Applications of Game Theory: Part II

1
Applications of Game Theory Part II(b)
John C.S. Lui Computer Science Eng. Dept The
Chinese University of Hong Kong
2
First course
On the interaction between Overlay Routing and
Underlay Routing Y. Liu, H. Zhang, W. Gong, D.
Towsley INFOCOM 2005
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Motivation Interactions Between Application
Level Network and Physical Network

• physical network control
• routing, congestion control,

• add an overlay

• and another

• Result?
• interactions?
• controllers mismatch?

4
Outline

• Problem Formulation
• Simulation Study
• Game-theoretic Study
• Conclusions

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Routing in Underlay Network

• Routing on physical network level
• Inter-domain BGP, etc.
• Intra-domain OSPF, MPLS, etc.
• determine routes for all source-destination
  traffic demand pairs
• minimize network-wide delay, cost, etc.

traffic demand pair A-gtB traffic demand pair
A-gtC traffic demand pair C-gtB
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Routing in Overlay Network

• An overlay network choose routes at application
  level to minimize its own delay or cost

Overlay demand A-gtB logical routes A-gtC-gtB and
A-gtB
C
A
B

• Overlay
• gains advantage
• better path delay, loss, throughput, etc
• is selfish
• potential performance degradation to other
  non-overlay traffic

C
D
E
A
B
demand pair A-gtC
demand pair C-gtB demand pair A-gtB
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Considering overlay and underlay together ?

• How do they interact with each other?
• How does selfish behavior of overlay routing
• affect overall network performance?
• affect non-overlay traffic performance?
• affect its own performance?

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Interactions Between Overlay Routing and
Underlay Routing
Overlay Routing Optimizer To minimize overlay
cost
Underlay Routing Optimizer To minimize overall
network cost
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Approach by authors

• Focusing interaction in a single AS
• Considering two routing models for overlay and
  one routing model for underlay
• Simulating the interaction dynamic process
• Studying this process in a Game-theoretic
  framework

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Routing Models

• Overlay routing model
• Selfish source routing
• Individual user controls infinitesimal amount of
  traffic, to minimize its own delay
• Optimal overlay routing
• A central entity minimizes the total delay of all
  overlay traffic demands

• Underlay routing model
• Optimal underlay routing
• A central entity minimizes the total delay of all
  network traffic, e.g. Traffic Engineering MPLS

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Simulation Study Optimal Overlay and Optimal
Underlay
14 node tier-1 POP network (Medina et.al.
2002) bimodal normal model of traffic demand 3
overlay nodes
Node without overlay
Node with overlay
Link
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Simulation Study ( case 1 8 overlay traffic)
Optimal Overlay and Optimal Underlay

• Iterative process
• Underlay takes turn at step 1, 3, 5,
• Overlay takes turn at step 2, 4, 6,

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Simulation Study (case 2 10 overlay traffic)
Optimal Overlay and Optimal Underlay

• Iterative process
• Underlay takes turn at step 1, 3, 5,
• Overlay takes turn at step 2, 4, 6,

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Game-theoretic Study

• Two-player non-zero sum game

Underlay
overlay
X strategy of overlay traffic allocation on
logical links Y strategy of underlay traffic
allocation on physical links
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Game-theoretic Study

• Best-reply dynamics
• Nash Equilibrium

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Optimal Underlay Routing v.s. Optimal Overlay
Routing

• Overlay
• One central entity calculates routes for all
  overlay demands, given current underlay routing
• Assumption it knows underlay topology and
  background traffic

X(k)
1-X(k)
Denote overlays routing decision with a single
variable X(k) overlays flow on path ACB after
round k
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Best-reply Dynamics

• There exists unique Nash equilibrium x,
• x globally stable x(k) ?x, from any initial
  x(1)

When x(1)0, overlay performance improves
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Best-reply Dynamics

• There exists unique Nash equilibrium x,
• x globally stable x(k) ?x, from any initial
  x(1)

