2015 01 22 - rende - unical - angelo fanelli: an overview of congestion games

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Non-Cooperative SystemsFundamental of Congestion Games

The Complexity of NashMatroid Congestion Games

An Overview of Congestion Games

Angelo Fanelli

CNRS (France)

Angelo Fanelli An Overview of Congestion Games

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Non-cooperative GamesNash DynamicsSolution Concept and Performance Metric

Non-cooperative Games

Non-cooperative Game

A non-cooperative game is defined by a tupleG = (N, (Σi )i∈N , (ci )i∈N), where

N = {1, 2, . . . , n} denotes the set of n players (agents)

Σi is the set of strategies of player i

ci : Σ 7→ R is the cost function for player i

A state of the game is given by an assignment of strategies to players

S = (s1, s2, . . . si , . . . , sn) si ∈ Σi .

The set of states is denoted by Σ = ×i∈NΣi

Each player acts selfishly: it aims at choosing the strategy loweringits cost, given the strategy choices of the other players


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Nash Dynamics - 1

S = (s1, s2, . . . , si , . . . , sn)If player i deviates from si to s ′i , the new resulting state is

S ′ = (S−i , s′i ) = (s1, s2, . . . , s

′i , . . . , sn)

Improvement move

An improvement move of player i in state S = (s1, s2, . . . , sn), is adeviation of player i from si to s ′i such that ci (S−i , s

′i ) < ci (S)


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Nash Dynamics - 2

Nash Dynamics Graph (ND Graph)

A Nash Dynamics Graph associated to a game G is a directed graphB = (V ,A) where

V , set of the states of G, i.e., V = Σ

(S , S ′) ∈ A corresponds to an improvement move

S ′ = (S−i , s′i ) and s ′i ∈ Σi

ci (S′) < ci (S)

Nash Dynamic

A path of the ND Graph

(Pure) Nash equilibrium (NE)

A sink of the ND Graph


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(Pure) Nash Equilibrium

(Pure) Nash Equilibrium (NE)

A (Pure) Nash equilibrium is a game state such that no player has animprovement move

Simple notion of a stable state

It does not always exists

Even if it exists, the game may not converge to it

Not necessarily good for the players


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Price of Anarchy

In the tradition of computer science, we want to analyze bounds onperformance ratios

How bad is the worst equilibrium that the game can settle into if theplayers are left to their own devices?

Social Cost C : Σ 7→ RLet Opt denote the minimum value of C

Usually C (S) =∑

i∈N ci (S)

Price of Anarchy (Koutsoupias and Papadimitriou, STACS ’99)

Let N be the set of all NE, PoA = maxS∈NC(S)Opt


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DefinitionExistence and Convergence to NEThe PoA for Linear Congestion Games

Congestion Games - formal definition

A Congestion Game is a tuple C = (N,E , (Σi )i∈N , (fe)e∈E , (ci )i∈N)where

N = {1, 2, . . . , n}, set of playersE = {e1, e2, . . . , em}, set of resourcesΣi ⊆ 2E , set of strategies of player i

fe : N 7→ N, latency (delay, payment) function of resource e ∈ E

ci (S) =∑

e∈sife(ne(S)), cost function of player i

ne(S) = # of players using e in S


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Subclasses of Congestion Games

network congestion games: the strategy space Σi of player icorresponds to the set of paths between a source ri and adestination ti in an underlying graph G = (V ,E )

symmetric congestion games: all players have the same strategyspace, otherwise called asymmetric

symmetric network congestion games: all players have the samesource and destination

singleton congestion games: each strategy consists only of asingle resource


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Example of Symmetric Network CG

Step

fe(1), fe(2), . . .

e

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2

In the example: two players(r , t)


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Step

Initial State

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2


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Step

Initial State

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2

Step 1...

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2


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Step

Step 1...

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2


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Step

Step 1...

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2

Step 2...

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2


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Step

Step 2...

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2


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Step

Step 2...

