Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Selfish Routing and Congestion Games:
Towards a Game Based Analysis of the Internet
Berthold Vöcking
Department of Computer Science
RWTH Aachen
Delis/Evergrow School, Santorini 2004
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Selfish Routing and the Price of Anarchy
Wardrop’s Traffic Model
Discrete and Atomic Models
Congestion and Crowding Games
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
On the Evolution of Selfish Routing
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Summary
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Definition of routing problem
Routing = multicommodity flow + latency functions
I
directed (multi)graph G = (V , E )
I
source-sink pairs si , ti
P
traffic rates ri ≥ 0 s.t.
ri = 1
latency functions `e (·), e ∈ E
I
I
I
I
flow on edge e is denoted by fe
latency on edge e is `e (fe )
global objective
minimize average latency C =
X
fe · `e (fe )
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Definition of routing problem
Routing = multicommodity flow + latency functions
I
directed (multi)graph G = (V , E )
I
source-sink pairs si , ti
P
traffic rates ri ≥ 0 s.t.
ri = 1
latency functions `e (·), e ∈ E
I
I
I
I
flow on edge e is denoted by fe
latency on edge e is `e (fe )
global objective
minimize average latency C =
X
fe · `e (fe )
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Definition of routing problem
Routing = multicommodity flow + latency functions
I
directed (multi)graph G = (V , E )
I
source-sink pairs si , ti
P
traffic rates ri ≥ 0 s.t.
ri = 1
latency functions `e (·), e ∈ E
I
I
I
I
flow on edge e is denoted by fe
latency on edge e is `e (fe )
global objective
minimize average latency C =
X
fe · `e (fe )
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Definition of routing problem
Routing = multicommodity flow + latency functions
I
directed (multi)graph G = (V , E )
I
source-sink pairs si , ti
P
traffic rates ri ≥ 0 s.t.
ri = 1
latency functions `e (·), e ∈ E
I
I
I
I
flow on edge e is denoted by fe
latency on edge e is `e (fe )
global objective
minimize average latency C =
X
fe · `e (fe )
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Definition of routing problem
Routing = multicommodity flow + latency functions
I
directed (multi)graph G = (V , E )
I
source-sink pairs si , ti
P
traffic rates ri ≥ 0 s.t.
ri = 1
latency functions `e (·), e ∈ E
I
I
I
I
flow on edge e is denoted by fe
latency on edge e is `e (fe )
global objective
minimize average latency C =
X
fe · `e (fe )
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Pigou’s Example
Assume r = r1 = 1.
Then optimal solution is
f1 = 1/2
x
s1
t1
1
f2 = 1/2
C = f1 · `1 (f1 ) + f2 · `2 (f2 ) = 43 .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Pigou’s Example
Assume r = r1 = 1.
Then optimal solution is
But flow on edge 2 is envious.
Moves onto edge 1.
f1 = 1/2
f1 = 1
x
s1
x
t1
1
f2 = 1/2
C = f1 · `1 (f1 ) + f2 · `2 (f2 ) = 43 .
Berthold Vöcking
s1
t1
1
Now C = 1.
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Braess paradox
Optimal Flow:
x
1
s
t
1
x
C = 32 .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Braess paradox
Optimal Flow:
x
Build a new road.
x
1
s
t
1
x
1
0
s
1
t
x
C = 32 .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Braess paradox
Build a new road.
Optimal Flow:
x
1
s
x
t
1
x
C = 32 .
1
0
s
1
t
x
Now C = 2.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Selfish flow - intuition
I
selfish flow = routes of many noncooperative, selfish agents
I
flow is at Nash equilibrium if no agent can improve its latency
by changing its path
Motivating Examples:
I
cars in a highway system
I
packets in a network at steady state
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Selfish flow - intuition
I
selfish flow = routes of many noncooperative, selfish agents
I
flow is at Nash equilibrium if no agent can improve its latency
by changing its path
Motivating Examples:
I
cars in a highway system
I
packets in a network at steady state
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Selfish flow - formal definition of equilibrium
I
I
I
I
For commodity i ∈ {1, . . . , k}, let Pi denote the set of paths
from si to ti .
P
For a path P and a flow f , define `P (f ) = e∈P `e (f ).
For a path P, let fP denote the flow along P.
A flow f is at Nash equilibrium if for all i ∈ {1, . . . , k} and
every P1 , P2 ∈ Pi and δ ∈ (0, fP1 ], we have `P1 (f ) ≤ `P2 (f 0 ),
where

 fP − δ if P = P1
0
fP =
f + δ if P = P2
 P
fP
else
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Selfish flow - formal definition of equilibrium
I
I
I
I
For commodity i ∈ {1, . . . , k}, let Pi denote the set of paths
from si to ti .
P
For a path P and a flow f , define `P (f ) = e∈P `e (f ).
For a path P, let fP denote the flow along P.
A flow f is at Nash equilibrium if for all i ∈ {1, . . . , k} and
every P1 , P2 ∈ Pi and δ ∈ (0, fP1 ], we have `P1 (f ) ≤ `P2 (f 0 ),
where

 fP − δ if P = P1
0
fP =
f + δ if P = P2
 P
fP
else
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Selfish flow - formal definition of equilibrium
I
I
I
I
For commodity i ∈ {1, . . . , k}, let Pi denote the set of paths
from si to ti .
P
For a path P and a flow f , define `P (f ) = e∈P `e (f ).
For a path P, let fP denote the flow along P.
A flow f is at Nash equilibrium if for all i ∈ {1, . . . , k} and
every P1 , P2 ∈ Pi and δ ∈ (0, fP1 ], we have `P1 (f ) ≤ `P2 (f 0 ),
where

 fP − δ if P = P1
0
fP =
f + δ if P = P2
 P
fP
else
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Selfish flow - formal definition of equilibrium
I
I
I
I
For commodity i ∈ {1, . . . , k}, let Pi denote the set of paths
from si to ti .
P
For a path P and a flow f , define `P (f ) = e∈P `e (f ).
For a path P, let fP denote the flow along P.
A flow f is at Nash equilibrium if for all i ∈ {1, . . . , k} and
every P1 , P2 ∈ Pi and δ ∈ (0, fP1 ], we have `P1 (f ) ≤ `P2 (f 0 ),
where