When x(1)0.5, overlay performance degrades
Overlay Delay Evolution
Overlay Routing Evolution
Underlays turn
delay
Overlays turn
x(k)
BAD INTERACTION!
x(k)gtx(k1)gtx
x
x(k)ltx(k1)ltx
Round k
Round k
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Conclusions Open Issues

• Selfish overlay routing can degrade performance
  of network as a whole
• Interactions between blind optimizations at two
  levels may lead to lose-lose situation
• Future work
• larger topology analysis/experimentation
• overlay routing and inter-domain routing
• interactions between multiple overlays ()
• implications on design overlay routing
• regulation between overlay and underlay ()

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Second course
On the Interaction of Multiple Overlay
Routings Joe W.J. Jiang, D.M. Chiu, John C.S.
Lui
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Questions

• These overlays tend to fully utilize available
  resource.
• So, is there any anarchy?
• How do overlay networks co-exist with each other?
• What is the implication of interactions?
• How to regulate selfish overlay networks via
  mechanism design?
• Can ISPs take advantage of this?

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Outline

• Motivation
• Mathematical Modeling
• Overlay Routing Game
• Implications of Interaction
• Pricing
• Conclusion

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Motivation

• Overlays provide a feasibility for users to
  control their own routing.
• Routing, possible multi-path, becomes an
  optimization problem.
• Interaction occurs (due to same underlay)
• Interaction between one overlay and underlay
  traffic engineering, Zhang et al, Infocom05.
• Interaction between co-existing overlays ?

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Performance Characteristics

• Objective minimize end-to-end delay (e.g., RON)
• Delay of a physical link e
• Performance Characteristics (Underlay)

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Performance Characteristics

• Objective minimize end-to-end delay
• Delay of a physical link e
• Performance Characteristics (Underlay)

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Performance Characteristics

• Objective minimize end-to-end delay
• Delay of a physical link e
• Performance Characteristics (Underlay)

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System Objectives

• Network Operators
• Min average delay in the whole underlay network
• Overlay Users
• Min average delay experienced by the overlay

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How do Overlays Interact?

• Overlapping physical links.
• Performance dependent on each other.
• Selfish routing optimization.
• Overlays are transparent to each other.
• Lack of information exchange between overlays.

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Contribution

• What is the form of interaction?
• Is there routing instability (oscillation), or
  there is an equilibrium ?
• Is the routing equilibrium efficient?
• What is the price of anarchy?
• Fairness issues
• Mechanism design can we lead the selfish
  behaviors to an efficient equilibrium?

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Mathematical Modeling

• Overlay routing An optimization problem

• Decision variable routing policy

s overlay f flow r path
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Mathematical Modeling

• Overlay routing An optimization problem

• Objective average weighted delay (matrix form)

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Overlay Routing Optimization

• Convex programming

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Algorithmic Solution

• Unique optimizer
• Convex programming
• feasible region convex
• delay function continuous, non-decreasing,
  strictly convex
• Solution
• Apply any convex programming techniques.
• Marginal cost network flow (probabilistic routing
  ICNP04).
• This is solved in an independent, and distributed
  fashion by each overlay.

But will independent optimization leads to system
instability (route flop)?
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Overlay Routing Game
Strategic Game GoverlayltN, (?s), (s)gt

• Nash Routing Game
• Player -- N
• all overlays
• Strategy -- ?s
• feasible routing policy feasible region of
  OVERLAY(s)
• Preference relation -- s
• low delay players utility function is -delay(s)

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Illustration of Interaction
Aggregate traffic on physical links
Routing decision on logical paths in overlay 1
Delay of logical paths in overlay 1
Overlay 1
Routing decision on logical paths in overlay 2
Delay of logical paths in overlay 2
Overlay 2
Overlay probing
Aggregate overlay traffic

?
Routing Underlay
Underlay (non-overlay) traffic

Overlay n
Routing decision on logical paths in overlay n
Delay of logical paths in overlay n
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Existence of Nash Equilibrium

• Definition Nash equilibrium point (NE)

A feasible strategy profile y(y(1),, y(s),,
y(n))T is a Nash equilibrium in the overlay
routing game if for every overlay s?N,
delay(s)(y(1),y(s),y(n))
delay(s)(y(1),y(s),y(n))for any other
feasible strategy profile y(s) .
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Existence of Nash Equilibrium

• Theorem

Good News NO ROUTE FLOP !!!
In the overlay routing game, there exists a Nash
equilibrium if the delay function
delay(s)(y(s) y(-s)) is continuous,
non-decreasing and convex.
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Fluid Simulation
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Overlay performance
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Overlay routing decisions
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The Price of Anarchy
Global Performance (average delay for all flows)
Efficiency Loss ?

