Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Nash Equilibria in Congestion Games:
Complexity and Convergence
Luca Moscardelli
Department of Science, University of Chieti-Pescara, Italy.
Rome, April 2011
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
1
Introduction and Preliminaries
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
2
The complexity of Pure Nash Equilibria
The Connection to Local Search
Results
3
The approximability of Pure Nash Equilibria
ǫ-Nash Equilibria
Results
4
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
1
Introduction and Preliminaries
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
2
The complexity of Pure Nash Equilibria
The Connection to Local Search
Results
3
The approximability of Pure Nash Equilibria
ǫ-Nash Equilibria
Results
4
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Congestion Games (CG) - 1
A CG is a tuple G = (N, E , (Σi )i ∈N , (fe )e∈E , (ci )i ∈N )
N, set of players let |N| = n
E , set of resources
Σi ⊆ 2E , set of strategies of player i
fe , delay function of resource e ∈ E
linear delays: fe (x) = ae x + be with ae , be ≥ 0
Pd
polynomial delays: fe (x) = j=1 ae,j x j with ae,j ≥ 0 for all j
ci , cost of player i
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Congestion Games (CG) - 2
For any state S = (s1 , s2 , . . . , sn ) ∈ (Σ1 ×Σ2 ×. . .×Σn )
ne (S) = |{i ∈ N | e ∈ si }|, congestion of resource e in S
fe (ne (S)), delay of resource e in S
P
ci (S) = e∈si fe (ne (S)), cost of player i in S
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Network Congestion Games
We are given a directed graph G
Each player is associated to a source and a destination node
of G
Resources are given by the edges of G
The strategies of a player are given by all the paths
connecting its source with its destination.
In the symmetric variant of the game, all players have the
same source and the same destination.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Nash Dynamics
Let S ⊕ si′ = (s1 , . . . , si −1 , si′ , si +1 , . . . , sn )
Improvement move
An improvement move of player i is a strategy si′ such that
ci (S ⊕ si′ ) < ci (S).
(Pure) Nash Equilibrium (NE)
A state in which no player has an improvement move.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Nash Dynamics
Let S ⊕ si′ = (s1 , . . . , si −1 , si′ , si +1 , . . . , sn )
Improvement move
An improvement move of player i is a strategy si′ such that
ci (S ⊕ si′ ) < ci (S).
(Pure) Nash Equilibrium (NE)
A state in which no player has an improvement move.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Best Response Dynamics
A very interesting dynamics is the one in which each move is a
best response
Best Response
A best response of player i in S is a strategy sib ∈ Σi such that
ci (S ⊕ sib ) ≤ ci (S ⊕ si′ ) for any other strategy si′ ∈ Σi , i.e., the
strategy yielding the minimum possible cost.
It is worth noticing that a Best Response in not necessarily an
Improvement Move (because a player’s best strategy could be
the one currently played)
A best response dynamics is any sequence of best responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Best Response Dynamics
A very interesting dynamics is the one in which each move is a
best response
Best Response
A best response of player i in S is a strategy sib ∈ Σi such that
ci (S ⊕ sib ) ≤ ci (S ⊕ si′ ) for any other strategy si′ ∈ Σi , i.e., the
strategy yielding the minimum possible cost.
It is worth noticing that a Best Response in not necessarily an
Improvement Move (because a player’s best strategy could be
the one currently played)
A best response dynamics is any sequence of best responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Questions
Existence: Does every CG posses a NE?
Complexity: Is it possible to efficiently compute a NE?
Convergence: Does any sequence of improvement moves
reaches a NE?
Time of Convergence: How many “steps” are needed to
reach a NE?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Existence and Convergence
Rosenthal ’73
For every CG any sequence of improvement moves is finite.
It follows by a potential function argument.
Φ(S) =
e (S)
X nX
fe (i )
e∈E i =1
Let si′ be and improvement move of player i in S, and S ′ = S ⊕ si′
the resulting state, then
ci (S) − ci (S ′ ) = Φ(S) − Φ(S ′ )
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
Existence and Convergence
Rosenthal ’73
For every CG any sequence of improvement moves is finite.
It follows by a potential function argument.
Φ(S) =
e (S)
X nX
fe (i )
e∈E i =1
Let si′ be and improvement move of player i in S, and S ′ = S ⊕ si′
the resulting state, then
ci (S) − ci (S ′ ) = Φ(S) − Φ(S ′ )
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
1
Introduction and Preliminaries
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
2
The complexity of Pure Nash Equilibria
The Connection to Local Search
Results
3
The approximability of Pure Nash Equilibria
ǫ-Nash Equilibria
Results
4
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
A preface: Complexity and Time of Convergence
Our interest in studying the complexity of computing NE is
twofold:
First of all, it is of course interesting to efficiently compute a
Nash Equilibrium
There are important implications on the time of convergence
of Nash dynamics:
if an equilibrium is not efficiently computable, then the time of
convergence to NE is not polynomial
of course, the converse does not hold
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
A preface: Complexity and Time of Convergence
Our interest in studying the complexity of computing NE is
twofold:
First of all, it is of course interesting to efficiently compute a
Nash Equilibrium
There are important implications on the time of convergence
of Nash dynamics:
if an equilibrium is not efficiently computable, then the time of
convergence to NE is not polynomial
of course, the converse does not hold
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Definition of a Local Search Problem
In general, a local search problem Π is given by
a set of instances IΠ
for every instance I ∈ IΠ , we have a finite set of feasible
solutions F(I) and an objective function c : F(I) → Z
for every feasible solution S ∈ F(I), a neighborhood
N (S, I) ⊆ F(I).
