Class3

Algorithms and Economics
of Networks
Abraham Flaxman and Vahab Mirrokni,
Microsoft Research
Outline
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Network Congestion Games
Congestion Games
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Rosenthal’s Theorem: Congestion games are
potential Games:
PoA for Congestion Games.
Market Sharing Games.
Network Design Games.
Network Congestion Games
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A directed graph G=(V,E) with n users,
Each edge e in E(G) has a delay function fe,
Strategy of user i is to choose a path Aj from a
source si to a destination ti,
The delay of a path is the sum of delays of
edges on the path,
Each user wants to minimize his own delay by
choosing the best path.
Example: Network Congestion Game
s1
s2
s3
x 6  2 x t3
2x  6
2 ln x sin x
t2
2x
3
8x  2
x
4x
t1
Example: Network Congestion Game
s1
s2
s3
Agent 2
path 2
t3
t2
Agent 2
path 1
t1
Congestion Games
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n players, a set of facilities E,
Strategy of player i is to choose a subset of
facilities (from a given family of subsets Ti),
Facility i have a cost (delay) function fe which
depends on the number of players playing this
facility,
Each player minimizes its total cost,
Example: Congestion Games
f1
f2
f3
f4
F5
1   f 1, f 3,  f 2, f 3, f 4
2   f 2, f 4,  f 5, f 6
3   f 1, f 4
Picture from Kapelushnik Lior
F6
Example: Congestion Games
f1
f2
f3
f4
F5
1
1   f 1, f 3,  f 2, f 3, f 4
2   f 2, f 4,  f 5, f 6
3   f 1, f 4
F6
Example: Congestion Games
f1
f2
f3
f4
F5
2
1   f 1, f 3,  f 2, f 3, f 4
2   f 2, f 4,  f 5, f 6
3   f 1, f 4
F6
Example: Congestion Games
f1
f2
f3
f4
F5
3
1   f 1, f 3,  f 2, f 3, f 4
2   f 2, f 4,  f 5, f 6
3   f 1, f 4
F6
Congestion Games: Pure NE
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Rosenthal’s Theorem (1979): Any congestion game
is an exact potential game.
Proof is based on the following Potential Function
ne  A 
  f t 
eE t 1
e
Classes of Congestion Games
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Every network congestion game is a
congestion game
Symmetric and Asymmetric Players
Network Design Games.
Maximizing Congestion Games: Each player wants to
maximize his payoff (instead of minimizing his
delay)  Market Sharing Games.
Generalizations:
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Weighted Congestion Games
Player-specific Congestion Games
Congestion Games: Social Cost
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Two social Cost functions:
Consider a pure Strategy A = (A1, A2, …,
An).
Defintion 1:
Max A  max iN ci  A
Defintion 2:
Sum A   ci  A
iN
Congestion Games: PoA
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PoA for two social Cost functions:
Defintion 1:
max A is a NE
Max  A
opt
max A is a NE
Sum A
opt
Defintion 2:
We Prove bounds for
Sum A   ci  A
iN
Congestion Games: PoA
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PoA for congestion game with linear delay
functions is at most 5/2.
Proof:
Lemma 1: for a pair of nonnegative
integers a,b:
1 2 5 2


b a 1  a  b
3
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Proof: …
3
Congestion Games: Lower Bound
S1
t1
S2
t2
S3
t3
fe( x)  0
fe( x)  x
opt
NE
Congestion Games: PoA for mixed NE
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Theorem: PoA for mixed Nash equilibria in
congestion games with linear latency function is
2.61.
Theorem: PoA for mixed Nash equilibria of
weighted congestion games with linear latency
function is 2.61.
Theorem: PoA for polynomial delay functions of
constant degree is a constant.
Other Variants
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Atomic Congestion Games: Many infinitesimal
users. The load of each user is very small.
Theorem: PoA for non-atomic congestion games
with linear latency functions is 4/3.
Splittable Network Congestion Games
Market Sharing Games
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Congestion Game
Facilities are Markets.
Cost function  Profit Function.
Players share the profit of markets (equally).
Each player has some packing constraint for the
set of markets he can satisfy.
PoA: 1/2.
Network Design Games
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Players want to construct a network.
They share the cost of buying links in the
network.
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Known Results: Price of Stability, Convergence…
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Next Lecture
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Coordination Mechanism Design

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