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Congestion games

Congestion games

Laurent Gourves

LAMSADE, CNRS - PSL, Universite Paris Dauphine

September 8, 2015

Congestion games

Outline

The setting

Pure Nash equilibria

Strong equilibria

Congestion games with player-specific payoffs

Capacitated congestion games

Conclusion

Congestion games

The setting

Table of contents

The setting


Strong equilibria



Conclusion

Congestion games

The setting

Congestion effect

◮ driving on the highway◮ traffic jams :(◮ empty highway :)

◮ price of French books in◮ Reykjavik :(◮ Paris :)

Congestion games

The setting

Congestion effect



◮ playing football◮ alone :(◮ with 9 kids :)◮ with 45 kids :(

Congestion games

The setting

Congestion effect




◮ playing chess◮ odd number of players :(◮ even number of players :)

Congestion games

The setting

Congestion effect




◮ playing chess◮ odd number of players :(◮ even number of players :)

It can be good :) or bad :(, it depends on the number of users butnot on their identity

Congestion games

The setting

Congestion model

〈N,M, (Ai )i∈N , (dj )j∈M〉

◮ N = {1, 2, . . . , n} is the set of players

◮ M = {1, 2, . . . ,m} is the set of resources

◮ Ai is the strategy space of player i◮ Ai is a non-empty subset of 2M

◮ dj is a delay function associated with resource j

◮ dj : {1, 2, . . . , n} → IR

If x players have resource j in their strategy then the delayassociated with j is dj(x)

Congestion games

The setting

Congestion game

Strategic game

◮ σ is a strategy profile and σi is the strategy of player i

◮ A is the set of all strategy profiles (×i∈NAi)

◮ for j ∈ M, ℓj(σ) = |{i ∈ N : j ∈ σi}| is the load of resource j

− congestion model + individual costs of the players (ci )i∈N− each player i strives to minimize her individual costci (σ) =

∑

j∈σidj(ℓj(σ))

Congestion games

The setting

Congestion game

1

2

3

A

B

C

D

dA := 4/5/7

dB := 43/30/28

dC := 16/20/55

dD := 1/10/52

players resources delays

Congestion games

The setting

1

2

3

A

B

C

D

dA := 4/5/7

dB := 43/30/28

dC := 16/20/55

dD := 1/10/52

cost of player 1 : 5 + 30 = 35 ; cost of player 2 : 5 ;cost of player 3 : 30 + 16 = 46

Congestion games


Table of contents

The setting


Strong equilibria



Conclusion

Congestion games


Pure Nash equilibrium

TheoremEvery congestion game admits a pure strategy Nash equilibrium

ProofEvery state σ is associated with a potential function Φ : A→ IR

Φ(σ) =∑

j∈M

ℓj(σ)∑

k=1

dj(k)

Reminder: ℓj(σ) is the number of players using resource j understrategy profile σ

Congestion games


Suppose player i modifies her strategy

The initial state is σ and the new state is σ′

Φ(σ)− Φ(σ′) =∑

j∈M

ℓj(σ)∑

k=1

dj(k)−∑

j∈M

ℓj (σ′)

∑

k=1

dj(k)

=∑

j∈σi\σ′

i

dj(ℓj(σ)) −∑

j∈σ′

i\σi

dj(ℓj (σ′))

= ci (σ)− ci (σ′)

The individual cost of player i and the potential decrease by thesame amount

Congestion games


1

2

3

A

B

C

D

cA := 4/5/7

cB := 43/30/28

cC := 16/20/55

cD := 1/10/52

Φ(σ) = 4 + 5 + 43 + 30 + 16 = 98 cost of player 3 : 46

Congestion games


1

2

3

A

B

C

D

cA := 4/5/7

cB := 43/30/28

cC := 16/20/55

cD := 1/10/52

Φ(σ) = 4 + 5 + 43 + 30 + 1 = 83 cost of player 3 : 31

Congestion games


If the players sequentially perform unilateral profitable deviationsthen we get a finite sequence of strategy profiles (σ0, σ1, · · · , σr )called improvement path

It must be Φ(σ0) > Φ(σ1) > · · · > Φ(σr )

If the improvement path is finite then its last state is a pure Nashequilibrium

A game has the finite improvement property (FIP) if everyimprovement path of the game is finite