 fP − δ if P = P1
0
fP =
f + δ if P = P2
 P
fP
else
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Price of anarchy
What is the ratio between Nash cost and optimal cost?
Theorem 1 [Roughgarden & Tardos, 2000]
For linear latency functions the cost of a Nash flow is at most
times that of the optimal cost.
4
3
Theorem 2 [Roughgarden & Tardos, 2000]
Suppose
I
the latency functions are continuous and non-decreasing, and
I
the cost functions Ce (f ) = f · `e (f ) are convex.
Then the cost of a Nash flow with rate r is at most the optimal
cost for rates 2r .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Price of anarchy
What is the ratio between Nash cost and optimal cost?
Theorem 1 [Roughgarden & Tardos, 2000]
For linear latency functions the cost of a Nash flow is at most
times that of the optimal cost.
4
3
Theorem 2 [Roughgarden & Tardos, 2000]
Suppose
I
the latency functions are continuous and non-decreasing, and
I
the cost functions Ce (f ) = f · `e (f ) are convex.
Then the cost of a Nash flow with rate r is at most the optimal
cost for rates 2r .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Price of anarchy
What is the ratio between Nash cost and optimal cost?
Theorem 1 [Roughgarden & Tardos, 2000]
For linear latency functions the cost of a Nash flow is at most
times that of the optimal cost.
4
3
Theorem 2 [Roughgarden & Tardos, 2000]
Suppose
I
the latency functions are continuous and non-decreasing, and
I
the cost functions Ce (f ) = f · `e (f ) are convex.
Then the cost of a Nash flow with rate r is at most the optimal
cost for rates 2r .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Characterization of optimal flow
I
Flow problem can be described by convex program with
convex objective function:
P
Minimize e∈E fe · `e (fe )
subject to:
∀i
P
P∈Pi fP
∀e fe =
∀P
I
P
= ri
P:e∈P fP
fP ≥ 0
Thus, local optimum = global optimum.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Characterization of optimal flow
I
Flow problem can be described by convex program with
convex objective function:
P
Minimize e∈E fe · `e (fe )
subject to:
∀i
P
P∈Pi fP
∀e fe =
∀P
I
P
= ri
P:e∈P fP
fP ≥ 0
Thus, local optimum = global optimum.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Characterization of optimal flow
Lemma: A flow f is optimal if and only if for each source-sink pair
i and paths P1 , P2 ∈ Pi with fP1 > 0,
X
X
Ce0 (fe ) ≤
Ce0 (fe ) .
e∈P1
Berthold Vöcking
e∈P2
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Characterization of Nash flow
Lemma: [Wardrop’s Principle]
A flow f is in Nash equilibrium if and only if for each source-sink
pair i and paths P1 , P2 ∈ Pi with fP1 > 0,
X
X
`e (fe ) ≤
`e (fe ) .
e∈P1
Berthold Vöcking
e∈P2
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Consequences for Linear Latency Functions
Suppose `e (x) = ae x + be .
Characterization of Nash flow f :
X
X
ae fe + b e ≤
ae fe + b e
e∈P1
e∈P2
Ce (x) = ae x 2 + be x, Ce0 (x) = 2ae x + be .
Characterization of optimal flow f ∗ :
X
X
2ae fe∗ + be ≤
2ae fe∗ + be
e∈P1
e∈P2
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Consequences for Linear Latency Functions
Suppose `e (x) = ae x + be .
Characterization of Nash flow f :
X
X
ae fe + b e ≤
ae fe + b e
e∈P1
e∈P2
Ce (x) = ae x 2 + be x, Ce0 (x) = 2ae x + be .
Characterization of optimal flow f ∗ :
X
X
2ae fe∗ + be ≤
2ae fe∗ + be
e∈P1
e∈P2
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Consequences for Linear Latency Functions
Suppose `e (x) = ae x + be .
Characterization of Nash flow f :
X
X
ae fe + b e ≤
ae fe + b e
e∈P1
e∈P2
Ce (x) = ae x 2 + be x, Ce0 (x) = 2ae x + be .
Characterization of optimal flow f ∗ :
X
X
2ae fe∗ + be ≤
2ae fe∗ + be
e∈P1
e∈P2
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Consequences for Linear Latency Functions
Suppose `e (x) = ae x + be .
Characterization of Nash flow f :
X
X
ae fe + b e ≤
ae fe + b e
e∈P1
e∈P2
Ce (x) = ae x 2 + be x, Ce0 (x) = 2ae x + be .
Characterization of optimal flow f ∗ :
X
X
2ae fe∗ + be ≤
2ae fe∗ + be
e∈P1
e∈P2
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Corollary 1: Suppose be = 0, for every e ∈ E . Then
Nash flow f = optimal flow f ∗ .
Corollary 2: The Nash flow f /2 is optimal for rates r i /2.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Corollary 1: Suppose be = 0, for every e ∈ E . Then
Nash flow f = optimal flow f ∗ .
Corollary 2: The Nash flow f /2 is optimal for rates r i /2.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Applying Corollary 2 gives
f
C (f ) ≥ C
2
X fe 2
fe
=
ae
+ be
2
2
e
X
1
≥
ae fe2 + be fe
4 e
∗
=
1
C (f )
4
Thus C (f ) ≤ 4C (f ∗ ).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Applying Corollary 2 gives
f
C (f ) ≥ C
2
X fe 2
fe
=
ae
+ be
2
2
e
X
1
≥
ae fe2 + be fe
4 e
∗
=
1
C (f )
4
Thus C (f ) ≤ 4C (f ∗ ).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Applying Corollary 2 gives
f
C (f ) ≥ C
2
X fe 2
fe
=
ae
+ be
2
2
e
X
1
≥
ae fe2 + be fe
4 e
∗
=
1
C (f )
4
Thus C (f ) ≤ 4C (f ∗ ).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Applying Corollary 2 gives
f
C (f ) ≥ C
2
X fe 2
fe
=
ae
+ be
2
2
e
X
1
≥
ae fe2 + be fe
4 e
∗
=
1
C (f )
4
Thus C (f ) ≤ 4C (f ∗ ).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Applying Corollary 2 gives
f
C (f ) ≥ C
2
X fe 2
fe
=
ae
+ be
2
2
e
X
1
≥
ae fe2 + be fe
4 e
∗
=
1
C (f )
4
Thus C (f ) ≤ 4C (f ∗ ).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Makespan scheduling
I
m servers with bandwidths b1 ≥ b2 ≥ · · · ≥ bm
I
n tasks with weights w1 , . . . , wn should be assigned to servers
I
decision variables xij = 1 if i → j, else 0
I
cost of server j
=
Cj
X wi · x j
i
bj
i ∈[n]
I
global objective: minimize makespan
C
Berthold Vöcking
= max Cj
j∈[m]
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Makespan scheduling
I
m servers with bandwidths b1 ≥ b2 ≥ · · · ≥ bm
I
n tasks with weights w1 , . . . , wn should be assigned to servers
I
decision variables xij = 1 if i → j, else 0
I
cost of server j
=
Cj
X wi · x j
i
bj
i ∈[n]
I
global objective: minimize makespan
C
Berthold Vöcking
= max Cj
j∈[m]
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Makespan scheduling
I
m servers with bandwidths b1 ≥ b2 ≥ · · · ≥ bm
I
n tasks with weights w1 , . . . , wn should be assigned to servers
I
decision variables xij = 1 if i → j, else 0
I
cost of server j
=
Cj
X wi · x j
i
bj
i ∈[n]
I
global objective: minimize makespan
C
Berthold Vöcking
= max Cj
j∈[m]
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Makespan scheduling
I
m servers with bandwidths b1 ≥ b2 ≥ · · · ≥ bm
I
n tasks with weights w1 , . . . , wn should be assigned to servers
I
decision variables xij = 1 if i → j, else 0
I
cost of server j
=
Cj
X wi · x j
i
bj
i ∈[n]
I
global objective: minimize makespan
C
Berthold Vöcking
= max Cj
j∈[m]
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Makespan scheduling
I
m servers with bandwidths b1 ≥ b2 ≥ · · · ≥ bm
I
n tasks with weights w1 , . . . , wn should be assigned to servers
I
decision variables xij = 1 if i → j, else 0
I
cost of server j
=
Cj
X wi · x j
i
bj
i ∈[n]
I
global objective: minimize makespan
C
Berthold Vöcking
= max Cj
j∈[m]
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
A routing game based on makespan scheduling
[Koutsoupias & Papadimitriou, 1999]
I every task is managed by a selfish agent
I
mixed strategy: probability distribution, p ij
I
expected cost of server j
E [Cj ] =
h
i
= Pr xij = 1
X wi · p j