• GOR Global Optimal Routing
• NOR Nash equilibrium for Overlay Routing Game
• NSR Nash equilibrium for Selfish Routing

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Selfish Routing

• (User) selfish routing a single packets
  selfishness
• Every single packet chooses to route via a
  shortest (delay) path.
• A flow is at Nash equilibrium if no packet can
  improve its delay by changing its route.

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Selfish Routing

• Also a Nash equilibrium of a mixed strategic game
• Player flow f
• Strategy p ? Pf
• Preference low delay
• System Optimization Problem

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Performance Comparison
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Inspiration

• Is the equilibrium point efficient (at least
  Pareto optimal) ?
• Fairness issues of resource competition between
  overlays.

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Example Network
y1
1-y1
y2
1-y2
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Sub-Optimality
y1
y2
Non Pareto-optimal !
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Fairness Paradox
y1
y2

• a, b, c, ? are non-negative parameters
• Everything is symmetric except two private links
  a c

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Fairness Paradox
y1
y2
a lt c
Overlay 1 has a better private link !
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Fairness Paradox
y1
y2
Unbounded Unfairness
a lt c ? delay1 lt delay2
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War of Resource Competition
1 unit
1 unit
USA
China
poil(y1y2)
pusa(1-y1)
pchn(1-y2)
pusalt pchn
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War of Resource Competition
1 unit
1 unit
y1
USA
China
poil(y1y2)
Min Costchn(y2 y1) y2poil(y1y2)(1-y2)pchn(
1-y2)
pusa(1-y1)
pchn(1-y2)
pusalt pchn
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War of Resource Competition
1 unit
1 unit
USA
poil(y1y2)
China
pusalt pchn ? Costusa gt Costchn
pusa(1-y1)
pchn(1-y2)
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Pricing (opportunity for ISP)
Mechanism Design
Inefficient Nash equilibrium
Desired equilibrium
payment
new Nash equilibrium
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Pricing I Improve Delay

• Objective to achieve global optimality

• NE of overlay routing game

• Global optimal

le(s) traffic of overlay s le(-s) traffic
other than overlay s
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Pricing I Improve Delay

• Objective to achieve global optimality

• New NE of overlay routing game

• Global optimal

Heterogeneous pricing
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Pricing I Improve Delay

• Global optimal

• New NE of overlay routing game

KKT condition
KKT condition
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Pricing II improve fairness

• Cause of unfairness
• Over-utilize good common resources
• Unfair resource (bandwidth) allocation
• Pricing Scheme

price p
ISP
Overlay
Improve performance Reduce cost
maximize profit
routing decision
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Incentive Resource Allocation

• For overlays

new Nash equilibrium ? le
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Revenue Distribution

• For ISPs (links)

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Effectiveness of Pricing
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Conclusion

• Study the interaction between multiple
  co-existing overlays.
• Non-cooperative Nash routing game.
• Prove the existence of NEP.
• Show the anomalies and implications of the NEP.
• Present two distributed pricing schemes to
  address the anomalies.

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Third Course
Interaction of ISPs Distributed Resource
Allocation and Revenue Maximization Sam C.M.
Lee, Joe W.J. Jiang, D.M Chiu, John C.S. Lui
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View of ISPs
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Tier-2 ISP
Local ISP
ISP
Peer
Peer
Peer
Peering link
ISP link
ISP
Peer
Peer
Peer
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Optimization problem of peers
Issues to consider
Tier-2 ISP (ISP)
Peer i
1. performance of the link
2. charge of the link
Peer k
Peer j
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Optimization problem of peers
Happiness obtained from sending traffic to peers
Delay cost in ISP link
Payment to ISP
Delay costs in peering links
Payments to peers
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Constraints of peers
1.
2.
3.
4.
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Solution to the peers

• Objective function is strictly concave in every
  transmission rate
• The optimal transmission rates and maximum
  utility are unique and can be found by the
  Lagrangian method.