Congestion games have the FIP

A state σ that minimizes Φ(σ) is a pure Nash equilibrium

Congestion games


Finite but exponentially long improvement path

Consider a congestion game with only one player

Her strategy set is every nonempty subset of M

Using resource j induces a cost of 2j−1, for all j

state 3 2 1 cost

σ0 1 1 1 7←σ1 1 1 0 6σ2 1 0 1 5σ3 1 0 0 4σ4 0 1 1 3σ5 0 1 0 2σ6 0 0 1 1

Congestion games






state 3 2 1 cost

σ0 1 1 1 7σ1 1 1 0 6←σ2 1 0 1 5σ3 1 0 0 4σ4 0 1 1 3σ5 0 1 0 2σ6 0 0 1 1

Congestion games






state 3 2 1 cost

σ0 1 1 1 7σ1 1 1 0 6σ2 1 0 1 5←σ3 1 0 0 4σ4 0 1 1 3σ5 0 1 0 2σ6 0 0 1 1

Congestion games



We get a sequence of 2|M| − 1 states

state 3 2 1 cost

σ0 1 1 1 7σ1 1 1 0 6σ2 1 0 1 5σ3 1 0 0 4σ4 0 1 1 3σ5 0 1 0 2σ6 0 0 1 1←

Each step is a better response, not a best response

However, even with best-response, there exists an instance havingan exponentially long improvement path

Congestion games


Singleton congestion games

DefinitionIn a singleton congestion game, every player selects a singleresource at a time, i.e. Ai ⊆ M

TheoremIn a singleton congestion game, the length of every improvementpath is at most n2m

Congestion games


Singleton congestion games

DefinitionIn a singleton congestion game, every player selects a singleresource at a time, i.e. Ai ⊆ M

TheoremIn a singleton congestion game, the length of every improvementpath is at most n2m

Proof sketchThe size of F = {dj (k) : 1 ≤ k ≤ n and 1 ≤ j ≤ m} is at most nm

Thus Φ(σ) =∑

j∈M

∑ℓj (σ)k=1 dj(k), which consists of n terms, can

take at most n2m different values

The potential sctrictly decreases in an improvement path

Congestion games


Symmetric network congestion gamesA directed graph and a set of n players

A common source s and a common destination t

The cost of each arc j is dj(x) where x is the number of playersusing j

Every player wants to find the cheapest directed path from s to t

•

•

•

•

•

•

•s

a

b

c

e

d

t

Congestion games


Symmetric network congestion gamesA directed graph and a set of n players

A common source s and a common destination t

The cost of each arc j is dj(x) where x is the number of playersusing j

Every player wants to find the cheapest directed path from s to t

•

•

•

•

•

•

•

3 players

s

a

b

c

e

d

t

1/4/5

6/7/92/3/4

1/6/7

7/9/123/6/10

4/5/92/5/7

1/8/9

Congestion games


Symmetric network congestion games

TheoremThere exists a polynomial time algorithm for computing a Nashequilibrium in symmetric network congestion games when (dj)j∈Mis monotone non decreasing

ProofReplace each arc ab by n arcs with costs dab(1), . . . dab(n) andcapacity 1

• •a b1/4/5

• •a b

1

4

5

By hypothesis dab(1) ≤ dab(2) ≤ . . . ≤ dab(n). A flow of n unitswith minimum cost corresponds to a strategy profile σ minimizingφ(σ)

Congestion games

Strong equilibria

Table of contents

The setting


Strong equilibria



Conclusion

Congestion games

Strong equilibria

Strong equilibria

Deviation from σ: some players Γ ⊆ N change their strategy, i.e.they play s ∈ AΓ instead of σΓ

◮ unilateral deviation when |Γ| = 1

◮ group deviation when |Γ| ≥ 1

Profitable deviation: for every i ∈ Γ, ci (s, σ−Γ) < ci (σ)

Nash equilibrium: no profitable unilateral deviation

Strong equilibrium: no profitable group deviation

Congestion games

Strong equilibria

Strong equilibria in singleton congestion games

2 players and 2 resources

• •s t

1/3

4/2

top bottom

top3 4

3 1

bottom1 2

4 2

Congestion games

Strong equilibria


2 players and 2 resources

• •s t

1/3

4/2

top bottom

top3 4

3 1

bottom1 2

4 2

(top,top) is a Nash equilibrium but there is no strong equilibrium

Congestion games

Strong equilibria


Suppose every delay function dj is monotone non-decreasing

Congestion has a negative impact

TheoremEvery Nash equilibrium is also a strong equilibrium

Congestion games

Strong equilibria





Proof by contradiction

Take a Nash equilibrium σ and suppose a coalition of players wantsto deviate to σ′