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
The Transition Graph
The transition graph for an instance IΠ contains
a node v (S) for every feasible solution S ∈ F(I)
a directed edge from a node v (S1 ) to a node v (S2 ) if S2 is in
the neighborhood of S1 and if the objective value c(S2 ) is
strictly better than c(S1 ).
The sinks of this graph are the local optima.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Local Search vs Nash Equilibria
Rosenthal’s potential function allows to interpret CGs as Local
Search Problems.
The set of feasible solutions ] The set of strategy profiles
(states)
The objective function ] Rosenthal’s potential function Φ
The neighborhood of a state S ] Those states that deviate
from S only in one player’s strategy.
Paths in the transition graph ] Sequences of improvement
moves
Sinks of the transition graph ] Nash equilibria of the game.
Notice that the classical Local Search paradigm corresponds to a
sequence of improvement moves leading to a NE.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Local Search vs Nash Equilibria
Rosenthal’s potential function allows to interpret CGs as Local
Search Problems.
The set of feasible solutions ] The set of strategy profiles
(states)
The objective function ] Rosenthal’s potential function Φ
The neighborhood of a state S ] Those states that deviate
from S only in one player’s strategy.
Paths in the transition graph ] Sequences of improvement
moves
Sinks of the transition graph ] Nash equilibria of the game.
Notice that the classical Local Search paradigm corresponds to a
sequence of improvement moves leading to a NE.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Local Search vs Nash Equilibria
Rosenthal’s potential function allows to interpret CGs as Local
Search Problems.
The set of feasible solutions ] The set of strategy profiles
(states)
The objective function ] Rosenthal’s potential function Φ
The neighborhood of a state S ] Those states that deviate
from S only in one player’s strategy.
Paths in the transition graph ] Sequences of improvement
moves
Sinks of the transition graph ] Nash equilibria of the game.
Notice that the classical Local Search paradigm corresponds to a
sequence of improvement moves leading to a NE.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
The Polynomially Local Search (PLS) class
A local search problem Π belongs to PLS if the following
polynomial time algorithms exist:
an algorithm A which computes for every instance I of Π an
initial feasible solution S ∈ F(I)
an algorithm B which computes for every instance I of Π and
every feasible solution S ∈ F(I) the objective value c(S)
an algorithm C which determines for every instance I of Π
and every feasible solution S ∈ F(I) whether S is locally
optimal or not and finds a better solution in the neighborhood
of S in the latter case
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
PLS-Reductions
In 1988, Johnosn, Papadimitriou and Yannakakis introduced the
notion of PLS-reduction.
A problem Π1 ∈ PLS is PLS-reducible to a problem Π2 ∈ PLS if
there are polynomial time computable functions g and h such that:
g maps instances I of Π1 to instances g (I ) of Π2
h maps pairs (S2 , I ) with S2 being a solution of g (I ) to
solutions S1 of I
for all instances I of Π1 , if S2 is a local optimum of instance
g (I ), then h(S2 , I ) is a local optimum of I
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
PLS-Completeness
A local search problem Π ∈ PLS is PLS-complete if every
problem in PLS is PLS-reducible to Π
It has been proved that many problems, including MaxCut1 ,
are PLS-Complete
A polynomial time algorithm for computing local optimal of
PLS-complete problems is “unlikely” to exist
Of course, it cannot just use the local search paradigm,
because there are instances of PLS-problems whose
transaction graph has nodes (solutions) exponentially far from
any sink (local optimum).
1
Schaffer and Yannakakis, 1991.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Negative Results - Sketch of proof 1
Fabrikant, Papadimitriou, Talwar, STOC ’04
Computing a NE in (symmetric) CGs is PLS-complete.
We first show a reduction from MaxCut to an asymmetric CG
In MaxCut we have an edge-weighted graph G = (V , E ) and
we want to partition its nodes in two sets VA and VB such
that the sum Wcut of the weights of the edges (u, v ) with
u ∈ VA and v ∈ VB is maximized
g function: For each node v ∈ V we have a player
For each edge e ∈ E we have two resources eA and eB
Each player v has 2 strategies, the A-strategy one containing
all eA for edges e incident to v , and the B-strategy containing
all eB for the same edges
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Negative Results - Sketch of proof 1
Fabrikant, Papadimitriou, Talwar, STOC ’04
Computing a NE in (symmetric) CGs is PLS-complete.
We first show a reduction from MaxCut to an asymmetric CG
In MaxCut we have an edge-weighted graph G = (V , E ) and
we want to partition its nodes in two sets VA and VB such
that the sum Wcut of the weights of the edges (u, v ) with
u ∈ VA and v ∈ VB is maximized
g function: For each node v ∈ V we have a player
For each edge e ∈ E we have two resources eA and eB
Each player v has 2 strategies, the A-strategy one containing
all eA for edges e incident to v , and the B-strategy containing
all eB for the same edges
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Negative Results - Sketch of proof 1
Fabrikant, Papadimitriou, Talwar, STOC ’04
Computing a NE in (symmetric) CGs is PLS-complete.