i
bj
i ∈[n]
I
I
every agent aims to minimize its expected cost
expected cost of task i on server j
P
h
i
wi + k6=i wk · pkj
j
j
ci = E Cj | x i = 1 =
bj
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
A routing game based on makespan scheduling
[Koutsoupias & Papadimitriou, 1999]
I every task is managed by a selfish agent
I
mixed strategy: probability distribution, p ij
I
expected cost of server j
E [Cj ] =
h
i
= Pr xij = 1
X wi · p j
i
bj
i ∈[n]
I
I
every agent aims to minimize its expected cost
expected cost of task i on server j
P
h
i
wi + k6=i wk · pkj
j
j
ci = E Cj | x i = 1 =
bj
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
A routing game based on makespan scheduling
[Koutsoupias & Papadimitriou, 1999]
I every task is managed by a selfish agent
I
mixed strategy: probability distribution, p ij
I
expected cost of server j
E [Cj ] =
h
i
= Pr xij = 1
X wi · p j
i
bj
i ∈[n]
I
I
every agent aims to minimize its expected cost
expected cost of task i on server j
P
h
i
wi + k6=i wk · pkj
j
j
ci = E Cj | x i = 1 =
bj
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
A routing game based on makespan scheduling
[Koutsoupias & Papadimitriou, 1999]
I every task is managed by a selfish agent
I
mixed strategy: probability distribution, p ij
I
expected cost of server j
E [Cj ] =
h
i
= Pr xij = 1
X wi · p j
i
bj
i ∈[n]
I
I
every agent aims to minimize its expected cost
expected cost of task i on server j
P
h
i
wi + k6=i wk · pkj
j
j
ci = E Cj | x i = 1 =
bj
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
A routing game based on makespan scheduling
[Koutsoupias & Papadimitriou, 1999]
I every task is managed by a selfish agent
I
mixed strategy: probability distribution, p ij
I
expected cost of server j
E [Cj ] =
h
i
= Pr xij = 1
X wi · p j
i
bj
i ∈[n]
I
I
every agent aims to minimize its expected cost
expected cost of task i on server j
P
h
i
wi + k6=i wk · pkj
j
j
ci = E Cj | x i = 1 =
bj
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Cost of mixed allocation
C
= E max Cj
j∈[m]
Nash equilibrium
Allocation is in Nash equilibrium if
pij > 0 ⇒ ∀k ∈ [m] : cij ≤ cik
Coordination ratio
R
=
C (X )
X is Nash allocation opt
Berthold Vöcking
max
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Easy lower bounds – two servers
R≥
3
2
Example: 2 servers, 2 tasks, b1 = b2 = w1 = w2 = 1
opt = 1
C =1·
Berthold Vöcking
1
2
+2·
1
2
=
3
2
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Easy lower bounds – m identical servers
R =Ω
log m
log log m
Example:
I
n = m tasks
I
all bandwidths and weights equal to one
I
one task for each server gives opt = 1
I
fully mixed Nash
pij =
assignment:
gives C = Θ logloglogmm
Berthold Vöcking
1
m,
for i ∈ [n], m ∈ [m],
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Results on the coordination ratio
identical servers
2−
pure
mixed
Θ
1
m
log m
log log m
general model
Θ
Θ
log m
log log m
log m
log log log m
[Koutsoupias & Papadimitriou, 1999]
[Mavronicolas & Spirakis, 2001]
[Czumaj & Vöcking, 2002]
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Proof of upper bound for pure strategies
I
Fix an arbitrary pure Nash equilibrium.
I
Suppose the maximum cost over all servers is r · opt, for r ≥ 1.
I
To simplify notation, assume r is integral.
We need to show: r = O
log m
log log m
.
I
Recall b1 ≥ b2 ≥ · · · ≥ bm .
I
For 0 ≤ k ≤ r − 1, define group Gk to be the maximum
length prefix of servers with cost ≥ k · opt.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Proof of upper bound for pure strategies
I
Fix an arbitrary pure Nash equilibrium.
I
Suppose the maximum cost over all servers is r · opt, for r ≥ 1.
I
To simplify notation, assume r is integral.
We need to show: r = O
log m
log log m
.
I
Recall b1 ≥ b2 ≥ · · · ≥ bm .
I
For 0 ≤ k ≤ r − 1, define group Gk to be the maximum
length prefix of servers with cost ≥ k · opt.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Proof of upper bound for pure strategies
I
Fix an arbitrary pure Nash equilibrium.
I
Suppose the maximum cost over all servers is r · opt, for r ≥ 1.
I
To simplify notation, assume r is integral.
We need to show: r = O
log m
log log m
.
I
Recall b1 ≥ b2 ≥ · · · ≥ bm .
I
For 0 ≤ k ≤ r − 1, define group Gk to be the maximum
length prefix of servers with cost ≥ k · opt.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Proof of upper bound for pure strategies
I
Fix an arbitrary pure Nash equilibrium.
I
Suppose the maximum cost over all servers is r · opt, for r ≥ 1.
I
To simplify notation, assume r is integral.
We need to show: r = O
log m
log log m
.
I
Recall b1 ≥ b2 ≥ · · · ≥ bm .
I
For 0 ≤ k ≤ r − 1, define group Gk to be the maximum
length prefix of servers with cost ≥ k · opt.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Proof of upper bound for pure strategies
I
Fix an arbitrary pure Nash equilibrium.
I
Suppose the maximum cost over all servers is r · opt, for r ≥ 1.
I
To simplify notation, assume r is integral.
We need to show: r = O
log m
log log m
.
I
Recall b1 ≥ b2 ≥ · · · ≥ bm .
I
For 0 ≤ k ≤ r − 1, define group Gk to be the maximum
length prefix of servers with cost ≥ k · opt.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Proof of upper bound for pure strategies
I
Fix an arbitrary pure Nash equilibrium.
I
Suppose the maximum cost over all servers is r · opt, for r ≥ 1.
I
To simplify notation, assume r is integral.
We need to show: r = O
log m
log log m
.
I
Recall b1 ≥ b2 ≥ · · · ≥ bm .
I
For 0 ≤ k ≤ r − 1, define group Gk to be the maximum
length prefix of servers with cost ≥ k · opt.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Definition of groups
r
r−1
G r−1
G r−2
G r−3
G r−4
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
We will prove the following recurrence
|Gk | ≥ (k + 1) |Gk+1 |
, for 0 ≤ k ≤ r − 2
|Gr −1 | ≥ 1 .
As a consequence,
m = |G0 | ≥ (r − 1)! = Γ(r )
Solving for r gives
r
≤ Γ−1 (m) = O
Berthold Vöcking
log m
log log m
.
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
First, let us prove |Gr −1 | ≥ 1
r
r−1
G r−1
G r−2
G r−3
G r−4
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
I
For contradiction, assume |Gr −1 | = 0, that is, server 1 has
cost less than (r − 1)opt.
I
Let i ∈ [n] be a task placed on the server with maximum cost,
that is, task i has cost r in the Nash equilibrium.
I
Moving i to server 1 gives cost less than
(r − 1)opt +
I
wi
b1
≤ (r − 1)opt + opt
≤ r · opt .
This contradicts the Nash equilibrium assumption.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
I
For contradiction, assume |Gr −1 | = 0, that is, server 1 has
cost less than (r − 1)opt.
I
Let i ∈ [n] be a task placed on the server with maximum cost,
that is, task i has cost r in the Nash equilibrium.
I
Moving i to server 1 gives cost less than
(r − 1)opt +
I
wi
b1
≤ (r − 1)opt + opt
≤ r · opt .
This contradicts the Nash equilibrium assumption.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
I
For contradiction, assume |Gr −1 | = 0, that is, server 1 has
cost less than (r − 1)opt.
I
Let i ∈ [n] be a task placed on the server with maximum cost,
that is, task i has cost r in the Nash equilibrium.
I
Moving i to server 1 gives cost less than
(r − 1)opt +
I
wi
b1
≤ (r − 1)opt + opt
≤ r · opt .
This contradicts the Nash equilibrium assumption.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
I
For contradiction, assume |Gr −1 | = 0, that is, server 1 has
cost less than (r − 1)opt.
I