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Problems for an ISP
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Information exchange framework
Next period
Bandwidth allocation
ISP
Bid
peer
Compute resource distribution
Compute optimal rates
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ISP 1 Resource distribution
ISP
Bandwidth 600MBps
?
?
?
peer1
peer2
peer3
Bid 100MBps
Bid 150MBps
Bid 50MBps
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Proportional share algorithm
ISP
Bandwidth 600MBps
100MBps
200MBps
300MBps
peer1
peer2
peer3
Bid 100MBps
Bid 150MBps
Bid 50MBps
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Equal share algorithm
ISP
Bandwidth 600MBps
150MBps
200MBps
250MBps
peer1
peer2
peer3
Bid 100MBps
Bid 150MBps
Bid 50MBps
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Simulations

• When the happiness coefficients of peers are low

PSA
ESA
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Simulation

• When the happiness coefficients of peers are high

PSA
ESA
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ISP 2 Maximization of Revenue
Unit price
Demand by peer i
Total revenue from the peers
Determine the optimal price
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Solution Maximization of revenue

• Estimate the aggregate traffic ( ) from
  all
• peers in term of the price (P)

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Conclusions

• Utility maximization of a peer
• Resource distribution of ISP
• Revenue maximization of ISP

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Fourth Course
On the Access Pricing Issues of Wireless Mesh
Networks
Ray K. Lam Dah-Ming Chiu John C.S. Lui
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WMN Paints a Bright Future

• Wireless mesh network (WMN)
• Wireless nodes
• Multi-hop routing
• Form a wireless mesh
• More access to the Internet
• More people, rich or poor
• More ubiquitous, anywhere, anytime
• More opportunities to everyone

Internet
Internet
82
The Critical ThingCooperation

• Multi-hop routing
• Relay packets for each other
• My concerns bandwidth, CPU time, security
• Community network with symmetric traffic
• Help each other gt mutual benefit
• Access network with asymmetric traffic
• Geographically good VS poor
• Why help the poor?
• Incentive system neededpricing

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When AP Meets a Client

• Simple analysis by Musacchio and Walrand 1
• A game with 2 players
• Access point (AP) provides Internet access
• Client buys the service
• One deal per time slot

AP
Client
p1
accept
slot 1
p2
service duration
accept
slot 2
p3
reject
slot 3
p
AP
Client
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A Beautiful Equilibrium

• AP and client each maximizesher gain
• AP guess the right price
• Client compare the price p withservice utility
  U
• Web browsing utility function
• A beautiful equilibrium
• AP has the same optimal price in every time slot
• Client connects if her per-slot service utility
  is greater than slot price (U gt p)
• Encourages flat-rate pricing

85
To a Multi-hop Scenario
AP
Client
RS

• Adding a relaying node, or reseller (RS)
• RS tries to mark up APs price to the right
  level
• AP takes note of RSs action when setting her
  price
• Equilibrium is still flat-rate pricing
• Multi-hop gt multiple RSs

c1
p1
accept
accept
c
p
slot 1
service duration
c2
p2
accept
accept
slot 2
AP
Client
RS
c3
p3
reject
reject
slot 3
86
Drawbacks of the Simple Model

• Assuming unlimited network capacity
• 2-player game represents whole system
• Treat every incoming client the same
• Unlimited admission gt unlimited capacity
• Assuming a tree-like network
• 2-hop / multi-hop linear network extension
• Does not consider multiple paths
• Pricing competition may occur

AP
Client
RS
A tree-like network
A graph-like network
87
What If Capacity Limited?

