1 on the price of anarchy and stability of correlated equilibria of linear congestion games by...

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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

3

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

6

An Example

From a to c

ed

c

ab

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An Example

From a to c

ed

c

ab

8

An Example

From a to c

ed

c

ab

9

An Example

From e to c

ed

c

ab

10

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