Let i ∈ [n] be a task placed on the server with maximum cost,
that is, task i has cost r in the Nash equilibrium.
I
Moving i to server 1 gives cost less than
(r − 1)opt +
I
wi
b1
≤ (r − 1)opt + opt
≤ r · opt .
This contradicts the Nash equilibrium assumption.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
It remains to show |Gk | ≥ (k + 1) |Gk+1 |, for 0 ≤ k ≤ r − 2.
k+1
k
k−1
G k+1
Berthold Vöcking
Gk
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
server q
k+1
k
k−1
G k+1
Gk
all tasks in Gk+1 have weight larger than bq · opt
⇒ OPT must give these tasks to servers with speed > b q
⇒ OPT needs to assign these tasks to servers in G k
⇒ |Gk | ≥ (k + 1) |Gk+1 |.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
server q
k+1
k
k−1
G k+1
Gk
all tasks in Gk+1 have weight larger than bq · opt
⇒ OPT must give these tasks to servers with speed > b q
⇒ OPT needs to assign these tasks to servers in G k
⇒ |Gk | ≥ (k + 1) |Gk+1 |.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
server q
k+1
k
k−1
G k+1
Gk
all tasks in Gk+1 have weight larger than bq · opt
⇒ OPT must give these tasks to servers with speed > b q
⇒ OPT needs to assign these tasks to servers in G k
⇒ |Gk | ≥ (k + 1) |Gk+1 |.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
server q
k+1
k
k−1
G k+1
Gk
all tasks in Gk+1 have weight larger than bq · opt
⇒ OPT must give these tasks to servers with speed > b q
⇒ OPT needs to assign these tasks to servers in G k
⇒ |Gk | ≥ (k + 1) |Gk+1 |.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Construction of lower bound
Nash Equilibrium for r machines
r
jobs
of
size
2r
r−1
jobs
of
size
2 r−1
one
job
of
size
one
r−2
jobs
of
size
2 r−2
... ...
speed
2
r
2
r−1
Berthold Vöcking
2
r−2
1
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Construction of lower bound
Replicate machines and their jobs in such a way that opt = O(1).
r
jobs
of
size
2r
speed
2
r
r−1
jobs
of
size
2 r−1
2
r times
...
r−1
r−1
jobs
of
size
2 r−1
2
Berthold Vöcking
r−1
r−2
jobs
of
size
2 r−2
2
r(r−1)
times
...
r−2
Selfish Routing and Congestion Games
r−2
jobs
of
size
2 r−2
...
2
r−2
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Other latency functions
Theorem (Krysta, Czmuaj, Vöcking, 2002)
The coordination ratio for the server game can be bounded
polynomially in m if and only if the latency functions satisfy
∃α ≥ 1 :
∀λ > 0 :
`(2 λ) ≤ α · `(λ) .
Otherwise the coordination ratio is unbounded.
bounded
linear functions
polynomial functions
Berthold Vöcking
unbounded
exponential functions
queueing functions
Erlang loss
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Other latency functions
Theorem (Krysta, Czmuaj, Vöcking, 2002)
The coordination ratio for the server game can be bounded
polynomially in m if and only if the latency functions satisfy
∃α ≥ 1 :
∀λ > 0 :
`(2 λ) ≤ α · `(λ) .
Otherwise the coordination ratio is unbounded.
bounded
linear functions
polynomial functions
Berthold Vöcking
unbounded
exponential functions
queueing functions
Erlang loss
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Discrete and atomic routing games in networks
I
Price of anarchy for unweighted games with linear latency
functions is 2.5
I
Price of anarchy for weighted games with linear latency
functions is 2.618
I
If latency functions are polynomials of degree d then the price
of anarchy is O(d d ).
I
Results hold for atomic and non-atomic games.
[Awerbuch, Azar, Epstein, unpublished work]
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Discrete and atomic routing games in networks
I
Price of anarchy for unweighted games with linear latency
functions is 2.5
I
Price of anarchy for weighted games with linear latency
functions is 2.618
I
If latency functions are polynomials of degree d then the price
of anarchy is O(d d ).
I
Results hold for atomic and non-atomic games.
[Awerbuch, Azar, Epstein, unpublished work]
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Discrete and atomic routing games in networks
I
Price of anarchy for unweighted games with linear latency
functions is 2.5
I
Price of anarchy for weighted games with linear latency
functions is 2.618
I
If latency functions are polynomials of degree d then the price
of anarchy is O(d d ).
I
Results hold for atomic and non-atomic games.
[Awerbuch, Azar, Epstein, unpublished work]
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Wardrop’s Traffic Model
Discrete and Atomic Models
Discrete and atomic routing games in networks
I
Price of anarchy for unweighted games with linear latency
functions is 2.5
I
Price of anarchy for weighted games with linear latency
functions is 2.618
I
If latency functions are polynomials of degree d then the price
of anarchy is O(d d ).
I
Results hold for atomic and non-atomic games.
[Awerbuch, Azar, Epstein, unpublished work]
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Congestion Games
Notation:
I
number of agents is denoted by n
I
finite set of resources, edges E
I
strategy space of player i is Si ⊆ 2E
I
latency function for edge e ∈ E is `e : [n] → N
Subclasses:
I
network congestion games
I
symmetric congestion games (identical strategy spaces)
I
single-choice congestion games (like parallel links)
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
A symmetric network congestion game
2,3,6
2,3,5
s
1,2,8
2,3,5
Berthold Vöcking
t
4,6,7
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Congestion Games
Generalizations:
I
weighted congestion games
I
subjective congestion games
Studied variants:
I
server game = weighted single-choice game
I
crowding game = subjective single-choice congestion game
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Congestion games – pure equilibria
Theorem: (Rosenthal 1973)
Every congestion game admits a pure Nash equilibrium.
The theorem follows by a nice potential function argument:
φ(s) =
X
X n(e,s)
`e (i)
e∈E i =1
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Rosenthal’s potential function
2,3,6
2,3,5
s
1,2,8
2,3,5
Berthold Vöcking
t
4,6,7
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Rosenthal’s potential function
2,3,6
2,3,5
s
1,2,8
2,3,5
Berthold Vöcking
t
4,6,7
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Rosenthal’s potential function
2,3,6
2,3,5
s
1,2,8
2,3,5
Berthold Vöcking
t
4,6,7
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Rosenthal’s potential function
2,3,6
2,3,5
s
1,2,8
2,3,5
Berthold Vöcking
t
4,6,7
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Complexity issues:
I
Potential function directly implies a pseudopolynomial
algorithm for congestion games.
I
There is a polynomial time algorithm for computing pure
equilibria in symmetric network congestion games.
I
Computing pure equilibria for more general variants is
PLS-complete.
[Fabrikant, Papadimitriou, Talwar 2004]
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
PLS (Polynomial Local Search)
Problems in PLS:
I
optimization problem Π with neighborhood Γ
I
for every solution s, Γ(s) denotes neighborhood of s
I
neibhorhood can be evaluated in polynomial time“
”
objective: find local optimum wrt Γ
I
PLS reductions:
I
Given two PLS problems Π1 and Π2 find
I
poly-time mapping of an instance of Π 1 to an instance of Π2
I
such that local optima are preserved.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
NAE-SAT (Not-All-Equal-SAT)
I
Input: A list of 3-clauses c1 , . . . , cm , ci = (xi1 , xi2 , xi3 ), only
positite literals, each of which having a weight w i
