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On the price of anarchy and stability of correlated equilibria of linear congestion games
By George Christodoulou Elias Koutsoupias
Presented by Efrat Naim
Part of slides taken from George Christodoulou and Elias Koutsoupias web site
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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
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Congestion Games
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.
d c
ba
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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)
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An Example
ed
c
ab
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An Example
From a to c
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c
ab
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An Example
From a to c
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c
ab
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An Example
From a to c
ed
c
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An Example
From e to c
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c
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An Example
From e to c
ed
c
ab
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An Example
Nash Equilibrium
Player 1 has cost 1+1=2Player 2 has cost 1+1 =2
ed
c
ab
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An Example
Another Nash Equilibrium
Player 1 has cost 2+1+1=4Player 2 has cost 2+1+1=4
ed
c
ab
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Mixed Strategy
A mixed strategy for a player is a probability distribution over its pure strategy set.
Mixed strategy profile p = (p1,…….pN)
ed
c
ab
1/2
1/2
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Correlated Strategy
A correlated strategy q for a set of players is any probability distribution over the set S = X i€N Si
ed
c
ab
ed
c
ab
1/2 1/2
1/2 0
0 1/2
L
R
L R
1
2
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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.
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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
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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
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Price of Anarchy
ed
c
ab
Player 1 has cost 2+1+1=4
Player 2 has cost 2+1+1=4
PoA = (4+4)/ (2+2) = 2
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Price of Stability
Price of stability
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Price of Stability
ed
c
ab
Player 1 has cost 1+1=2
Player 2 has cost 1+1=2
PoS = (2+2)/ (2+2) = 1
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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
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Correlated PoS – Upper Bound
We consider linear latencies
fe(x) = aex+be
Lemma 1:For every pair of non negative integers it holds
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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
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Correlated PoS – Upper Bound
Proof:
Let X be a pure strategy profile X=(X1,………XN)
From the potential inequality
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Correlated PoS – Upper Bound
A is Nash equilibrium so
Summing for all the players we get
Adding the two inequalities and use Lemma 1
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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
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Price of Stability - Lower Bound
Theorem 2:There are linear congestion games whose dominantequilibrium have price of stability of the SUM social cost approachingas the number of players N tends to infinity.
So this holds for correlated equilibrium.
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Price of Stability - Lower Bound
A strategies type P strategies
(equilibrium) (optimal social cost)1
2
3
N-1
N
1
2
3
N-1
N
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Price of Stability - Lower Bound
A strategies type P strategies
(equilibrium) (optimal social cost)1
2
3
N-1
N
1
2
3
N-1
N
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Price of Stability - Lower Bound
A strategies type P strategies(equilibrium) (optimal social
cost)
1
2
3
N-1
N
1
2
3
N-1
N
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Price of Stability - Lower Bound
We will fix and m such that in every allocation (A1,……AK,PK+1……PN), players
prefer their Ai strategies.
In order to be dominant (A1, ……. AN)
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Price of Stability - Lower Bound
And it is satisfied by
For , the price of anarchy
tends to as N tends to
infinity.
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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 :
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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
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Correlated price of anarchy
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Correlated price of anarchy – cont.
Sum over all players i
We finally obtain
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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 :
Build so satisfy
Achieved by
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Asymmetric weighted games
Proof:
Q – correlated equilibrium,P - optimal allocation e(s) - total load on the facility e for allocation s
multiply with i
Nash inequality
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Asymmetric weighted games
And for all players:
PoA = C(q)/ C(P) = (3+√ 5)/2 ≈ 2.618
Using Lemma 3
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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]
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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
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The End