network congestion games

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Network Congestion GamesEvdokia Nikolova

Assistant ProfessorTexas A&M University

College Station, TX

Best route depends on others

Evdokia Nikolova Network Congestion Games

Network Congestion Games

Travel time increases with congestion

Evdokia Nikolova

• Highway congestion costs were$115 billion in 2009.

• Avg. commuter travels 100 minutes a day.


Example: Inefficiency of equilibria

Town A Town B

Suppose drivers (total 1 unit of flow) leave from town A towards town B.

What is the traffic on the network?Every driver wants to minimize her own travel time.

1/2

1/2

In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path.

Delay is 1.5 hours for everybody at the unique Nash equilibriumx hours

x hours

1 hour

1 hour

Evdokia Nikolova



Town A Town B

A benevolent mayor builds a superhighway connecting the fast highways of the network.

What is now the traffic on the network?

1

No matter what the other drivers are doing it is always better for me to follow the zig-zag path.

Delay is 2 hours for everybody at the unique Nash equilibriumx hours

x hours

1 hour

1 hour

0 hours

Evdokia Nikolova



A B

1

A B

1/2

1/2

vs

Adding a fast road on a road-network is not always a good idea! Braess’s paradox

In the RHS network there exists a traffic pattern where all players have delay 1.5 hours.

Price of Anarchy:

x hours

x hours

1 hour

1 hour

x hours

x hours

1 hour

1 hour

Evdokia Nikolovameasures the loss in system performance due to free-will


Game model

• Directed graph G = (V,E)Multiple source-dest. pairs (sk,tk), demand dk

• Players (users): nonatomic (infinitesimally small) • Strategy set: paths Pk between (sk,tk) for all k

Players’ decisions: flow vector Sometimes will use for path flow.

• Edge delay (latency) functions: typically assumed continuous and nondecreasing.

|||| or ERRx

)( ee x

Evdokia Nikolova

||Rf


Outline

• Wardrop Equilibrium • Social Optimum

• Price of Anarchy

Evdokia Nikolova

Wardrop’s First Principle• “Travel times on used routes are equal and no greater

than travel times on unused routes.”

• Definition: A flow x is a Wardrop Equilibrium (WE) if for every source-dest. pair k and for every path with positive flow between this pair,

where

Also called User Equilibrium or Nash Equilibrium.

• Equilibrium flow is called Nash flow.

),()( ' xx pathpath


.)()(

pathe

eepath xx kPpath ' allfor


Outline



Evdokia Nikolova

• “The average [total] journey time is minimum.”

• The cost of flow x is defined as the “total journey time”:

• Denote , assumed convex.

Wardrop’s Second Principle


paths all

)()( xxxC pathpath

epathpath

Eeee xx

)(

)(:)( eeeee xxxc

paths all

)(pathe

eepath xx

.)( eEe

ee xx

Wardrop’s Second Principle• “The average [total] journey time is

minimum.”

• Definition: A flow x is a Social Optimum if it minimizes total delay:


. allfor 0

, allfor

, allfor s.t.

)()(min

:

Ppf

Kkfd

Eefx

xxxc

p

Pppk

pePppe

Eeeee

Eeee

k

flow constraints

• “The average [total] journey time is minimum.”

• Definition: A flow x is a Social Optimum if it minimizes total delay:

Social Optimum


sconstraint flow s.t.

)()(min

Ee

eeeEe

ee xxxc

• Definition: A flow x is a Social Optimum if it solves

• Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’,

where

Social Optimum


Ee

eeeEe

ee xxxc )()(min

),(')(' ' xcxc pp

.)(')('

pe

eep xcxcProof sketch: marginal benefit of marginal cost of reducing traffic on p increasing traffic on p’



• Corollary 1: If costs are convex, local opt is a global opt, and lemma gives equivalent defn of Social Optimum.

Social Optimum (SO)


Ee

eeeEe

ee xxxc )()(min

.)(')(' ' xcxc pp

)( ee xc



• Corollary 2: If costs are convex, SO is an equilibrium with respect to modified latencies

Social Optimum (SO)


Ee

eeeEe

ee xxxc )()(min

.)(')(' ' xcxc pp

)( ee xc

.)(')()(')(*eeeeeeeee xxxxcx



• Corollary 2: If costs are convex, SO is an equilibrium with respect to modified latencies

Social Optimum (SO)


Ee

eeeEe

ee xxxc )()(min

.)(')(' ' xcxc pp

)( ee xc

.)(')()(')(*eeeeeeeee xxxxcx

Mechanism Design Interpretation: If users value time and money equally, imposing tolls per unit flow on each edge will cause selfish players to reach the Social Optimum!

)(' ee x


• Corollary 3: If costs are convex, SO exists and can be found efficiently by solving convex program above.