I
Objective: Maximize the weighted number of clauses in
which not all literals have the same value
I
Neibhorhood: Flips of single variables.
Theorem: NAE-SAT is PLS-complete.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
NAE-SAT (Not-All-Equal-SAT)
I
Input: A list of 3-clauses c1 , . . . , cm , ci = (xi1 , xi2 , xi3 ), only
positite literals, each of which having a weight w i
I
Objective: Maximize the weighted number of clauses in
which not all literals have the same value
I
Neibhorhood: Flips of single variables.
Theorem: NAE-SAT is PLS-complete.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Hardness of computing equilibria
Theorem: Finding a Nash equilibrium in a general congestion
game is PLS-complete.
Proof idea: Reduction from NAE-SAT.
I
Agents correspond to variables.
I
For each clause ci spent two resources ei (0) and ei (1).
I
All latency functions are of the form (0, 0, w i ).
I
An agent whose variable is contained in c i uses ei (0) if the
variable has value 0, otherwise ei (1).
This is a one-to-one mapping between the solution spaces.
Agent improves its latency ⇔ agent improves NAE value.
Thus, Nash equilibrium = local optimum of NAE.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Hardness of computing equilibria
Theorem: Finding a Nash equilibrium in a general congestion
game is PLS-complete.
Proof idea: Reduction from NAE-SAT.
I
Agents correspond to variables.
I
For each clause ci spent two resources ei (0) and ei (1).
I
All latency functions are of the form (0, 0, w i ).
I
An agent whose variable is contained in c i uses ei (0) if the
variable has value 0, otherwise ei (1).
This is a one-to-one mapping between the solution spaces.
Agent improves its latency ⇔ agent improves NAE value.
Thus, Nash equilibrium = local optimum of NAE.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Hardness of computing equilibria
Theorem: Finding a Nash equilibrium in a general congestion
game is PLS-complete.
Proof idea: Reduction from NAE-SAT.
I
Agents correspond to variables.
I
For each clause ci spent two resources ei (0) and ei (1).
I
All latency functions are of the form (0, 0, w i ).
I
An agent whose variable is contained in c i uses ei (0) if the
variable has value 0, otherwise ei (1).
This is a one-to-one mapping between the solution spaces.
Agent improves its latency ⇔ agent improves NAE value.
Thus, Nash equilibrium = local optimum of NAE.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Hardness of computing equilibria
Theorem: Finding a Nash equilibrium in a general congestion
game is PLS-complete.
Proof idea: Reduction from NAE-SAT.
I
Agents correspond to variables.
I
For each clause ci spent two resources ei (0) and ei (1).
I
All latency functions are of the form (0, 0, w i ).
I
An agent whose variable is contained in c i uses ei (0) if the
variable has value 0, otherwise ei (1).
This is a one-to-one mapping between the solution spaces.
Agent improves its latency ⇔ agent improves NAE value.
Thus, Nash equilibrium = local optimum of NAE.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Hardness of computing equilibria
Theorem: Finding a Nash equilibrium in a general congestion
game is PLS-complete.
Proof idea: Reduction from NAE-SAT.
I
Agents correspond to variables.
I
For each clause ci spent two resources ei (0) and ei (1).
I
All latency functions are of the form (0, 0, w i ).
I
An agent whose variable is contained in c i uses ei (0) if the
variable has value 0, otherwise ei (1).
This is a one-to-one mapping between the solution spaces.
Agent improves its latency ⇔ agent improves NAE value.
Thus, Nash equilibrium = local optimum of NAE.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Hardness of computing equilibria
Theorem: Finding a Nash equilibrium in a general congestion
game is PLS-complete.
Proof idea: Reduction from NAE-SAT.
I
Agents correspond to variables.
I
For each clause ci spent two resources ei (0) and ei (1).
I
All latency functions are of the form (0, 0, w i ).
I
An agent whose variable is contained in c i uses ei (0) if the
variable has value 0, otherwise ei (1).
This is a one-to-one mapping between the solution spaces.
Agent improves its latency ⇔ agent improves NAE value.
Thus, Nash equilibrium = local optimum of NAE.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Some open problems
I
Find an efficient algorithm for computing a
(1 + )-approximate equilibrium, i.e., no agent can improve its
latency by a factor of more than (1 + ).
I
Show existence of a polynomial length local improvement path
leading to a (1 + )-approximate equilibrium.
I
Find a simple (randomized) process that quickly converges to
a (1 + )-approximate equilibrium.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Some open problems
I
Find an efficient algorithm for computing a
(1 + )-approximate equilibrium, i.e., no agent can improve its
latency by a factor of more than (1 + ).
I
Show existence of a polynomial length local improvement path
leading to a (1 + )-approximate equilibrium.
I
Find a simple (randomized) process that quickly converges to
a (1 + )-approximate equilibrium.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Some open problems
I
Find an efficient algorithm for computing a
(1 + )-approximate equilibrium, i.e., no agent can improve its
latency by a factor of more than (1 + ).
I
Show existence of a polynomial length local improvement path
leading to a (1 + )-approximate equilibrium.
I
Find a simple (randomized) process that quickly converges to
a (1 + )-approximate equilibrium.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Definition and Classification
Relationship to Potential Games
Relationship to Wardrop’s traffic model
Relationship to Wardrop’s traffic model
I
Wardrop’s traffic model is a continuous variant of a network
congestion game.
I
Same kind of potential function
φ(f ) =
XZ
e∈E
f (e)
`e (s)ds
0
ensures existence and essential uniqueness of equilibria via
convex programming.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Critical thoughts
I
We studied large networks using game theory.
I
Our results are mainly concerned with properties of equilibria.
I
These equilibria are based on fundamental game theoretic
assumptions like unbounded rationality and full knowledge.
Both of these assumptions cannot be justified when considering
large networks like the Internet.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Evolutionary game theory
I
Suppose a symmetric two player game is played by a large
population.
I
Different players are paired up at random over and over again.
Examples for different behavioral models:
I
I
I
Success-driven: At a fixed rate each player selects another
player at random, and switches to other player’s strategy with
probability proportional to other player’s payoff.
Disappointment-driven: The smaller the payoff of a player, the
faster she switches to the strategy of another randomly
selected player. W.l.o.g., the rate at which players migrate
decreases linear in their payoff.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
The replicator dynamics
I
We model the dynamics of the population vector x(t) in a
so-called fluid limit model.
I
Both suggested behavorial models (and others) lead to the
same kind of dynamics!
ẋi = λ(x) · xi · ((Ax)i − x · Ax)
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Example: Paper-Scissors-Stone