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2

Step 3... Nash Equilibrum!!

r

a b

t

d

c

1, 1

0, 99

2, 2

6, 6 1, 1

1, 1 0, 0

0, 2


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Rosenthal’s potential function

Proposition (Rosenthal 1973)

For every CG any sequence of improvement moves is finite

Sketch of proof.It follows by a potential function argument

Φ : Σ 7→ N

Φ(S) =∑e∈E

ne(S)∑i=1

fe(i)

Let S ′ = (S−i , s′i ) the resulting state of an improvement move of player i

from si to s ′i , then

ci (S)− ci (S′) = Φ(S)− Φ(S ′)

The ND Graph has at least one sink and does not contain cycles


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Proof.

Φ(S) =∑e∈E

ne(S)∑i=1

fe(i)

Idea: Φ(S) can be obtained by inserting the players one after theother in any order, and summing the costs of the players at thepoint of time at their insertion

Assume that they are inserted in this order 1, 2, . . . , n

n(i)e (S); # players using e that have an index in {1, 2, . . . , i}

Let c ′i =∑

e∈sife(n

(i)e (S)) the cost of player i at insertion time, that

is a virtual cost that i would have if players with index {i + 1, . . . , n}would not exist∑

i∈N

c ′i =∑i∈N

∑e∈si

fe(n(i)e (S)) =

∑e∈E

ne(S)∑i=1

fe(i) = Φ(S)


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Proof. (cont.)

Φ(S) =∑e∈E

ne(S)∑i=1

fe(i)

∑i∈N

c ′i (S) =∑i∈N

∑e∈si

fe(n(i)e (S)) =

∑e∈E

ne(S)∑i=1

fe(i) = Φ(S)

For player n we have cn(S) = c ′n(S)

Suppose player n can decrease its cost by switching strategy, then

cn(S)− cn(S′) = Φ(S)− Φ(S ′)

It is not a special property of player n. The potential can beobtained by inserting the players in any order.


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PoA for Linear Congestion Games

Christodoulou and Koutsoupias, STOC ’05

For every linear congestion game, if C (S) =∑

i∈N ci (S), thenPoA ≤ 5/2.

Proof.

Let us assume for simplicity that fe(x) = x for every e ∈ E

Let S = (s1, s2, . . . , sn) be a Nash equilibrium

Let S∗ = (s∗1 , s∗2 , . . . , s

∗n ) be an optimal allocation

C (S) =∑i∈N

ci (S) =∑i∈N

∑e∈si

ne(S) =∑e∈E

n2e(S)

Opt = C (S∗) =∑i∈N

ci (S∗) =

∑i∈N

∑e∈si

ne(S∗) =

∑e∈E

n2e(S∗)


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Proof. (cont.)

Nash inequality for player i

ci (S) =∑e∈si

ne(S) ≤∑e∈s∗i

ne(S−i , s∗i ) ≤

∑e∈s∗i

(ne(S) + 1)

Let us sum the previous inequalities over all players i ∈ N

C (S) =∑i∈N

ci (S)

≤∑i∈N

∑e∈s∗i

(ne(S) + 1)

=∑e∈E

ne(S∗)(ne(S) + 1)


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Proof. (cont.)

C (S) ≤∑e∈E

ne(S∗)(ne(S) + 1)

Lemma

For every pair on non-negative integers α, β, it holdsβ(α+ 1) ≤ 1

3α2 + 5

3β2.

ne(S∗)(ne(S) + 1) ≤ 1

3n2e(S) +

5

3n2e(S

∗)∑e∈E

ne(S∗)(ne(S) + 1) ≤ 1

3

∑e∈E

n2e(S) +5

3

∑e∈E

n2e(S∗)

C (S) ≤ 1

3C (S) +

5

3Opt


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Potential GamesLocal Search ProblemsComplexity of General Congestion Games