We first show a reduction from MaxCut to an asymmetric CG
In MaxCut we have an edge-weighted graph G = (V , E ) and
we want to partition its nodes in two sets VA and VB such
that the sum Wcut of the weights of the edges (u, v ) with
u ∈ VA and v ∈ VB is maximized
g function: For each node v ∈ V we have a player
For each edge e ∈ E we have two resources eA and eB
Each player v has 2 strategies, the A-strategy one containing
all eA for edges e incident to v , and the B-strategy containing
all eB for the same edges
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Negative Results - Sketch of proof 1
Fabrikant, Papadimitriou, Talwar, STOC ’04
Computing a NE in (symmetric) CGs is PLS-complete.
We first show a reduction from MaxCut to an asymmetric CG
In MaxCut we have an edge-weighted graph G = (V , E ) and
we want to partition its nodes in two sets VA and VB such
that the sum Wcut of the weights of the edges (u, v ) with
u ∈ VA and v ∈ VB is maximized
g function: For each node v ∈ V we have a player
For each edge e ∈ E we have two resources eA and eB
Each player v has 2 strategies, the A-strategy one containing
all eA for edges e incident to v , and the B-strategy containing
all eB for the same edges
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Sketch of proof - 2
Every resource associated to an edge e with weight we has the
following delay function
0
if x ≤ 1
feA (x) = feB (x) =
we if x ≥ 2
h function: Given a state S, the corresponding partition of
MaxCut is obtained by putting in A all players selecting their
A-strategy
P
Let W = e∈E we . It is easy to see that for any state S,
Wcut = W − Φ(S).
Nash equilibria of the CG are local optima of MaxCut
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Sketch of proof - 2
Every resource associated to an edge e with weight we has the
following delay function
0
if x ≤ 1
feA (x) = feB (x) =
we if x ≥ 2
h function: Given a state S, the corresponding partition of
MaxCut is obtained by putting in A all players selecting their
A-strategy
P
Let W = e∈E we . It is easy to see that for any state S,
Wcut = W − Φ(S).
Nash equilibria of the CG are local optima of MaxCut
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Sketch of proof - 2
Every resource associated to an edge e with weight we has the
following delay function
0
if x ≤ 1
feA (x) = feB (x) =
we if x ≥ 2
h function: Given a state S, the corresponding partition of
MaxCut is obtained by putting in A all players selecting their
A-strategy
P
Let W = e∈E we . It is easy to see that for any state S,
Wcut = W − Φ(S).
Nash equilibria of the CG are local optima of MaxCut
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Sketch of proof - 3
It remains to show a PLS-reduction from the general to the
symmetric case
We are given a general congestion game with strategy spaces
Σ1 , . . . , Σn ⊆ 2E
We extent E by additional resources e1 , . . . , en with delay 0 if
used by one player and delay M otherwise, where M is a large
number.
For i ∈ N, let Σ′i = {σ ∪ {ei }|σ ∈ Σi }.
The symmetric game has the common strategy space
Σ = Σ′1 ∪ . . . ∪ Σ′n
If M is chosen sufficiently large, any equilibrium of this game
has one player using a strategy from Σ′i .
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Sketch of proof - 3
It remains to show a PLS-reduction from the general to the
symmetric case
We are given a general congestion game with strategy spaces
Σ1 , . . . , Σn ⊆ 2E
We extent E by additional resources e1 , . . . , en with delay 0 if
used by one player and delay M otherwise, where M is a large
number.
For i ∈ N, let Σ′i = {σ ∪ {ei }|σ ∈ Σi }.
The symmetric game has the common strategy space
Σ = Σ′1 ∪ . . . ∪ Σ′n
If M is chosen sufficiently large, any equilibrium of this game
has one player using a strategy from Σ′i .
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Negative Results
The negative result for generic congestion games also holds for the
restricted class of Network CGs:
Ackermann, Röglin, Vöcking, FOCS ’06 and J. ACM ’08
PLS-completeness holds even for asymmetric Network CGs with
linear delays.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Positive Results
Fabrikant, Papadimitriou, Talwar, STOC ’04
It is possible to Compute a NE in symmetric Network CGs with
non-decreasing delay function in polynomial time.
Reduction to Min-cost flow problem
The result can be also extended to network CGs in which all
the players share the same source (Multicast games).
But, . . .
Ackermann, Röglin, Vöcking, FOCS ’06 and J. ACM ’08
There exist symmetric Network CG in which any sequence of
improvement moves is exponentially long in n.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Positive Results
Fabrikant, Papadimitriou, Talwar, STOC ’04
It is possible to Compute a NE in symmetric Network CGs with
non-decreasing delay function in polynomial time.
Reduction to Min-cost flow problem
The result can be also extended to network CGs in which all
the players share the same source (Multicast games).
But, . . .
Ackermann, Röglin, Vöcking, FOCS ’06 and J. ACM ’08
There exist symmetric Network CG in which any sequence of
improvement moves is exponentially long in n.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
Positive Results
By restricting the combinatorial structure of CGs, it is possible to
prove a polynomial convergence (and thus a polynomial algorithm
for computing a NE).
Ackermann, Röglin, Vöcking, FOCS ’06 and J. ACM ’08
In Matroid CGs, starting from any state S a NE is reached after a
polynomial number of best responses.
In Matroid CGs the strategy set of each player is the basis of
a matroid.