1 2 0
A= 0 1 2 
2 0 1
Stone
I
Where is the Nash
equilibrium? What
should one expect?
I
Stable orbit for the
standard case
I
Convergent for > 0
I
Divergent for < 0
1
0,8
0,6
0,4
0,2
Scissors
0
0,2
Paper
0
0,4
0
0,6
0,2
0,4
0,8
0,6
0,8
1
1
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Example: Paper-Scissors-Stone

Stone
Paper

1 2 0
A= 0 1 2 
2 0 1
Scissors
Berthold Vöcking
I
Where is the Nash
equilibrium? What
should one expect?
I
Stable orbit for the
standard case
I
Convergent for > 0
I
Divergent for < 0
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Example: Paper-Scissors-Stone

Stone
Paper

1 2 0
A= 0 1 2 
2 0 1
Scissors
Berthold Vöcking
I
Where is the Nash
equilibrium? What
should one expect?
I
Stable orbit for the
standard case
I
Convergent for > 0
I
Divergent for < 0
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Example: Paper-Scissors-Stone

Stone
Paper

1 2 0
A= 0 1 2 
2 0 1
Scissors
Berthold Vöcking
I
Where is the Nash
equilibrium? What
should one expect?
I
Stable orbit for the
standard case
I
Convergent for > 0
I
Divergent for < 0
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Example: Paper-Scissors-Stone

Stone
Paper

1
2+
0
A= 0
1
2+ 
2+
0
1
Scissors
Berthold Vöcking
I
Where is the Nash
equilibrium? What
should one expect?
I
Stable orbit for the
standard case
I
Convergent for > 0
I
Divergent for < 0
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Example: Paper-Scissors-Stone


