Algorithmic Game
Theoretic Perspectives
in Networking
Dr. Liane Lewin-Eytan
1
Prisoner’s Dilemma
Scenario:
Two suspects in a crime are put into separate cells. If they both
confess, each will be sentenced into 3 years in prison. If only one
of them confesses, he will be freed and used as a witness against
the other, who will receive a sentence of 4 years. If neither
confesses, they will both be convicted of a minor offense and
spend one year in prison.
don’t confess
confess
don’t confess
1,1
4,0
confess
0,4
3,3
1 equilibrium:
(confess, confess)
2
Games in Networks
Users with a multitude of
diverse economic
interests sharing a
Network (Internet)



browsers
routers
servers
Selfishness:
Parties deviate from their
protocol if it is in their
interest
3
Motivation
1. Networking


Traditional networks – single entity with
single control objective.
Modern networking – interaction of many
entities controlled by different parties.
4
Motivation
2. Games


Users act according to their individual interests so as
to maximize their own objective functions.
A user makes selfish decisions based on the state of
the network, which depends on the behavior of the
other users.
 non-cooperative network games.
5
Motivation
3. Algorithms

Computational and algorithmic issues arise in
large and complex games motivated by large
decentralized computer networks
(the Internet).
6
Networking
Algorithms
Games
Non-Cooperative
Network Games
Algorithmic
Game Theory
Algorithmic Perspectives
of Game Theory
in (Large Scale) Networks
7
A Simple Game: Load Balancing
Each job wants to be on a lightly loaded
machine.
With coordination we
can arrange them to
minimize load
1
2
3
2
Example: load of 4
machine 1
machine 2
8
A Simple Game: Load Balancing
Each job wants to be on a lightly loaded
machine.
• Without coordination?
• Stable arrangement:
No job has incentive to switch
2
1
3
2
• Example: some have load of 5
9
Games: Setup


A set of players (in example: jobs)
for each player, a set of strategies
(which machine to choose)
Game: each player picks a strategy
For each strategy profile (a strategy for
each player)  a payoff to each player
(load on selected machine)
10
Nash Equilibrium
A set of actions (strategy choices), one
per player, where no player can unilaterally
improve its performance by changing its
strategy.

The Nash equilibrium solutions of a game are
its stable operating points (stable strategy
profile).
11
Quality of Outcome:
Goal of the Game
Personal objective for player i:
min load Li
Overall objective?
• Social Welfare: i Li
• Makespan: maxi Li
12
Load Balancing and Routing
Load balancing:
Delay as a function
of load:
ℓe(x) = x
x unit of load 
causes delay ℓe(x)
jobs
machines
Routing network:
ℓe(x) = x
s
t
Allow more complex
networks
s
x
1
0
1
x
t
13
Framework




A network shared by selfish users.
Each resource has a cost to be paid by its
users.
Performance of a user = its total payment =
sum of payments for all the resources it uses.
Two fundamental models:
The congestion model.
 The cost sharing model.

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Congestion Model

Resource cost:
Cost Sharing Model

Modeled by a load
dependent function.
 Non-decreasing in the
load of the resource.
Cost sharing mechanism
determines how the cost
is shared by the users.

Each user has a
negative effect on the
performance of other
users.

Resource cost is fixed.

Each user has a
favorable effect on the
performance of other
users.

15
The Game Perspective

Strategy space of each player: subsets of resources.

Cost allocation method defines the rules of the game:
 determines the mutual influence among the players.

Each player knows the rules of the underlying game.

Players are rational: a player chooses a strategy that
minimizes its total payment.
16
The Congestion Model
17
Routing with Delay


Edge-delay is a function ℓe(•) of the load on
the edge e
Assume ℓe(x) continuous and non-decreasing
in load x on edge e.
x
s
1
t
0
1
x
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Example on two links

One unit of flow sent from s to t
x
s
Flow = .5
t
1
Traffic on upper
edge is envious.
Flow = .5
A stable solution:
No-one is
better off
x
s
Flow = 1
1
t
Flow = 0
Users control an infinitesimally small amount of flow.
19
Model of Routing Game



A directed graph G = (V,E)
source–sink pairs si,ti for
i=1,..,k
rate ri  0 of traffic
between si and ti for each
i=1,..,k
s
r1 =1
x
.5
1 .5
.5 1
.5 x
t
• Load-balancing jobs wanted min load
• Here want minimum delay:
delay adds along path
20
Goal of the Game
Personal objective: choose a path minimizing
ℓP(f) = sum of latencies of edges along path P
Overall objective: minimize
C(f) = total latency of a flow f = e fe•ℓe(f)
=social welfare
21
Network Routing Game
Flow represents
 cars on highways
 packets on the Internet
User goal: Find a path minimizing delay
 true for cars,
packets?: users do not choose paths on the Internet:
routers do!
With delay as primary metric  router protocols
choose shortest path!
22
Braess’s Paradox
Original Network
s
x .5
.5
1
Added edge:
s
Effect?
.5
.5
x
1
.5
.5
1
x
0
t
Cost of Nash flow
=2(1.5*0.5)=1.5
.5
.5
1
x
t
23
Braess’s Paradox
Original Network
s
x .5
.5
.5
.5
1
Added edge:
s
x 1
1
1
1
x
0
t
1
Cost of Nash flow
= 2(1.5*0.5)=1.5
1
x
t
Cost of Nash flow = 2
All the flow has increased delay!
24
Some Results
Theorem (Roughgarden-Tardos’00)


In a network with linear latency functions
 i.e., of the form ℓe(x)=aex+be
the cost of a Nash flow is at most 4/3
times that of the minimum-latency flow
 Price of Anarchy = 3/4
25
Some Results
Theorem 1 (Roughgarden-Tardos’00)


In a network with linear latency functions
 i.e., of the form ℓe(x)=aex+be
the cost of a Nash flow is at most 4/3
times that of the minimum-latency flow
x
s
Flow = .5
1
t
Flow = .5
Nash cost 1
optimum 3/4
s
x
1
0
1
x
r=1
t
Nash cost 2 optimum 1.5
26
The Price of Anarchy

Typically, Nash equilibrium outcomes do not
optimize the overall network performance.
Price of Anarchy: The ratio between the cost of
the worst Nash equilibrium and the (social) optimum.