A special case of Matroid CGs are Load Balancing (also called
Singleton) Games
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
An interesting open problem
We know that
PLS-completeness holds
for (general) symmetric CGs (in the reduction the delays are
not linear)
for network asymmetric CGs with linear delays
A polynomial time algorithm exists for computing a NE in
network symmetric CGs
Question
Is it possible to compute in polynomial time a NE for symmetric
CGs with linear delays, or is such a problem PLS-complete?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The Connection to Local Search
Results
Open questions
An interesting open problem
We know that
PLS-completeness holds
for (general) symmetric CGs (in the reduction the delays are
not linear)
for network asymmetric CGs with linear delays
A polynomial time algorithm exists for computing a NE in
network symmetric CGs
Question
Is it possible to compute in polynomial time a NE for symmetric
CGs with linear delays, or is such a problem PLS-complete?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
1
Introduction and Preliminaries
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
2
The complexity of Pure Nash Equilibria
The Connection to Local Search
Results
3
The approximability of Pure Nash Equilibria
ǫ-Nash Equilibria
Results
4
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Motivation
Question
Starting from an arbitrary initial state, does the Nash dynamics
converge rapidly?
We have seen that fast convergence can be guaranteed only
for special classes of CGs, having a particular combinatorial
structure of strategy spaces.
Thus if we want a notion of Nash equilibrium that is selfishly and
efficiently realizable, the best we can hope for is some kind of
approximation.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Some Definitions
Let S = (s1 , s2 , . . . , sn ).
An ǫ-improvement move of player i is a strategy si′ such that
ci (S ⊕ si′ ) < (1 − ǫ)ci (S).
S is an ǫ-Nash equilibrium if no player has an ǫ-improvement
move
An ǫ-better response of player i in S is
an ǫ-improvement move, if it is available
the current strategy si , otherwise.
An ǫ-better response dynamics is any sequence of ǫ-better
responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Some Definitions
Let S = (s1 , s2 , . . . , sn ).
An ǫ-improvement move of player i is a strategy si′ such that
ci (S ⊕ si′ ) < (1 − ǫ)ci (S).
S is an ǫ-Nash equilibrium if no player has an ǫ-improvement
move
An ǫ-better response of player i in S is
an ǫ-improvement move, if it is available
the current strategy si , otherwise.
An ǫ-better response dynamics is any sequence of ǫ-better
responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Some Definitions
Let S = (s1 , s2 , . . . , sn ).
An ǫ-improvement move of player i is a strategy si′ such that
ci (S ⊕ si′ ) < (1 − ǫ)ci (S).
S is an ǫ-Nash equilibrium if no player has an ǫ-improvement
move
An ǫ-better response of player i in S is
an ǫ-improvement move, if it is available
the current strategy si , otherwise.
An ǫ-better response dynamics is any sequence of ǫ-better
responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Some Definitions
Let S = (s1 , s2 , . . . , sn ).
An ǫ-improvement move of player i is a strategy si′ such that
ci (S ⊕ si′ ) < (1 − ǫ)ci (S).
S is an ǫ-Nash equilibrium if no player has an ǫ-improvement
move
An ǫ-better response of player i in S is
an ǫ-improvement move, if it is available
the current strategy si , otherwise.
An ǫ-better response dynamics is any sequence of ǫ-better
responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Preliminaries - 1
Bounded Jump Condition
For any α > 1, we say that a resource e in a CG satisfies the
α-bounded jump condition if its delay function satisfies
fe (x + 1) ≤ αfe (x) for all x ≥ 1.
This is a weak condition: notice that also fe (x) = αx satisfies
the α-bounded jump condition; for instance, resources with
polynomial delays of degree d satisfy the 2d -bounded jump
condition.
It does not make sense to restrict to symmetric CGs without
assuming such a condition
It is still PLS-complete to find an exact Nash Equilibrium in
Symmetric CGs with 2-bounded jump resources.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Preliminaries - 1
Bounded Jump Condition
For any α > 1, we say that a resource e in a CG satisfies the
α-bounded jump condition if its delay function satisfies
fe (x + 1) ≤ αfe (x) for all x ≥ 1.
This is a weak condition: notice that also fe (x) = αx satisfies
the α-bounded jump condition; for instance, resources with
polynomial delays of degree d satisfy the 2d -bounded jump
condition.
It does not make sense to restrict to symmetric CGs without
assuming such a condition
It is still PLS-complete to find an exact Nash Equilibrium in
Symmetric CGs with 2-bounded jump resources.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Preliminaries - 2
T -Minimum Liveness Condition
Given any T ≥ n, a dynamics satisfies the T -Minimum Liveness
Condition if and only if each player performs at least a response
every T consecutive responses.
