On the price of anarchy and stability of
correlated equilibria of linear congestion
games
By George Christodoulou
Elias Koutsoupias
Part of slides taken from George Christodoulou and Elias Koutsoupias web site
Presented by Efrat Naim
1
Agenda

Congestion Games

An example

Definitions

Bounds for correlated Price of stability of
congestions games

Bounds for correlated price of anarchy of
congestion games

Related Work

Results
2
Congestion Games
a
b
d
c

Introduced in [Rosenthal, 1973]

Each player has a source and destination.
Pure strategies are the path from source to
destination

The cost on each edge depends on the number of
the players using it.
3
Congestion Games

N players

M facilities (edges)

Pure strategy (path) is a subset of facilities.
Each player can select among a collection of pure
strategies (pure strategy set)

Cost of facility depends on the number of players
using it

The objective of each player is to minimize its
own total cost

Pure strategy profile s = (s1,……..sN)
4
An Example
e
d
c
b
a
5
An Example
From a to c
e
d
c
b
a
6
An Example
From a to c
e
d
c
b
a
7
An Example
From a to c
e
d
c
b
a
8
An Example
From e to c
e
d
c
b
a
9
An Example
From e to c
e
d
c
b
a
10
An Example
Nash Equilibrium
e
d
c
b
a
Player 1 has cost 1+1=2
Player 2 has cost 1+1 =2
11
An Example
Another Nash Equilibrium
e
d
c
b
a
Player 1 has cost 2+1+1=4
Player 2 has cost 2+1+1=4
12
Mixed Strategy
e
d
1/2
c
b
a
1/2
A mixed strategy for a player is a probability
distribution over its pure strategy set.
Mixed strategy profile p = (p1,…….pN)
13
Correlated Strategy
1/2
1/2
e
e
d
d
c
c
b
b
a
a
A correlated strategy q for a set of players is any
probability distribution over the set S = X i€N Si
2
L
L
1
R
1/2
0
R
0
1/2
14
Correlated Equilibrium

Introduced in [Auman,1974]

Consider a mediator that makes a random experiment
with a probability distribution q over the strategy space
S.

q is common knowledge to the players

The mediator, with respect to the outcome s€S ,
announces privately the strategy si to the player i.

Player i is free to obey or disobey to the mediator’s
recommendation, with respect to his own profit.

Player i doesn’t know the outcome of the experiment
If the best for every player is to follow mediator’s
recommendation , then q is a correlated equilibrium.
15
Correlated Equilibrium
Cost for player i for pure strategy A is
ne(A) =
number of players using e in A
Given a correlated strategy q, the expected cost of
a player i€N is
A correlated strategy q is a correlated equilibrium if
it satisfies the following
16
Price of Anarchy
Price of anarchy
A social cost (objective) of a pure strategy profile A
is the sum of players costs in A:
and
17
Price of Anarchy
e
d
c
b
a
Player 1 has cost 2+1+1=4
Player 2 has cost 2+1+1=4
PoA = (4+4)/ (2+2) = 2
18
Price of Stability
Price of stability
19
Price of Stability
e
d
c
b
a
Player 1 has cost 1+1=2
Player 2 has cost 1+1=2
PoS = (2+2)/ (2+2) = 1
20
PoS and PoA of congestion games

By PoS(PoA) for a class of games, we mean the
worst case Pos(PoA) over this class.

UpperBound: must hold for every congestion
game

LowerBound: Find such a congestion game
21
Correlated PoS – Upper Bound
We consider linear latencies
fe(x) = aex+be
Lemma 1:
For every pair of non negative integers a,b it holds
22
Correlated PoS – Upper Bound
Theorem 1:
Let A be a pure Nash equilibrium and P be any pure strategy
profile such that P(A)<= P(P), then SUM(A)<=8/5AUM(P)
Where P is the potential of strategy profile
This show that the correlated price of stability is at most 1.6
23
Correlated PoS – Upper Bound
Proof:
Let X be a pure strategy profile X=(X1,………XN)
From the potential inequality
24
Correlated PoS – Upper Bound
A is Nash equilibrium so
Summing for all the players we get
Adding the two inequalities and use Lemma 1
25
Price of Stability - Lower Bound
Dominant Strategies:
Each player prefers a particular strategy (dominant), no matter what
the other players will choose.
Dominant Strategies
Nash Equilibrium
Correlated Equilibrium
26
Price of Stability - Lower Bound
Theorem 2:
There are linear congestion games whose dominant
equilibrium have price of stability of the SUM social
cost approaching
as the number of players N tends to infinity.
So this holds for correlated equilibrium.
27
Price of Stability - Lower Bound
A strategies
(equilibrium)
type a
P strategies
(optimal social cost)
1
1
2
2
3
3
N-1
N-1
N
N
28
Price of Stability - Lower Bound
A strategies
(equilibrium)
type b
P strategies
(optimal social cost)
1
1
2
2
3
3
N-1
N-1
N
N
29
Price of Stability - Lower Bound
A strategies
(equilibrium)
type g
P strategies
(optimal social cost)
1
1
2
2
3
3
N-1
N-1
N
N
30
Price of Stability - Lower Bound
We will fix a,b and m such that in every
allocation (A1,……AK,PK+1……PN), players prefer
their Ai strategies.
In order to be dominant (A1, ……. AN)
31
Price of Stability - Lower Bound
And it is satisfied by
For
tends to
infinity.
, the price of anarchy
as N tends to
32
Correlated PoA- Upper Bound
Theorem 4:
The correlated price of anarchy of the average
social cost is 5/2.
Lemma 2:
For every non negative integers a, b :
33
Correlated PoA- Upper Bound
Proof:
Let q be a correlated equilibrium and P be an optimal
allocation.
Summing for all players
The optimal cost is
34
Correlated price of anarchy
35
Correlated price of anarchy – cont.
Sum over all players i
We finally obtain
36
Asymmetric weighted games
Theorem 6:
For Linear weighted congestions games, the correlated
price of anarchy of the total latency is at most
Lemma 3:
For every non negative real a, b :
Build so
satisfy
Achieved by
37
Asymmetric weighted games
Proof:
Q – correlated equilibrium,
P - optimal allocation
Qe(s) - total load on the facility e for allocation s
Nash inequality
multiply with wi
38
Asymmetric weighted games
And for all players:
Using Lemma 3
PoA = C(q)/ C(P) = (3+√ 5)/2 ≈ 2.618
39
Related Work

Max social cost, parallel links
[Mavronicolas, Spirakis,2001], [Czumaj, Vocking,
2002]

Max social cost, Single-Commodity Network
[Fotakis, Kontogiannis,Spirakis,2005]

Sum social cost, parallel links
[Lucking, Mavronicals, Monien, Rode, 2004]

Splittable, General Network
[Roughgarden, Tardos, 2002]

Max, Sum social cost, General Network
[Awebuch, Azar, Epstein, 2005]
[Christodoulou, Koutsoupias, 2005]
40
Results
For linear congestion games
Correlaed PoS = [1.57,1.6]
Correlatted PoA = 2.5
For weighted congestion games
Correlated PoA = (3+√ 5)/2 ≈ 2.618
41
The End
42