Computing Social Optimum (SO)


)( ee xc


)()(min

Ee

eeeEe

ee xxxc


Outline

• Revisit Wardrop Equilibrium • Social Optimum


Evdokia Nikolova

Equilibrium existence• WE Definition: A flow x is a Wardrop

Equilibrium if for every source-dest. pair k and for every path with positive flow between this pair,

compare with:

• SO Definition: A flow vector x is a Social Optimum for every source-dest. pair k and for every path with positive flow between this pair,

' allfor ),()( ' pathxx pathpath


' allfor ),(')(' ' pathxcxc pathpath



where

• SO Definition: A flow vector x is a Social Optimum for every source-dest. pair k and for every path with positive flow between this pair,

' allfor ),(')(' ' pathxhxh pathpath


' allfor ),(')(' ' pathxcxc pathpath

).(:)(' eeee xxh



where

• Alternative SO Definition: A flow vector x is a Social Optimum if it solves:

' allfor ),(')(' ' pathxhxh pathpath


).(:)(' eeee xxh

Ee

ee xc )(min

Equilibrium existence• Alternative WE Definition: A flow vector x is

a Wardrop Equilibrium if it solves:

where



).(:)(' eeee xxh

Ee

ee xc )(min

,)(min Ee

ee xh

ex

eee dzzxh0

)()( E.g.,

Equilibrium existence• Alternative WE Definition: A flow vector x is

a Wardrop Equilibrium if it solves:

where



).(:)(' eeee xxh

Ee

ee xc )(min

,)(min0

Ee

x

e

e

dzz

Equilibrium existence• Alternative WE Definition: A flow vector x is a

Wardrop Equilibrium if it solves:

• Theorem: A Wardrop Equilibrium exists and can be computed in polynomial time. Also, if program above is strictly convex, equilibrium is unique, up to same flow cost.



)(min0

Ee

x

e

e

dzz


Outline



Evdokia Nikolova



Town A Town B

Suppose drivers (total 1 unit of flow) leave from town A towards town B.

What is the traffic on the network?Every driver wants to minimize her own travel time.

1/2

1/2

In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path.

Delay is 1.5 hours for everybody at the unique Nash equilibriumx hours

x hours

1 hour

1 hour

Evdokia Nikolova



Town A Town B

A benevolent mayor builds a superhighway connecting the fast highways of the network.

What is now the traffic on the network?

1

No matter what the other drivers are doing it is always better for me to follow the zig-zag path.

Delay is 2 hours for everybody at the unique Nash equilibriumx hours

x hours

1 hour

1 hour

0 hours

Evdokia Nikolova



A B

1

A B

1/2

1/2

vs

Adding a fast road on a road-network is not always a good idea! Braess’s paradox

In the RHS network there exists a traffic pattern where all players have delay 1.5 hours.

Price of Anarchy:

x hours

x hours

1 hour

1 hour

x hours

x hours

1 hour

1 hour

Evdokia Nikolovameasures the loss in system performance due to free-will

Price of Anarchy

• Cost of Flow: total user cost

• Social optimum: flow minimizing total user cost

• Price of anarchy: (Koutsoupias, Papadimitriou ’99)

Cost Optimum SocialCost mEquilibriusup

instancesproblem


Variational Inequality representation of equilibria

Theorem: Equilibria in nonatomic games are solutions to the Variational Inequality (VI)

where

VI Solution exists over compact convex set with ℓ(x) continuous [Hartman, Stampacchia ‘66]. ∎

• VI Solution unique if ℓ(x) is monotone: (ℓ(x)-ℓ (x’))(x-x’) ≥ 0. [Exercise: verify]

' flows feasible allfor 0)'()( xxxx

ector][latency v ))(),...,(()(

vector][flow ),...,(

||1

||1

xxx

xxx

P

P


Proof: Flow x is an equilibrium if and only if ℓ(x).x <= ℓ(x).x’ .

Proof: (=>) Equilibrium flow routes along minimum-cost paths ℓ(x). Fixing path costs at ℓ(x), any other flow x’ that assigns flow to higher-cost paths will result in higher overall cost ℓ(x).x’.

(<=) Suppose x is not an eq. Then there is a flow-carrying path p with ℓp(x) > ℓp’(x). Shifting flow from p to p’ will obtain flow x’ withℓ(x).x’ < ℓ(x).x, contradiction.

Theorem**: The price of anarchy (PoA) is 4/3 in general graphs and latencies i.e. where x is WE and x* is SO flow.

Pf:

,0, ,)( eeeeeee babxaxl

Price of Anarchy with linear latencies


*)()()( eEe

eeeEe

ee xxxxxC

),(34)( *xCxC

])41([)( *2*** 2

eEe

eeeeeEe

eeee xbxxaxbxxa

Ee Ee

eeeeee xaxbxa 2**

41)(

**References: Roughgarden, Tardos ’02; Correa, Schulz, Stier-Moses ‘04, ‘08

QED. ,)()(43 *xCxC

22**

41 xxxx

2* )21(0 xx

)(41)( * xCxC

*)( eEe

eee xbxa

Take-away points• Equilibrium and Social Optimum in

nonatomic routing games exist and can be found efficiently via convex programs.

• Social optimum is an equilibrium with respect to modified latencies = original latencies plus toll.

• Price of anarchy: 4/3 for linear latencies, can be found similarly for more general classes of latency functions.


References

• Wardrop ‘52, Beckmann et al. ’56, …

• A lot of work in AGT community and others. Surveys of recent work:

• AGT Book Nisan et al. ‘07• Correa, Stier-Moses ’11


Some open questions

• What is the price of anarchy with respect to other Social Cost functions?

• Dynamic (time-changing) latency functions?

• Uncertain delays?


network congestion games

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