Without such a condition, one or more players could be
“locked out” for arbitrarily long and we could not expect to
bound the rate of convergence
If T = n, the players move in a round-robin fashion
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Positive Results
Chien and Sinclair, SODA 2007
In any symmetric CG whose resources satisfy the α-bounded jump
condition, any ǫ-better dynamics satisfying the T -Minimum
Liveness Condition converges
from anyminitial state to an ǫ-Nash
l
Φmax
equilibrium in at most n(α+1)
ǫ(1−ǫ) log Φmin T ǫ-better responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Proof Idea: a Potential Function Argument
If the player performing the better response is the one, say
player i , with the highest cost, we have ci (S) ≥ Φ(S)
n
If player i ’s better response is an ǫ-improvement move, c(i ),
and also Φ, decrease by more than ǫΦ(S)
n
After more than nǫ log ΦΦmax
steps it would be reached a state
min
′
′
S such that Φ(S ) < Φmin
Therefore, an ǫ-Nash equilibrium has to be reached in at most
that number of steps
By exploiting the symmetry and the α-bounded jump
resources, it is possible to show that actually high-cost player
move “frequently”
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Negative Results - 1
Skopalik and Vöcking, STOC 2008
Finding an ǫ-Nash equilibrium in a CG with increasing delay
functions is PLS-complete.
The reduction exploits asymmetric CGs with unbounded jumps.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Negative Results - 2
Nevertheless, the following result shows that the polynomial
convergence of symmetric games cannot be extended to the
asymmetric case, even considering bounded jump resources.
Skopalik and Vöcking, STOC 2008
For any ǫ < 12 there exists an α > 1 for which, given any n, there
are CGs with n players, with resources satisfying the α-bounded
jump condition, such that, starting from a state S, the length of
every ǫ-better response dynamics leading to an ǫ-Nash equilibrium
is exponential in n.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
ǫ-Nash Equilibria
Results
Open questions
Computing Approximate NE
Question
Is it possible to compute in polynomial time an ǫ-NE for any ǫ in
CG with resources satisfying the α-bounded jump condition?
It is not possible to obtain such an algorithm by letting the
dynamics evolve
A very recent (yet unpublished) result provides a (partial)
answer to this question:
Fanelli, Gravin, 2011
For any ǫ > 0, there exists a polynomial time algorithm computing
a 12 + ǫ -NE for asymmetric CGs with linear latencies.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
1
Introduction and Preliminaries
Definition of Congestion Games
Nash Dynamics
Existence of and Convergence to NE
2
The complexity of Pure Nash Equilibria
The Connection to Local Search
Results
3
The approximability of Pure Nash Equilibria
ǫ-Nash Equilibria
Results
4
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Why are we interested in NE?
CGs are one of the most studied class of non-cooperative
games in algorithmic game theory.
Of course, a very interesting property they have, given by the
Rosenthal’s potential function, is the existence of NE
Another strong point of CGs is that the quality of NE can be
proved to be high in many interesting settings
Question
What do we mean by quality of a Nash Equilibrium?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Why are we interested in NE?
CGs are one of the most studied class of non-cooperative
games in algorithmic game theory.
Of course, a very interesting property they have, given by the
Rosenthal’s potential function, is the existence of NE
Another strong point of CGs is that the quality of NE can be
proved to be high in many interesting settings
Question
What do we mean by quality of a Nash Equilibrium?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
The price of anarchy
A social value C (S) =
state S
Pn
i =1 ci (S)
can be associated to each
We are interested in minimizing S, but unfortunately a NE, in
general, does not provide the best possible social cost
Opt = minS C (S)
The price of anarchy is defined as the worst case ratio between
the cost of a Nash Equilibrium and the social optimum.
Price of Anarchy
PoA =
S
Luca Moscardelli
max
being a NE
C (S)
Opt
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
The price of anarchy
A social value C (S) =
state S
Pn
i =1 ci (S)
can be associated to each
We are interested in minimizing S, but unfortunately a NE, in
general, does not provide the best possible social cost
Opt = minS C (S)
The price of anarchy is defined as the worst case ratio between
the cost of a Nash Equilibrium and the social optimum.
Price of Anarchy
PoA =
S
Luca Moscardelli
max
being a NE
C (S)
Opt
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
The importance of complexity and convergence
For many classes of CGs, the PoA is “small”: for instance, if
the delays are linear it is 52 2
In the linear delay case, also the ǫ-NE have been proved3 to
have good performances, approaching 52 as ǫ tends to 0.
So, if the convergence to (ǫ-)NE is guaranteed to be fast, we
know that after a polynomial amount of time the players
reach a good solution.
Question
Do we really need Nash Equilibria in order to guarantee a good
social behavior of the non cooperative system?
2
Christodoulou and Koutsoupias, and also Awerbuch, Azar and Epstein,
STOC 2005
3
Christodoulou, Koutsoupias, and Spirakis, ESA 2009
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
The importance of complexity and convergence
For many classes of CGs, the PoA is “small”: for instance, if
the delays are linear it is 52 2
In the linear delay case, also the ǫ-NE have been proved3 to
have good performances, approaching 52 as ǫ tends to 0.
So, if the convergence to (ǫ-)NE is guaranteed to be fast, we
know that after a polynomial amount of time the players
reach a good solution.
Question
Do we really need Nash Equilibria in order to guarantee a good
social behavior of the non cooperative system?
2
Christodoulou and Koutsoupias, and also Awerbuch, Azar and Epstein,
STOC 2005
3
Christodoulou, Koutsoupias, and Spirakis, ESA 2009
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
A new idea
Question
Do we really need Nash Equilibria in order to guarantee a good
social behavior of the non cooperative system?
Maybe the polynomial convergence time to (ǫ-)NE is a too
strong requirement.