1
2−
0
A= 0
1
2− 
2−
0
1
Stone
Paper
Scissors
Berthold Vöcking
I
Where is the Nash
equilibrium? What
should one expect?
I
Stable orbit for the
standard case
I
Convergent for > 0
I
Divergent for < 0
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
The replicator dynamics
ẋi = xi · λ(x)((Ax)i − x · Ax)
{z
}
|
=:gi (x)
Desirable properties:
I
payoff monotonicity: (Ax)i > (Ax)j ⇔ gi (x) > gj (x)
I
aggregate monotonicity: y · Ax > z · Ax ⇔ y · g (x) > z · g (x)
Any aggregate monotone dynamics can be represented in form of
the replicator dynamics!
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
The replicator dynamics
ẋi = xi · λ(x)((Ax)i − x · Ax)
{z
}
|
=:gi (x)
Further observations:
I
Replicator dynamics do not discover initially unused strategies.
I
A strategy that is initially in the support will never die out
completely.
In the following, we assume that all strategies are in the initial support and, hence, belong to the support at any time.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Evolutionary stability
Definition (evolutionary stable)
A strategy x is evolutionary stable if for any other strategy y 6= x
there exists an > 0 such that for w = y + (1 − )x it holds
xAw > yAw .
Proposition
A strategy x is evolutionary stable if there exists > 0 such that
for every y with ||y − x|| ≤ it holds
xAw > yAw .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Evolutionary stability
Definition (evolutionary stable)
A strategy x is evolutionary stable if for any other strategy y 6= x
there exists an > 0 such that for w = y + (1 − )x it holds
xAw > yAw .
Proposition
A strategy x is evolutionary stable if there exists > 0 such that
for every y with ||y − x|| ≤ it holds
xAw > yAw .
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Theorem (Taylor and Jonker, 1987)
Evolutionary stability implies asymptotic stability in the local sense,
that is, starting close at a Nash equilibrium, the replicatior
dynamics converges towards the Nash equilibrium.
Proof by Luapunov method.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Selfish routing and evolutionary game theory
Single commodity flow – symmetric games:
I
Every strategy corresponds to a path p ∈ P
I
Payoff corresponds to negative latency −` p (x)
¯
Average latency is denoted by `(x)
I
Behavioral model: Suppose agents change their strategy at a rate
proportional to their latency, and adopt the strategy of another
randomly selectd player.
This results in the following network dynamics:
¯ − `p (x)) .
ẋp = λ(x) · xp · (`(x)
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Selfish routing and evolutionary game theory
Single commodity flow – symmetric games:
I
Every strategy corresponds to a path p ∈ P
I
Payoff corresponds to negative latency −` p (x)
¯
Average latency is denoted by `(x)
I
Behavioral model: Suppose agents change their strategy at a rate
proportional to their latency, and adopt the strategy of another
randomly selectd player.
This results in the following network dynamics:
¯ − `p (x)) .
ẋp = λ(x) · xp · (`(x)
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Selfish routing and evolutionary game theory
Single commodity flow – symmetric games:
I
Every strategy corresponds to a path p ∈ P
I
Payoff corresponds to negative latency −` p (x)
¯
Average latency is denoted by `(x)
I
Behavioral model: Suppose agents change their strategy at a rate
proportional to their latency, and adopt the strategy of another
randomly selectd player.
This results in the following network dynamics:
¯ − `p (x)) .
ẋp = λ(x) · xp · (`(x)
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Let us assume strictly increasing latency functions.
Then there is a unique Wardrop equilibrium x ∗ .
Lemma
For every y 6= x∗ , it holds x∗ · `(y) < y · `(y).
Corollary
The Wardrop equilibrium is evolutionary stable.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof of Lemma:
I
In a Wardrop equilibrium x, all latencies of used paths are
equal, implying x · `(x) ≤ y · `(x) and, hence,
y · `(y) ≥ y · `(y) + x · `(x) − y · `(x)
= x · `(x) + y · (`(y) − `(x))
X
= x · `(x) +
ye (`e (y) − `e (x)).
e∈E
I
For each edge e ye (`e (y) − `e (x)) ≥ xe (`e (y) − `e (x)), at
least once with strict inequality.
I
Thus, y · `(y) > x · `(x) +
X
xe (`e (y) − `e (x)) = x · `(y).
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof of Lemma:
I
In a Wardrop equilibrium x, all latencies of used paths are
equal, implying x · `(x) ≤ y · `(x) and, hence,
y · `(y) ≥ y · `(y) + x · `(x) − y · `(x)
= x · `(x) + y · (`(y) − `(x))
X
= x · `(x) +
ye (`e (y) − `e (x)).
e∈E
I
For each edge e ye (`e (y) − `e (x)) ≥ xe (`e (y) − `e (x)), at
least once with strict inequality.
I
Thus, y · `(y) > x · `(x) +
X
xe (`e (y) − `e (x)) = x · `(y).
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof of Lemma:
I
In a Wardrop equilibrium x, all latencies of used paths are
equal, implying x · `(x) ≤ y · `(x) and, hence,
y · `(y) ≥ y · `(y) + x · `(x) − y · `(x)
= x · `(x) + y · (`(y) − `(x))
X
= x · `(x) +
ye (`e (y) − `e (x)).
e∈E
I
For each edge e ye (`e (y) − `e (x)) ≥ xe (`e (y) − `e (x)), at
least once with strict inequality.
I
Thus, y · `(y) > x · `(x) +
X
xe (`e (y) − `e (x)) = x · `(y).
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof of Lemma:
I
In a Wardrop equilibrium x, all latencies of used paths are
equal, implying x · `(x) ≤ y · `(x) and, hence,
y · `(y) ≥ y · `(y) + x · `(x) − y · `(x)
= x · `(x) + y · (`(y) − `(x))
X
= x · `(x) +
ye (`e (y) − `e (x)).
e∈E
I
For each edge e ye (`e (y) − `e (x)) ≥ xe (`e (y) − `e (x)), at
least once with strict inequality.
I
Thus, y · `(y) > x · `(x) +
X
xe (`e (y) − `e (x)) = x · `(y).
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof of Lemma:
I
In a Wardrop equilibrium x, all latencies of used paths are
equal, implying x · `(x) ≤ y · `(x) and, hence,
y · `(y) ≥ y · `(y) + x · `(x) − y · `(x)
= x · `(x) + y · (`(y) − `(x))
X
= x · `(x) +
ye (`e (y) − `e (x)).
e∈E
I
For each edge e ye (`e (y) − `e (x)) ≥ xe (`e (y) − `e (x)), at
least once with strict inequality.
I
Thus, y · `(y) > x · `(x) +
X
xe (`e (y) − `e (x)) = x · `(y).
e∈E
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
The lemma implies asymptotical stability in the global sense.
Theorem
The networks dynamics converge to the Wardrop equilibrium x ∗
starting from any point of the interior of the simplex.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof by Luapunov method
I
As Luapunov function use relative entropy
Hx∗ (y) :=
X
i
xi∗ ln
xi∗
yi
I
Entropy is continuous and differentiable.
I
Hx∗ (x∗ ) = 0 and Hx∗ (y) > 0, for all y 6= x∗ .
P
Derivative is Ḣx∗ (y) = − i xi∗ ẏyii .
I
I
Substituting the replicator dynamics yields the continuous
function Ḣx∗ (y) = λ(y) · (x∗ − y) · `(y).
I
Our lemma shows that this term is negative.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof by Luapunov method
I
As Luapunov function use relative entropy
Hx∗ (y) :=
X
i
xi∗ ln
xi∗
yi
I
Entropy is continuous and differentiable.
I
Hx∗ (x∗ ) = 0 and Hx∗ (y) > 0, for all y 6= x∗ .
P
Derivative is Ḣx∗ (y) = − i xi∗ ẏyii .
I
I
Substituting the replicator dynamics yields the continuous
function Ḣx∗ (y) = λ(y) · (x∗ − y) · `(y).
I
Our lemma shows that this term is negative.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof by Luapunov method
I
As Luapunov function use relative entropy
Hx∗ (y) :=
X
i
xi∗ ln
xi∗
yi
I
Entropy is continuous and differentiable.
I
Hx∗ (x∗ ) = 0 and Hx∗ (y) > 0, for all y 6= x∗ .
P
Derivative is Ḣx∗ (y) = − i xi∗ ẏyii .
I
I
Substituting the replicator dynamics yields the continuous
function Ḣx∗ (y) = λ(y) · (x∗ − y) · `(y).
I
Our lemma shows that this term is negative.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof by Luapunov method
I
As Luapunov function use relative entropy
Hx∗ (y) :=
X
i
xi∗ ln
xi∗
yi
I
Entropy is continuous and differentiable.
I
Hx∗ (x∗ ) = 0 and Hx∗ (y) > 0, for all y 6= x∗ .
P
Derivative is Ḣx∗ (y) = − i xi∗ ẏyii .
I
I
Substituting the replicator dynamics yields the continuous
function Ḣx∗ (y) = λ(y) · (x∗ − y) · `(y).
I
Our lemma shows that this term is negative.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof by Luapunov method
I
As Luapunov function use relative entropy
Hx∗ (y) :=
X
i
xi∗ ln
xi∗
yi
I
Entropy is continuous and differentiable.
I
Hx∗ (x∗ ) = 0 and Hx∗ (y) > 0, for all y 6= x∗ .
P
Derivative is Ḣx∗ (y) = − i xi∗ ẏyii .
I
I
Substituting the replicator dynamics yields the continuous
function Ḣx∗ (y) = λ(y) · (x∗ − y) · `(y).
I
Our lemma shows that this term is negative.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Proof by Luapunov method
I
As Luapunov function use relative entropy
Hx∗ (y) :=
X
i
xi∗ ln
xi∗
yi
I
Entropy is continuous and differentiable.
I
Hx∗ (x∗ ) = 0 and Hx∗ (y) > 0, for all y 6= x∗ .
P
Derivative is Ḣx∗ (y) = − i xi∗ ẏyii .
I
I
Substituting the replicator dynamics yields the continuous
function Ḣx∗ (y) = λ(y) · (x∗ − y) · `(y).
I
Our lemma shows that this term is negative.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Multicommodity Flow – Asymmetric Games
˙ 2 ∪˙ · · · ∪P
˙ k
Recall: P = P1 ∪P
Definition (Asymmetric Network dynamics)
For p ∈ Pi , set ẋp = λi (x) · xp · (`¯i (x) − `p (x)).
Entropy argument works only if we assume that the dynamics for
all commodities behave in exactly the same way, that is, if the
leading λ-factors are identical for all commodities.
However, there is a Luapunov function that yields convergence also
for heterogenous mix of dynamics.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Multicommodity Flow – Asymmetric Games
˙ 2 ∪˙ · · · ∪P
˙ k
Recall: P = P1 ∪P
Definition (Asymmetric Network dynamics)
For p ∈ Pi , set ẋp = λi (x) · xp · (`¯i (x) − `p (x)).
Entropy argument works only if we assume that the dynamics for
all commodities behave in exactly the same way, that is, if the
leading λ-factors are identical for all commodities.
However, there is a Luapunov function that yields convergence also
for heterogenous mix of dynamics.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Multicommodity Flow – Asymmetric Games
˙ 2 ∪˙ · · · ∪P
˙ k
Recall: P = P1 ∪P
Definition (Asymmetric Network dynamics)
For p ∈ Pi , set ẋp = λi (x) · xp · (`¯i (x) − `p (x)).
Entropy argument works only if we assume that the dynamics for
all commodities behave in exactly the same way, that is, if the
leading λ-factors are identical for all commodities.
However, there is a Luapunov function that yields convergence also
for heterogenous mix of dynamics.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Luapunov method based on Rosenthals potential function
Φ(x) :=
XZ
e∈E
I
`e (s) ds
0
!
Calculate derivative and plug in replicator dynamics:


X
X
Φ̇ =
λi (x) ¯
`i (x)2 −
xp `p (x)2 
i
I
xe
p∈Pi
This derivative is negative because of Jensen’s inequality,
unless x satisfies the Wardrop equilibrium property.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Luapunov method based on Rosenthals potential function
Φ(x) :=
XZ
e∈E
I
`e (s) ds
0
!
Calculate derivative and plug in replicator dynamics:


X
X
Φ̇ =
λi (x) ¯
`i (x)2 −
xp `p (x)2 
i
I
xe
p∈Pi
This derivative is negative because of Jensen’s inequality,
unless x satisfies the Wardrop equilibrium property.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Luapunov method based on Rosenthals potential function
Φ(x) :=
XZ
e∈E
I
`e (s) ds
0
!
Calculate derivative and plug in replicator dynamics:


X
X
Φ̇ =
λi (x) ¯
`i (x)2 −
xp `p (x)2 
i
I
xe
p∈Pi
This derivative is negative because of Jensen’s inequality,
unless x satisfies the Wardrop equilibrium property.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Analyzing the time of convergence
Suppose every player relates its latency to the average latency
within each commodity, and switches her strategy at a rate
proportional to the ratio between her own latency and the average
latency.
This gives the following variant of the network dynamics.
ẋp = xp ·
`¯i (x) − `p (x)
`¯i (x)
How long does it take to come close to equilibrium?
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Analyzing the time of convergence
Suppose every player relates its latency to the average latency
within each commodity, and switches her strategy at a rate
proportional to the ratio between her own latency and the average
latency.
This gives the following variant of the network dynamics.
ẋp = xp ·
`¯i (x) − `p (x)
`¯i (x)
How long does it take to come close to equilibrium?
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Analyzing the time of convergence
Suppose every player relates its latency to the average latency
within each commodity, and switches her strategy at a rate
proportional to the ratio between her own latency and the average
latency.
This gives the following variant of the network dynamics.
ẋp = xp ·
`¯i (x) − `p (x)
`¯i (x)
How long does it take to come close to equilibrium?
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Approximate equilibria
I
Need to define right notion of “approximate equilibrium”.
I
Euclidian distance not suitable here as very poor strategies,
i.e., very slow paths, will never die out completely.
Definition (relaxed (1 + )-equilibrium)
A network game is in a relaxed (1 + )-equilibrium if less than an
¯
-fraction of the flow uses paths with latency larger than (1 + ) · `.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Approximate equilibria
I
Need to define right notion of “approximate equilibrium”.
I
Euclidian distance not suitable here as very poor strategies,
i.e., very slow paths, will never die out completely.
Definition (relaxed (1 + )-equilibrium)
A network game is in a relaxed (1 + )-equilibrium if less than an
¯
-fraction of the flow uses paths with latency larger than (1 + ) · `.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Approximate equilibria
I
Need to define right notion of “approximate equilibrium”.
I
Euclidian distance not suitable here as very poor strategies,
i.e., very slow paths, will never die out completely.
Definition (relaxed (1 + )-equilibrium)
A network game is in a relaxed (1 + )-equilibrium if less than an
¯
-fraction of the flow uses paths with latency larger than (1 + ) · `.
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Bounds on the time of convergence
Starting from any point of the simplex the network dynamics
reaches a relaxed (1 + )-equilibrium in time
I
O(−3 · ln(`max /`∗ )), for symmetric games on general networks
I
O(−3 · `max /`∗ ), for asymmetric games on general networks
There is a symmetric game on parallel links in which the time to
reach an n -equilibrium is Ω(−2 · ln(`max /`∗ )).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Bounds on the time of convergence
Starting from any point of the simplex the network dynamics
reaches a relaxed (1 + )-equilibrium in time
I
O(−3 · ln(`max /`∗ )), for symmetric games on general networks
I
O(−3 · `max /`∗ ), for asymmetric games on general networks
There is a symmetric game on parallel links in which the time to
reach an n -equilibrium is Ω(−2 · ln(`max /`∗ )).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Evolutionary Game Theory
Stability of Symmetric Routing Games
Stability of Asymmetric Routing Games
Time of Convergence
Bounds on the time of convergence
Starting from any point of the simplex the network dynamics
reaches a relaxed (1 + )-equilibrium in time
I
O(−3 · ln(`max /`∗ )), for symmetric games on general networks
I
O(−3 · `max /`∗ ), for asymmetric games on general networks
There is a symmetric game on parallel links in which the time to
reach an n -equilibrium is Ω(−2 · ln(`max /`∗ )).
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Summary
I
different models for selfish routing
I
I
I
price of anarchy
I
I
I
almost independent of the network structure
heaviliy depending on the class of latency functions
congestion games
I
I
I
I
Wardrop’s continuous traffic model
discrete server game
local improvement steps lead to pure Nash equilibria
symmetric network games solvable via min-cost flow
other variants are PLS-complete
evolutionary games
I
I
I
I
adaption to selfish routing (joint work with Simon Fischer)
evolutionary stability implies asymptotic stability
very fast convergence in case of symmetric games
fast convergence for asymmetric games
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Summary
I
different models for selfish routing
I
I
I
price of anarchy
I
I
I
almost independent of the network structure
heaviliy depending on the class of latency functions
congestion games
I
I
I
I
Wardrop’s continuous traffic model
discrete server game
local improvement steps lead to pure Nash equilibria
symmetric network games solvable via min-cost flow
other variants are PLS-complete
evolutionary games
I
I
I
I
adaption to selfish routing (joint work with Simon Fischer)
evolutionary stability implies asymptotic stability
very fast convergence in case of symmetric games
fast convergence for asymmetric games
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Summary
I
different models for selfish routing
I
I
I
price of anarchy
I
I
I
almost independent of the network structure
heaviliy depending on the class of latency functions
congestion games
I
I
I
I
Wardrop’s continuous traffic model
discrete server game
local improvement steps lead to pure Nash equilibria
symmetric network games solvable via min-cost flow
other variants are PLS-complete
evolutionary games
I
I
I
I
adaption to selfish routing (joint work with Simon Fischer)
evolutionary stability implies asymptotic stability
very fast convergence in case of symmetric games
fast convergence for asymmetric games
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Summary
I
different models for selfish routing
I
I
I
price of anarchy
I
I
I
almost independent of the network structure
heaviliy depending on the class of latency functions
congestion games
I
I
I
I
Wardrop’s continuous traffic model
discrete server game
local improvement steps lead to pure Nash equilibria
symmetric network games solvable via min-cost flow
other variants are PLS-complete
evolutionary games
I
I
I
I
adaption to selfish routing (joint work with Simon Fischer)
evolutionary stability implies asymptotic stability
very fast convergence in case of symmetric games
fast convergence for asymmetric games
Berthold Vöcking
Selfish Routing and Congestion Games
Selfish Routing and the Price of Anarchy
Congestion and Crowding Games
On the Evolution of Selfish Routing
Summary
Thanks ...
Berthold Vöcking
Selfish Routing and Congestion Games