = 2 Social Welfare = 3/2 John Musacchio – Allerton 07
SLIDE 27 Convex Latency
- Provided a pure strategy equilibrium exists...
- Linearize at equilibrium
SLIDE 28
Conclusions
Analysis of network pricing game can be
d d t l i f i it reduced to analysis of a circuit.
Potential for using method for extended
d l model.
Analysis of circuit a bit more tedious than
desired desired.
John Musacchio – Allerton 07