In an highly decentralized and self-evolving system, what we
want is that after a “small” amount of time, the selfish
(S)
is small
players are able to reach a solution S such that COpt
(i.e. constant), or at least close to the PoA of the game.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Positive Results
Awerbuch, Azar, Epstein, Mirrokni, Skopalik, EC 2008
In any CG whose resources satisfy the α-bounded jump condition,
any ǫ-better dynamics satisfying the T -Minimum Liveness
Condition converges from any initial state
to a state
S such that
C (S)
Φmax
nα
Opt ≤ PoA(1 + O(ǫ)) in at most O ǫ2 log Φmin T ǫ-better
responses.
The exploited technique again relies on a potential function
argument.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Negative Results
Why the authors considered ǫ-better dynamics and not best
response dynamics?
Awerbuch, Azar, Epstein, Mirrokni, Skopalik, EC 2008
There exists a CG with linear delays and an initial state S 0 such
that a best response dynamics starting from S 0 is such that for a
number of best responses
exponential
in n the cost of the reached
√
n
states is always Ω log n Opt .
Nevertheless, in the following we will prove that, for best response
dynamics, the situation is actually not so bad . . .
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Negative Results
Why the authors considered ǫ-better dynamics and not best
response dynamics?
Awerbuch, Azar, Epstein, Mirrokni, Skopalik, EC 2008
There exists a CG with linear delays and an initial state S 0 such
that a best response dynamics starting from S 0 is such that for a
number of best responses
exponential
in n the cost of the reached
√
n
states is always Ω log n Opt .
Nevertheless, in the following we will prove that, for best response
dynamics, the situation is actually not so bad . . .
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Preliminaries - 1
Following the framework introduced by Mirrokni and Vetta in
Approx-Random 2004, some basic Nash Dynamics can be defined.
round: a best-response dynamics in which each player plays
exactly once.
covering: a best-response dynamics in which each player
plays at least once.
β-bounded covering: a best-response dynamics in which
each player plays at least once and at most β times.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Preliminaries - 2
β-Fairness Condition
Given any β ≥ 1, a dynamics satisfies the β-Fairness Condition if it
can be decomposed into a sequence of sub-dynamics being
minimal β-bounded coverings.
β is a sort of (un)fairness index.
If β is constant, it means that every player plays at most a
constant number of times in each sub-dynamics and therefore
the dynamics can be considered fair
Such a condition is somewhat related to the T -Minimum
Liveness Condition
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Positive Results
Fanelli, Flammini, Moscardelli, ICALP 2008 and TALG 2011
In any CG with polynomial delays, any fair best response dynamics
(satisfying the β-Fairness Condition with β = O(1)) converges
(S)
= O(PoA) in at
from any initial state to a state S such that COpt
most O (n log log n) best responses.
The exploited technique does not rely on a potential function
argument
The result can be extended to weighted CGs with polynomial
delays (not being potential games)
The result is tight: there exist CGs in which Ω (n log log n)
best responses are needed in order to reach a good solution
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Positive Results
Fanelli, Flammini, Moscardelli, ICALP 2008 and TALG 2011
In any CG with polynomial delays, any fair best response dynamics
(satisfying the β-Fairness Condition with β = O(1)) converges
(S)
= O(PoA) in at
from any initial state to a state S such that COpt
most O (n log log n) best responses.
The exploited technique does not rely on a potential function
argument
The result can be extended to weighted CGs with polynomial
delays (not being potential games)
The result is tight: there exist CGs in which Ω (n log log n)
best responses are needed in order to reach a good solution
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 1
Restricted case for this sketch: linear delays and β = 1
W.l.o.g., we assume that for every e ∈ E , fe (x) = x.
Let hR1 , . . . , Rk i be the rounds (1-bounded coverings) forming
the considered best response dynamics
Each R = S 0 , S 1 , S 2 , . . . , S n
The “immediate” cost ci (S i ) of player i can be upper
bounded by ci (S i −1 ⊕ si∗ )
P
ci (S i −1 ⊕ si∗ ) ≤ e∈s ∗ ne (S i −1 ) + 1
i
We define
Γ(R) =
n X
X
i =1 e∈si∗
ne (S i −1 ) + 1
It can be interpreted as an upper bound to
Luca Moscardelli
Pn
i =1 ci (S
i)
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 1
Restricted case for this sketch: linear delays and β = 1
W.l.o.g., we assume that for every e ∈ E , fe (x) = x.
Let hR1 , . . . , Rk i be the rounds (1-bounded coverings) forming
the considered best response dynamics
Each R = S 0 , S 1 , S 2 , . . . , S n
The “immediate” cost ci (S i ) of player i can be upper
bounded by ci (S i −1 ⊕ si∗ )
P
ci (S i −1 ⊕ si∗ ) ≤ e∈s ∗ ne (S i −1 ) + 1
i
We define
Γ(R) =
n X
X
i =1 e∈si∗
ne (S i −1 ) + 1
It can be interpreted as an upper bound to
Luca Moscardelli
Pn
i =1 ci (S
i)
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 1
Restricted case for this sketch: linear delays and β = 1
W.l.o.g., we assume that for every e ∈ E , fe (x) = x.