Congestion games

Strong equilibria





Proof by contradiction

Take a Nash equilibrium σ and suppose a coalition of players wantsto deviate to σ′

If there exists j such that ℓj(σ′) > ℓj(σ) then σ is not a Nash

equilibrium

Congestion games

Strong equilibria

ℓj(σ) = ℓj(σ′), ∀j because n =

∑

j∈M ℓj(σ) =∑

j∈M ℓj(σ′)

Same multi-set of individual costs → at least one player of thecoalition does not decrease her individual cost

�

Congestion games

Strong equilibria


Suppose every delay function dj is monotone non-increasing

Congestion has a positive impact

• •s t

3/1

4/2

Congestion games

Strong equilibria


Suppose every delay function dj is monotone non-increasing

Congestion has a positive impact

• •s t

3/1

4/2

top bottom

top1 4

1 3

bottom3 2

4 2

2 Nash equilibria and 1 strong equilibrium

Congestion games

Strong equilibria


Congestion has a positive impact, dj is monotone non-increasing

TheoremEvery instance admits a strong equilibrium

Algorithmic construction of the strong equilibrium

1. Initialisation◮ No player is assigned a resource at the beginning◮ P = set of pending players (= N)◮ R = set of pending resources (= M)

Congestion games

Strong equilibria


Congestion has a positive impact, dj is monotone non-increasing

TheoremEvery instance admits a strong equilibrium

Algorithmic construction of the strong equilibrium

1. Initialisation◮ No player is assigned a resource at the beginning◮ P = set of pending players (= N)◮ R = set of pending resources (= M)

2. While P 6= ∅ do◮ ∀j ∈ R , let xj = |{i ∈ P : j ∈ Ai}|◮ j∗ = argminj∈R dj(xj)◮ Assign the xj∗ players of P to resource j∗

◮ Remove these xj∗ players from R and remove j∗ from R

Congestion games

Strong equilibria


•

•

•

•

•

•

•

• players

resources5/2

1

7/6/5/3

2

3/2

3

8/1

4

Congestion games

Strong equilibria


•

•

•

•

•

•

•

• players

resources5/2

1

7/6/5/3

2

3/2

3

8/1

4

P = {1, 2, 3, 4}R = {1, 2, 3, 4}

Congestion games

Strong equilibria


•

•

•

•

•

•

•

• players

resources5/2

1

7/6/5/3

2

3/2

3

1

4

P = {1, 2, 3, 4}R = {1, 2, 3, 4}

Congestion games

Strong equilibria


•

•

•

•

•

•

•

• players

resources5/2

1

7/6/5/3

2

3/2

3

1

4

P = {1, 3}R = {1, 2, 3}

Congestion games

Strong equilibria


•

•

•

•

•

•

•

• players

resources5/2

1

7/6/5/3

2

3

3

1

4

P = {1, 3}R = {1, 2, 3}

Congestion games

Strong equilibria


•

•

•

•

•

•

•

• players

resources5/2

1

7/6/5/3

2

3

3

1

4

P = {1}R = {1, 2}

Congestion games

Strong equilibria


•

•

•

•

•

•

•

• players

resources5

1

7/6/5/3

2

3

3

1

4

P = ∅R = {2}

Congestion games

Strong equilibria


ProofThe first group of players assigned by the algorithm gets the lowestpossible cost, they have no incentive to deviate, and no extraplayer can join them on their resource

The second group of players assigned by the algorithm gets thesecond-lowest possible cost, they have no incentive to deviate, andno extra player can join them on their resource

etc.