Potential Games

Potential Game

A potential game is a game that admits a potential function, that is, afunction Φ : Σ 7→ N always decreasing each time a player changes itsstrategy, that is

ci (S)− ci (S−i , s′i ) > 0 ⇒ Φ(S)− Φ(S−i , s

′i ) > 0

Proposition

Every potential game admits a Nash equilibrium corresponding to a localminimum of Φ


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Exact Potential Games

Exact Potential Game

∀S ∈ Σ, i ∈ N and ∀s ′i ∈ Σi , it holds

ci (S)− ci (S−i , s′i ) = Φ(S)− Φ(S−i , s

′i )

Monderer and Shapley, ’96

The class of Exact Potential Games is isomorphic to the class ofCongestion Games


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The relationship to Local Search

The potential function allows us to interpret Potential Games asLocal Search Problems

Local Search Problem

A Local Search Problem Π is given by its set of instances IΠ and it iseither a maximization or a minimization problem. For every instanceI ∈ IΠ we are given

a set of feasible solutions F(I )

an objective function c : F(I ) 7→ Zfor every S ∈ F(I ), a neighborhood N (S , I ) ⊆ F(I )

Given an instance IΠ, the problem is to find local optimal solution S .That is c(S) ≤ c(S ′) for all S ′ ∈ N (S , I ) (for minimization)


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Polynomial Local Search Problems (PLS)

A local search problem Π belongs to PLS if the following polynomialalgorithms exist

an algorithm A which computes for every instance I ∈ IΠ an initialfeasible solution S ∈ F(I )

an algorithm B which computes for every instance I ∈ IΠ and everyfeasible solution S ∈ F(I ) the objective value c(S)

an algorithm C which determines for every instance I ∈ IΠ and everyfeasible solution S ∈ F(I ) whether S is locally optimal or not andfinds a better solution in the neighborhood of S in the latter case


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PLS-reduction and PLS-completeness

A problem Π1 from PLS is PLS-reducible to Π2 from PLS if there arepolynomial computable functions f and g such that

f maps instances I ∈ Π1 to instances f (I ) of Π2

g maps pairs (S2, I ) with S2 denoting a solution of f (I ) to solutionsS1 of I

for all instances I ∈ Π1, if S2 is a local optimum of instance f (I )then g(S2, I ) is a local optimum of I

PLS-complete

A local seach problem Π from PLS is PLS-complete if every problem inPLS is PLS-reducible to Π


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PLS-complete

102 6. Time Complexity

POPLS

NPO

NP-hard problems(TSP,STG,MGC,...)

PLS-complete problems

Figure 6.2. Positioning of the classes PO, PLS, and NPO in the case that P NP,NP co-NP, and PO PLS.

B in Definition 6.3 shows that this polynomial-time procedure exists. This provescD NP, which completes the proof of the theorem.

6.1.1 *First PLS-Complete Problem: A Starting PointSuppose we want to apply local search to the problem of finding an integer s with0 s 2n for which a given function f : 0 1 2n 1 0 1 2m 1 isminimized. We can use the neighborhood function in which s 0 1 2n 1is a neighbor of s if s can be obtained from s by flipping exactly one of the n bits inthe binary representation of s. The corresponding local search problem, formulatedin terms of Boolean circuits, is the first problem that has been proved to be PLS-complete.

Boolean circuits are theoretical counterparts of the digital circuits from whichcomputers are made. They compute Boolean functions f : 0 1 n 0 1 m and,conversely, each Boolean function is computed by a circuit.

Definition 6.6. A Boolean circuit is a directed acyclic graph D V A . The nodeset V consists of n input nodes and V n gates. The input nodes have indegreezero and are labeled by the binary variables x1 x2 xn. The labels of the gates aretaken from the set of Boolean functions. The gates with outdegree zeroare called the output nodes and they are additionally labeled by the binary variablesy1 y2 ym. Boolean circuit D computes a Boolean function f : 0 1 n 0 1 m

by deriving for given values of the input variables x1 x2 xn corresponding valuesfor the output variables y1 y2 ym in the following way.