Let hR1 , . . . , Rk i be the rounds (1-bounded coverings) forming
the considered best response dynamics
Each R = S 0 , S 1 , S 2 , . . . , S n
The “immediate” cost ci (S i ) of player i can be upper
bounded by ci (S i −1 ⊕ si∗ )
P
ci (S i −1 ⊕ si∗ ) ≤ e∈s ∗ ne (S i −1 ) + 1
i
We define
Γ(R) =
n X
X
i =1 e∈si∗
ne (S i −1 ) + 1
It can be interpreted as an upper bound to
Luca Moscardelli
Pn
i =1 ci (S
i)
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 2
ne (S i −1 ) + 1
P P
0
H(R) = ni=1
e∈s ∗ ne (S )
Γ(R) =
Pn
i =1
P
e∈si∗
i
H(R) represents the sum over all the moves in the covering
walk R of the delay that the moving player would experience
in the initial state S 0 on its optimal strategy si∗ .
Clearly, H(Rj ) and Γ(Rj ) are correlated, and if players never
move on resources used in E ∗ , H(Rj ) + Opt would be an
upper bound to Γ(Rj ) .
Actually, since players during the round can load servers used
in S ∗ , some technical issues have to be fixed, and it holds that
Γ(Rj ) ≤ 2H(Rj ) + 6Opt
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 2
ne (S i −1 ) + 1
P P
0
H(R) = ni=1
e∈s ∗ ne (S )
Γ(R) =
Pn
i =1
P
e∈si∗
i
H(R) represents the sum over all the moves in the covering
walk R of the delay that the moving player would experience
in the initial state S 0 on its optimal strategy si∗ .
Clearly, H(Rj ) and Γ(Rj ) are correlated, and if players never
move on resources used in E ∗ , H(Rj ) + Opt would be an
upper bound to Γ(Rj ) .
Actually, since players during the round can load servers used
in S ∗ , some technical issues have to be fixed, and it holds that
Γ(Rj ) ≤ 2H(Rj ) + 6Opt
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 2
ne (S i −1 ) + 1
P P
0
H(R) = ni=1
e∈s ∗ ne (S )
Γ(R) =
Pn
i =1
P
e∈si∗
i
H(R) represents the sum over all the moves in the covering
walk R of the delay that the moving player would experience
in the initial state S 0 on its optimal strategy si∗ .
Clearly, H(Rj ) and Γ(Rj ) are correlated, and if players never
move on resources used in E ∗ , H(Rj ) + Opt would be an
upper bound to Γ(Rj ) .
Actually, since players during the round can load servers used
in S ∗ , some technical issues have to be fixed, and it holds that
Γ(Rj ) ≤ 2H(Rj ) + 6Opt
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 3
Now we know that
Γ(Rj )
Opt
≤2
H(Rj )
Opt
+6
The key point in the proof
q is the relation between H(Rj+1 )
H(Rj+1 )
Γ(R )
and Γ(Rj ): Opt ≤ 2 Optj
It is worth noting that by combining the above 2 inequalities,
both H and Γ fast decrease between two consecutive
coverings; by repeating the argument
qfor allthe consecutive
coverings, we obtain
Γ(Rk )
Opt
=O
2k−1
Γ(R1 )
Opt
Since it also holds that
The final cost C (S) is at most 2Γ(Rk )
Γ(R1 ) = O(n)Opt
(S)
we finally obtain that COpt
= O(1) after k = log log n
coverings (and thus n log log n best responses)
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 3
Now we know that
Γ(Rj )
Opt
≤2
H(Rj )
Opt
+6
The key point in the proof
q is the relation between H(Rj+1 )
H(Rj+1 )
Γ(R )
and Γ(Rj ): Opt ≤ 2 Optj
It is worth noting that by combining the above 2 inequalities,
both H and Γ fast decrease between two consecutive
coverings; by repeating the argument
qfor allthe consecutive
coverings, we obtain
Γ(Rk )
Opt
=O
2k−1
Γ(R1 )
Opt
Since it also holds that
The final cost C (S) is at most 2Γ(Rk )
Γ(R1 ) = O(n)Opt
(S)
we finally obtain that COpt
= O(1) after k = log log n
coverings (and thus n log log n best responses)
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Sketch of proof - 3
Now we know that
Γ(Rj )
Opt
≤2
H(Rj )
Opt
+6
The key point in the proof
q is the relation between H(Rj+1 )
H(Rj+1 )
Γ(R )
and Γ(Rj ): Opt ≤ 2 Optj
It is worth noting that by combining the above 2 inequalities,
both H and Γ fast decrease between two consecutive
coverings; by repeating the argument
qfor allthe consecutive
coverings, we obtain
Γ(Rk )
Opt
=O
2k−1
Γ(R1 )
Opt
Since it also holds that
The final cost C (S) is at most 2Γ(Rk )
Γ(R1 ) = O(n)Opt
(S)
we finally obtain that COpt
= O(1) after k = log log n
coverings (and thus n log log n best responses)
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
An interesting open problem
How the last result can hold given the negative result of Awerbuch
et al.?
Question
Witch is the maximum value of β for which a fast convergence to
“good” solutions is guaranteed?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
Fanelli, Flammini, Moscardelli, ICALP 2008 and TALG 2011
In any CG with polynomial delays, any fair best response dynamics
(satisfying the β-Fairness Condition with β = O(1)) converges
(S)
from any initial state to a state S such that COpt
= O(PoA) in at
most O (n log log n) best responses.
Awerbuch, Azar, Epstein, Mirrokni, Skopalik, EC 2008
There exists a CG with linear delays and an initial state S 0 such
that a best response dynamics starting from S 0 is such that for a
number of best responses
exponential
in n the cost of the reached
√
n
states is always Ω log n Opt .