Congestion games


Table of contents

The setting


Strong equilibria



Conclusion

Congestion games


Payoff maximization instead of cost minimization

So far dj(x) was interpreted as: if x players use resource j theneach of them incurs a cost of dj(x)

Moreover ci (σ) =∑

j∈σidj(ℓj(σ)) is the cost of player i

Actually dj(x) can be interpreted as: if x players use resource j

then each of them receives a payoff of dj(x)

Accordingly, ui (σ) =∑

j∈σidj(ℓj (σ)) is the payoff of player i

All previous results remain valid

Congestion games



Each delay function dj is replaced by dij , a delay function that isspecific to player i

If x players use resource j then player i receives a payoff of dij(x)

It can be dij(x) 6= di ′j(x) for (i , i′) ∈ N × N

Restrictions:

◮ singleton strategy (one resource at a time)

◮ symmetric strategy space (Ai = M, ∀i ∈ N)

◮ the dij ’s are monotone non increasing (congestion has anegative impact)

Congestion games


Congestion games with player-specific payoffsThe finite improvement property (FIP) is not preserved (unlessthere are 2 resources)

2 players and 3 resources:

resource 1 resource 2 resource 3

1 user (6, 5) (5, 6) (3, 3)2 users (1, 4) (4, 1) (2, 2)

the payoff of player 1 is 6 if he uses resource 1 alone


resource 1 (1, 4) (6, 6) (6, 3)resource 2 (5, 5) (4, 1) (5, 3)resource 3 (3, 5) (3, 6) (2, 2)

Congestion games



A cycle of better moves



Congestion games



Pure Nash equilibria in blue



An instance with a cycle of best moves exists (3 players)

Congestion games


TheoremEvery congestion game with player-specific payoffs has a pure Nashequilibrium

Proof by induction

Every instance with n − 1 players has a pure NE

Congestion games



Proof by induction


Put player n apart and take a pure NE σ for players 1, . . . , n − 1

Then introduce player n and let her take her best response

Congestion games



Proof by induction




Only a player with the same strategy as player n may want todeviate

◮ diσn is non increasing ⇒ no one wants to join resource σn

Suppose player i0 moves from resource σn to resource j0

Then only a player with strategy j0, say i1, may want to move toresource j1

Congestion games



Proof by induction




Only a player with the same strategy as player n may want todeviate

◮ diσn is non increasing ⇒ no one wants to join resource σn

Suppose player i0 moves from resource σn to resource j0

Then only a player with strategy j0, say i1, may want to move toresource j1

and so on...

Congestion games


Each circle is a resource

1 ℓ1

2

ℓ2

3

ℓ3

4

ℓ4

5

ℓ5

6ℓ6

ℓj is the load of resource j (number of users) in the pure Nashequilibrium reached by the n − 1 first players

Congestion games


Player n is introduced and her best response is to play resource 2

1 ℓ1

2

ℓ2 + 1

3

ℓ3

4

ℓ4

5

ℓ5

6ℓ6

Congestion games


A player from resource 2 may leave it

1 ℓ1

2

ℓ2

3

ℓ3

4

ℓ4 + 1

5

ℓ5

6ℓ6

Congestion games


A player from resource 4 may leave it

1 ℓ1

2

ℓ2

3

ℓ3 + 1

4

ℓ4

5

ℓ5

6ℓ6

Congestion games


The load vector is always (ℓ1, ..., ℓm) with a +1 somewhere

No player moves more than once

◮ never move twice in a row because a move is a best response

Congestion games


The load vector is always (ℓ1, ..., ℓm) with a +1 somewhere

No player moves more than once

◮ never move twice in a row because a move is a best response

◮ if a player has moved to resource j then she is happy with theload vector (ℓ1, ..., ℓj + 1, ..., ℓm) and no other load vector canpush her to reconsider her choice

�

Congestion games


Table of contents

The setting


Strong equilibria



Conclusion

Congestion games


Congestion game with capacities

Each resource j is associated with a capacity level κj

κj = maximum number of users that resource j maysimultaneously accommodate

Each resource j has a ranking posj : N → [1, n], prescribing thepriority of accommodation of the users

Congestion games


Congestion game with capacities

Each resource j is associated with a capacity level κj

κj = maximum number of users that resource j maysimultaneously accommodate

Each resource j has a ranking posj : N → [1, n], prescribing thepriority of accommodation of the users

Nj (σ) = set of players using resource j in the strategy profile σ

Congestion games


Player i ∈ Nj(σ) is accommodated by resource j iff

|{i ′ ∈ Nj(σ) : posj(i′) < posj(i)}| < κj

The delay of player i ∈ Nj (σ) on resource j is:

d ij (σ) =

{

dj(min{ℓj(σ), κj}) if i is accommodated,+∞ otherwise.

Congestion games


Player i ∈ Nj(σ) is accommodated by resource j iff

|{i ′ ∈ Nj(σ) : posj(i′) < posj(i)}| < κj

The delay of player i ∈ Nj (σ) on resource j is:

d ij (σ) =

{

dj(min{ℓj(σ), κj}) if i is accommodated,+∞ otherwise.