Let l be the label of a gate g. If l is given by , then the value of g is oneminus the value of the node from which the only incoming edge of g is incident.Next, suppose that l . Then g has exactly two incoming edges. Let o1 o2be the values of the nodes from which these two edges are incident. If g is labeled

, then it is assigned the value one if o1 o2 1 and it is assigned the value zero,otherwise. If g is labeled , then it is assigned the value zero if o1 o2 0 and itis assigned the value one otherwise. The size of Boolean circuit D V A is givenby the number of nodes in V .


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The relationship to Local Search

Transition Graph associated to I ∈ IΠa node v(S) for every feasible solution S ∈ F(I )

a directed edge (v(S1), v(S2)) if S2 ∈ N (S1, I ) and c(S2) < c(S1)

The sinks of this graph are local optima

Potential Games ⇐⇒ Local Search Problems

feasible solutions ⇐⇒ states

neighborhood of S ⇐⇒ states who deviate from S only in oneplayer’s strategy

objective function ⇐⇒ potential function

Transition Graph ⇐⇒ Nash Dynamic Graph

local optima ⇐⇒ Nash equilibria


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Complexity of Nash for General CGs

Fabrikant, Papadimitriou and Talwar, STOC ’04

Computing a Nash equilibrium of a Congestion Game is PLS-complete

Proof.

We show a reduction from MAX-CUT with Flip-Neighborhood

MAX-CUT/Flip

Instance: G = (V ,E ) undirected with a weight w{i,j} for each{i , j} ∈ E

Feasible solution: partition (A,B) of V

Objective function: Max U(A,B) =∑

{i,j}|i∈A,j∈B w{i,j};

Neighborhood function: (A′,B ′) is a neighbor of (A,B) iff it canbe obtained from moving a single node from one side to the otherone and U(A,B) < U(A′,B ′)


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Proof. (cont.)

MAX-CUT/Flip as Party Affiliation Game

Nodes correspond to players. The strategies of a node are

A: choose the set A

B: choose the set B

The costs for these strategies are

A: sum of the weights of the incident edges to set A

B: sum of the weights of the incident edges to set B


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Proof. (cont.)

We represent the party affiliation game in the form of a quadraticthreshold congestion game

Threshold congestion games

E = Ein ∪ Eout , with Ein ∩ Eout = ∅ and Eout = {e1, e2, . . . , en}Every player i has two strategies

in: an arbitrary subset si ⊆ Ein. Each resource in Ein is contained in thestrategy of two playersout: a subset s′i = {ei} for a unique resource ei ∈ Eout with a fixed latency,the so-called threshold Ti , i.e., fei (x) = Ti


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Proof. (cont.)

MAX-CUT/Flip as a quadratic threshold congestion game

Nodes correspond to players

With each edges {i , j} we associate the resources e{i,j} ∈ Ein whichcan be used only by i and j

A resource e{i,j} ∈ Ein has the delay function

fe{i,j}(1) = 0 fe{i,j}(2) = w{i,j}

For each player i the threshold is Ti =12

∑j :{i,j}∈E w{i,j}

Players’ preferences in both games are identical


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Summary

Fabrikant, Papadimitriou and Talwar, STOC ’04

Network General

Symmetric P PLS-completeAsymmetric PLS-complete PLS-complete *


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Easy Cases

Under which conditions computing a pure Nash equilibrium inCongestion Games is tractable?

Question 1

Which combinatorial property of the players’ strategy spaces garanteesthe problem to be tractable? ... Matroid Congestion Games

Question 2

Can we relax the property of the equilibrium in order to guarantee theproblem to be tractable? ... Approximate Nash equilibria


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MatroidMatroid Congestion GamesComputational Complexity of Matroid Congestion Games

Matroid

Matroid

A matroid M is a pair (E , I ), where E is a finite set and I is a collectionof subsets of E , i.e, I ⊆ 2E (called the independent sets) with thefollowing properties:

∅ ∈ I .