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
An interesting open problem
How the last result can hold given the negative result of Awerbuch
et al.?
Question
Witch is the maximum value of β for which a fast convergence to
“good” solutions is guaranteed?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
A new result - 1
Fanelli, Moscardelli, Skopalik, 2011
In any CG with polynomial delays, any best response dynamics
satisfying the β-Fairness Condition converges from any initial state
(S)
to a state S such that COpt
= O(β · PoA) in at most βn log log n
best responses.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
A new result - 2
Fanelli, Moscardelli, Skopalik, 2011
For any ǫ > 0, there exist CGs with linear delays in which, for
infinite values of β, the response dynamics satisfying the β-Fairness
Condition is such that for a number of best responses exponential
in n the cost of the reached states is always Ω(β 1−ǫ · PoA).
Fair dynamics (β = O(1)) is a necessary condition in order to
fast obtain good solutions.
What about symmetric games?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
A new result - 2
Fanelli, Moscardelli, Skopalik, 2011
For any ǫ > 0, there exist CGs with linear delays in which, for
infinite values of β, the response dynamics satisfying the β-Fairness
Condition is such that for a number of best responses exponential
in n the cost of the reached states is always Ω(β 1−ǫ · PoA).
Fair dynamics (β = O(1)) is a necessary condition in order to
fast obtain good solutions.
What about symmetric games?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
A new result - 2
Fanelli, Moscardelli, Skopalik, 2011
For any ǫ > 0, there exist CGs with linear delays in which, for
infinite values of β, the response dynamics satisfying the β-Fairness
Condition is such that for a number of best responses exponential
in n the cost of the reached states is always Ω(β 1−ǫ · PoA).
Fair dynamics (β = O(1)) is a necessary condition in order to
fast obtain good solutions.
What about symmetric games?
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
A new result - 3
Fanelli, Moscardelli, Skopalik, 2011
In any symmetric CG with polynomial delays, any best response
dynamics satisfying the T -Minimum Liveness Condition converges
(S)
from any initial state to a state S such that COpt
= O(PoA) in at
most T log log n best responses.
The technique exploited for the asymmetric case (not relying
on a potential function argument) is joined to a potential
function argument.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
A new result - 3
Fanelli, Moscardelli, Skopalik, 2011
In any symmetric CG with polynomial delays, any best response
dynamics satisfying the T -Minimum Liveness Condition converges
(S)
from any initial state to a state S such that COpt
= O(PoA) in at
most T log log n best responses.
The technique exploited for the asymmetric case (not relying
on a potential function argument) is joined to a potential
function argument.
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
We have not spoken about...
Probabilistic results
Goemans, Mirrokni and Vetta (FOCS 2005) studied the
expected approximation ratio after a polynomial number of
best responses
Concurrent moves
Fotakis, Kaporis, Spirakis. Atomic congestion games: fast,
myopic and concurrent (SAGT 2008)
Berenbrink, Friedetzky, L. A. Goldberg, P. Goldberg, Hu, and
Martin. Distributed selfish load balancing. (SODA 2006)
Non-increasing delay functions (Shapley cost sharing games)
Syrgkanis, PLS-completeness (WINE 2010)
Albers and Lenzner, a logarithmic approximated Nash is given
by optimal solutions (WINE 2010)
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
We have not spoken about...
Probabilistic results
Goemans, Mirrokni and Vetta (FOCS 2005) studied the
expected approximation ratio after a polynomial number of
best responses
Concurrent moves
Fotakis, Kaporis, Spirakis. Atomic congestion games: fast,
myopic and concurrent (SAGT 2008)
Berenbrink, Friedetzky, L. A. Goldberg, P. Goldberg, Hu, and
Martin. Distributed selfish load balancing. (SODA 2006)
Non-increasing delay functions (Shapley cost sharing games)
Syrgkanis, PLS-completeness (WINE 2010)
Albers and Lenzner, a logarithmic approximated Nash is given
by optimal solutions (WINE 2010)
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
We have not spoken about...
Probabilistic results
Goemans, Mirrokni and Vetta (FOCS 2005) studied the
expected approximation ratio after a polynomial number of
best responses
Concurrent moves
Fotakis, Kaporis, Spirakis. Atomic congestion games: fast,
myopic and concurrent (SAGT 2008)
Berenbrink, Friedetzky, L. A. Goldberg, P. Goldberg, Hu, and
Martin. Distributed selfish load balancing. (SODA 2006)
Non-increasing delay functions (Shapley cost sharing games)
Syrgkanis, PLS-completeness (WINE 2010)
Albers and Lenzner, a logarithmic approximated Nash is given
by optimal solutions (WINE 2010)
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence
Introduction and Preliminaries
The complexity of Pure Nash Equilibria
The approximability of Pure Nash Equilibria
Convergence to Nearly Optimal Solution
The quality of Nash Equilibria
ǫ-better response dynamics
Best response dynamics
Open questions
What is missing in the literature (and thus open)
Besides the interesting main open problems stated during this talk,
the max social function has not been studied in the literature
with respect to the convergence to good approximated
solutions.
Other kind of dynamics (for instance coalitional responses)
could be considered
...
Luca Moscardelli
NE in Congestion Games: Complexity and Convergence