Quantifies the penalty incurred by lack of
cooperation.
27
The Cost Sharing Model
28
Multicast
• A source simultaneously transmits the same data to a group of
destinations.
• Messages are transmitted over each link of the network only once.
• Multicast nodes create copies when the links to the destinations split.
• Multicast routing increases network efficiency.
r
t1
t2
t4
t5
t6
t3
29
A Cost Sharing Multicast Game



A special source node (root) r, and a set N of
n receivers (players).
A player’s strategy is a routing decision –
the choice of a route from its terminal to r.
Egalitarian cost sharing mechanism: the cost of
each edge is evenly split among its downstream
receivers.
cei(s) = ce / ne(s)
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Egalitarian Cost Sharing Mechanism
c1
Payment of t1: c1/4
Payment of t2: c1/4 + c2
Payment of t3: c1/4 + c3/2
Payment of t4: c1/4 + c3/2 + c4
c2
t1
t2
c3
t3
t4
r
c4
t5
t6
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Goal of the Game
Personal objective: choose a path to the
root minimizing payment.
Overall objective: minimize
C(T) = total cost of T = eT ce
= social welfare
= Steiner tree !
32
Potential Function


Multicast game admits a potential function.
Potential function Φ of a solution T [Rosenthal `73]:
ne (T )
ce
(T )   
eT k 1 k


Exact potential:
Change in potential = change in payoff of player
making a move
Global / Local optimum of Φ corresponds to a NE.
33
Price of Anarchy
 Price of anarchy can be as bad as (n).
OPT (= Best NE)
all players use cheap edge
each pays 1/n
total cost = 1
Worst NE
all players use expensive edge
each pays n/n=1
total cost = n
1
s
n
t
1
s
n
t
34
The Price of Stability

Price of anarchy:


Can be unbounded.
Also captures “non-interesting” equilibria.
Price of Stability: The ratio between the cost of
the best Nash solution and the cost of OPT.

Outcome of scenarios in the ‘middle ground’ between
centrally enforced solutions and selfish behavior.
 E.g.: central entity can enforce the initial operating
point.
35
Price of Stability

Price of stability – upper bound is O(log n).
c(TNash)  Φ(TNash)  Φ(Tinitial)  log n ∙c(Tinitial)
proof:
ce with ne > 0 users

edge cost

edge potential with ne > 0 users
e =ce·(1+1/2+1/3+…+1/ne)
 Ratio at most Hn=O(log n)
36
Example: Bound is Tight
t
1
1+
1
1
2
1
2
0
1
3
3
0
0
...
0
n-1
n-1
1
n
n
0
37
Example: Bound is Tight
cost(OPT) = 1+ε
t
1
1+
1
1
2
1
2
0
1
3
3
0
0
...
0
n-1
n-1
1
n
n
0
38
Example: Bound is Tight
t
1
1+
1
1
2
1
2
0
1
3
3
0
0
...
0
n-1
0
n-1
1
n
cost(OPT) = 1+ε
…but not a NE:
player n
pays (1+ε)/n,
n
could pay 1/n
39
Example: Bound is Tight
t
1
1+
1
1
2
1
2
0
1
3
3
0
0
...
0
n-1
n-1
1
so player n
would deviate
n
n
0
40
Example: Bound is Tight
t
1
1+
1
1
2
1
3
2
0
1
3
0
0
...
0
n-1
n-1
1
n
n
now player n-1
pays (1+ε)/(n-1),
could pay 1/(n-1)
0
41
Example: Bound is Tight
t
1
1+
1
1
2
1
2
0
1
3
3
0
0
...
0
n-1
n-1
1
n
n
so player n-1
deviates too
0
42
Example: Bound is Tight
Continuing this
process, all
players defect.
t
1
1+
1
1
2
1
2
0
1
3
3
0
0
...
0
n-1
0
n-1
1
n
n
This is a NE!
(the only Nash)
1
1
cost = 1 + 2 + … + n
Price of Stability is Hn = Θ(log n) !
43
Best Response Dynamics




Best response dynamics : each player, in its turn,
selects a strategy minimizing its cost (or maximizing
its profit).
Natural game course continues until a NE is reached.
PoA may depend on the initial game configuration.
A natural starting point: empty configuration.
44
NE
Cost of user 1:
c (r, x, 1) = 1+ε
c (r , 1) = 1
OPT
r
Greedy cost of 3, … ,n = 1
Cost of user 2:
c (r , x, 2) = 1+ε
c (r, 1, x, 2 ) = 1+2ε
c (r, 2) = 1
1
1
1
3/4
1
1
1
x
¼+ε
1
2
¼+ε
3
¼+ε
¼+ε
…
n-2
n-1
n
Price of anarchy = 4
Can a good equilibrium be achieved as a consequence of
best-response dynamics, starting from an empty configuration?
45
Some Results
(Chuzhoy et al. ‘06)

Upper bound of O( n log 2 n) on the PoA of
best-response dynamics in case players join
the game sequentially starting from an
‘empty’ configuration.
 was improved to O(log3 n) by Charikar et al.
 log n 
 Lower bound of 
 log log n  on the PoA of


this game.

Computing a NE minimizing Rosenthal’s
potential function is NP-hard.
46
Thank You 
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