(hereditary property). For each A′ ⊆ A ∈ E , if A ∈ I then A′ ∈ I

(exchange property). If A,B ∈ I and |A| > |B| then there existsa ∈ A \ B such that B ∪ {a} ∈ I

Element of I are called independent set

A maximal independent set is called basis of M, and its size rank(M)


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Property of Matroids

Proposition 1

Give two basis B1,B2, let r2 ∈ B2 \ B1, then there exists somer1 ∈ B1 \ B2 such that B1 ∪ {r2} \ {r1} is a basis

Consider G (B1∆B2) = (V ,E ) with V = (B1 \ B2) ∪ (B2 \ B1) andE = {{r1, r2}|r1 ∈ (B1 \ B2), r2 ∈ (B2 \ B1),B1 ∪ {r2} \ {r1} is a basis}

Proposition 2

There exists a perfect matching in the graph G (B1∆B2)


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Matroid Congestion Games

Matroid Congestion Games

We call a Congestion Game C = (N,E , (Σi )i∈N , (fe)e∈E , (ci )i∈N) aMatroid Congestion Game if for every i ∈ N, let Mi = (E , Ii ) withIi = {I ⊆ S |S ∈ Σi}

Mi is a matroid

Σi is the set of bases of Mi

rank(C) = maxi∈N rank(Mi )


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Example of Matroid Congestion Games

Singleton Congestion Games

rank(C) = 1; convergence in n2m moves

Spanning Tree Congestion Games; given a network G , the strategyset of each player is the set of spanning tree of G


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Polynomial convergence for Matroid Congestion Games

Achermann, Roglin and Vocking, FOCS ’06

In a Matroid Congestion Game, all sequences of best response moveshave polynomial length in the number of players and resources.

Proof.

Given S = (s1, s2, . . . , sn), let s′i be a (strictly) best response of i to

S w.r.t. fe , and let S ′ = (S−i , s′i )

Consider the bipartite graph G (s ′i∆si ) with a perfect matching PM

Since s ′i is a best response, it holds that, ∀{e′, e} ∈ PM ,

fe′(ne′(S′)) ≤ fe(ne(S))

and ∃{e′, e} ∈ PM

fe′(ne′(S′)) < fe(ne(S))


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Poof. (cont.)

The same property must hold if we replace fe with de for every e ∈ E

de is defined as follows

Assume that the list of all delays fe(i), with e ∈ E and 1 ≤ i ≤ |N|,is sorted in a non-decreasing orderLet de : N 7→ N a new delay function, where de(i) equals the rank offe(i) for 1 ≤ i ≤ |N|de(i) ≤ |N| · |E | for all resource e ∈ E and all 1 ≤ i ≤ |N|


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Proof. (cont.)

Given S = (s1, s2, . . . , sn), let s′i be a (strictly) best response of i to

S w.r.t. fe , and let S ′ = (S−i , s′i )

Consider the bipartite graph G (s ′i∆si ) with a perfect matching PM

Since s ′i is a best response, it holds that, ∀{e′, e} ∈ PM ,

de′(ne′(S′)) ≤ de(ne(S))

and ∃{e′, e} ∈ PM

de′(ne′(S′)) < de(ne(S))

It implies that s ′i is a (strictly) best response of i to S also w.r.t. de ,

c ′i (S) =∑e∈si

de(S)


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Proof. (cont.)

The decrease of cost of i w.r.t. de is equal to the decrease of thepotential

Φ′(S) =∑e∈E

ne(S)∑i=1

de(i)

Since de(i) ≤ |N| · |E |, for every e ∈ E and 1 ≤ i ≤ |N|, we have

Φ′(S) =∑e∈E

ne(S)∑i=1

de(i) ≤∑i∈N

∑e∈si

|N| · |E | ≤ |N|2 · |E | · rank(C)

Since Φ′(S) ≥ 0 and Φ decreases at each step by at least one unit,the theorem follows


2015 01 22 - rende - unical - angelo fanelli: an overview of congestion games

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