The cost of a player i under strategy profile σ is

ci (σ) =∑

j∈σi

d ij (σ)

Congestion games


x y

z

{x} {y , z}

{x}+∞ 3

2 2

{y , z}2 +∞

3 0

κx = κz = 1, κy = 2

Priority is always given to the line player

Existence of a pure Nash equilibrium not guaranteed

Congestion games


Symmetric singleton capacitated congestion games

TheoremA pure Nash equilibrium can be computed for any instance ofsymmetric singleton capacitated congestion game

Sketch of the algorithm

1. Find ℓj the number of players resource j accommodates◮ greedy strategy

Congestion games






2. Sort the resources by non decrasing delay◮ d1(ℓ1) ≤ . . . ≤ dm(ℓm) w.l.o.g.

Congestion games






2. Sort the resources by non decrasing delay◮ d1(ℓ1) ≤ . . . ≤ dm(ℓm) w.l.o.g.

3. For r = 1 to m, assign to resource r the players with highestpriority

Congestion games



1

κ1

2 3

κ2

κ3

8 players

Congestion games



1

κ1

2 3

κ2

κ3

6 players

Congestion games



1

κ1

2 3

κ2

κ3

3 players

Congestion games



1

κ1

2 3

κ2

κ3

1 player

Congestion games



1

κ1

2 3

κ2

κ3

0 player

Congestion games



replacements

1

κ1

2 3

κ2

κ3

Suppose d1(3) ≤ d2(4) ≤ d3(1)

Fill resource 1 with the 3 players with highest priority on resource 1

Congestion games



replacements

1

κ1

2 3

κ2

κ3

Suppose d1(3) ≤ d2(4) ≤ d3(1)

Fill resource 2 with the 4 players with highest priority on resource 2(the players assigned to resource 1 are ignored)

Congestion games



replacements

1

κ1

2 3

κ2

κ3

Suppose d1(3) ≤ d2(4) ≤ d3(1)

Fill resource 3 with the remaining player

Congestion games

Conclusion

Table of contents

The setting


Strong equilibria



Conclusion

Congestion games

Conclusion

Variations in congestion games

We have seen special cases of congestion games: symmetric,singleton and monotone

◮ Non-atomic congestion games: a continuum of players, eachof which is infinitesimally small

◮ Weighted congestion games◮ each player i has a weight wi

◮ ℓj(σ) is defined as∑

i∈N :j∈σiwi instead of |{i ∈ N : j ∈ σi}|

Congestion games

Conclusion

Variations in congestion games

We have seen special cases of congestion games: symmetric,singleton and monotone

◮ Non-atomic congestion games: a continuum of players, eachof which is infinitesimally small

◮ Weighted congestion games◮ each player i has a weight wi

◮ ℓj(σ) is defined as∑

i∈N :j∈σiwi instead of |{i ∈ N : j ∈ σi}|

◮ Bottleneck congestion games◮ ci(σ) = maxj∈σi

dj(ℓj(σ))

Congestion games

Conclusion

Important aspects not covered by this presentation

Complexity of computing an equilibrium in a congestion game

Potential games

Congestion games

Conclusion

Important aspects not covered by this presentation

Complexity of computing an equilibrium in a congestion game

Potential games

Price of anarchy/stability

Congestion games

Conclusion

Selected bibliography

1. R. Rosenthal. A class of games possessing pure-strategy Nashequilibria. International Journal of Game Theory, 1973.

2. D. Monderer and L. Shapley. Potential games. Games andEconomic Behavior, 1996.

3. I. Milchtaich. Congestion Games with Player-Specific PayoffFunctions. Games and Economic Behavior, 1996.

4. R. Holzman and N. Law-Yone. Strong equilibrium incongestion games. Games and Economic Behavior, 1997.

5. S. Ieong, R. McGrew, E. Nudelman, Y. Shoham and Q. Sun.Fast and compact: A simple class of congestion games. AAAI2005.

6. L. Gourves, J. Monnot, S. Moretti and Nguyen Kim T.Congestion games with capacitated resources. Theory ofcomputing systems.

Congestion games

Conclusion

Thank you!

congestion games - uniupo.itpeople.unipmn.it/fragnelli/stud/gtaa/congestion.pdfcongestion games pure...

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