• Cannot admit unlimited clients
• Client demands bandwidth guarantee
• AP admission control
• APs system capacity m
• 2-player game not enough
• AP deals with each client differently
• Client arrival model Poisson process
• Like an M/M/m/m/M queuing system

88
Flat-rate Pricing Fails

• Failure scenario
• AP is full m clients admitted
• An admitted client a is paying 5/slot
• A new client b arrives
• AP asks b for 6/slot
• If b accepts
• AP raises price for a to 6/slot, OR
• Simply kicks a out
• Flat-rate pricing is not optimal!

89
Everybody Loves Flat Rate

• Unrealistic for variable rate
• More practicalfixed-rate, non-interrupted
  service
• AP charges a client a fixed rate p over time
• AP cannot disconnect a client unilaterally
• AP can still charge different clients at
  different fixed rates
• How to set the optimal rate on different
  occasions?

90
Best Strategy in New Service Model

• AP sets price based on remaining capacity
• Raises price when becoming full
• State price at state k, AP charges next
  to-be-admitted client at fixed rate pk
• Policy of AP characterized by price vector
• Clients best strategy
• Connect AP if service utility per unit time gt
  price per unit time (U gt p)

91
System Dynamics

• State transition
• Adding a factor P(U gt pk) to regular arrival rate
  in M/M/m/m/M model
• Reward structure
• Simplification immediate expected profit when a
  client connects

? M P(U gt p0)
? (M-1) P(U gt p1)
? (M-m1) P(U gt pm-1)

0
1
2
m
m-1
?
2?
m?
State transition diagram
92
System Dynamics

• State transition
• Adding a factor P(U gt pk) to regular arrival rate
  in M/M/m/m/M model
• Reward structure
• Simplification immediate expected profit when a
  client connects

p0/?
p1/?
pm-1/?

0
1
2
m
m-1
0
0
0
Reward Structure
93
Finding Optimal Price Vector

• Classical optimization
• Solution for queuing system gives limiting state
  probability for each state k, ?k
• Gain of AP is a function of price vector
• Complicated to optimize with classical techniques
• Policy-iteration method in Markovian decision
  theory
• Reduces computational complexity by iterative
  algorithm
• Guarantees convergence to the best policy

94
Numerical Results

• Capacity m5, population M10, departure rate ?1
• Vary arrival rate ? from 0.2 to 10
• Utility U uniformly distributed on 0,10

• U normally distributed with mean 5, s.d.
  1.67
• Price increases number of clients in AP and with
  ?

95
Limited Capacity in Multi-hop Case

• Simplification
• Traffic merges at AP
• AP is the bottleneck
• Only AP controlsadmission
• APs policy specified by a price matrix
• At each state, different prices for requests from
  different distances
• pki price at state k for a client i-hop away

Internet
bandwidth bottleneck
AP
Client
RS
96
System Dynamics

• Removing finite population
• Complicates state information
• Different arrival rates for clients at different
  distances

?n P(U gt mn(p0,n))
?n P(U gt mn(pm-1,n))
Client n-hop away arrives
Client 2-hop away arrives


Client 1-hop away arrives
?2 P(U gt m2(p0,2))
?2 P(U gt m2(pm-1,2))
?1 P(U gt m1(p0,1))
?1 P(U gt m1(pm-1,1))

0
1
m-1
m
m?
?
State transition diagram
97
System Dynamics

• Removing finite population
• Complicates state information
• Different arrival rates for clients at different
  distances

p0,n/?
pm-1,n/?
Client n-hop away arrives
Client 2-hop away arrives


Client 1-hop away arrives
p0,2/?
pm-1,2/?
p0,1/?
pm-1,1/?

0
1
m-1
m
0
0
Reward Structure
98
Conclusion

• Contributions
• Show that fixed-rate pricing fails with limited
  capacity
• Generalize unlimited capacity model into limited
  capacity model
• Devise optimal pricing for fixed-rate,
  non-interrupted service with Markovian decision
  theory
• References
• 1 J. Musacchio and J. Walrand. WiFi access
  point pricing as a dynamic game. IEEE/ACM Trans.
  Networking. to appear in.