Game-Theoretic Analysis of Network
Quality-of-Service Pricing
Introduction
David R.M. Thompson
Albert Xin Jiang
Kevin Leyton-Brown
daveth@cs.ubc.ca
jiang@cs.ubc.ca
kevinlb@cs.ubc.ca
TCP/IP Back-off
Network System
Paris Metro Pricing
Network System
Introduction
Different users have different values for quality of
service:
Q: How does a tiered QoS system compare with
Paris Metro pricing?
•Consider the same network and assumptions as
in Paris Metro pricing example
•Add “Perfect” expedited service: Expedited
traffic unaffected by non-expedited
CONGESTION
First class
Economy class
Q: Why charge different prices for identical
service?
A: Because they’re expensive, first-class cars are
less crowded.
User 1, Normal back-off: utility = -1
Converges to
•Equal division of bandwidth
•Limited congestion
•User’s experienced QoS (e.g. latency)
influenced by other users’ actions (which cause
congestion)
•This interdependence means game theory
applies.
Black Box
GameTheoretic
Solver
Nash
Equilibrium
Definition: “Nash equilibrium”: a stable state
where no user wants to change their action,
given the actions of everyone else [Nash, 1950]
Game Solver
•Normally impractical: Nash equilibria are too
expensive to compute (O(22n) where n is number
of users)
•Action Graph Games exploit structure for
massive speed gain: [Bhat & Leyton-Brown,
2004; Jiang & Leyton-Brown, 2006]
•Anonymity: other users’ behavior affects my
QoS, not their identities
•Context specific independence: my QoS is
unaffected by traffic on links I’m not using
User Group 1 Action
Action
Action
User 2, Normal back-off: utility = -4
Latency
Same concept applied to highway traffic:
Toronto 407’s toll is tuned to control congestion
CONGESTION
s
t
v
Delay of a path is the sum of delays of link
segments along the path
20 users
Each can choose any path from s to t
At equilibrium: flow split between 2 paths
1mb/s, $0
Adding a link
2 users:
$1.00/s delay
u
Game Solver
•Iterate over a range of prices: $0.00 to $2.00 in
$0.01 increments
•AGG solver finds usage pattern given costs:
1mb/s, $1
User 1, Hacked back-off: utility = 0
Linear, additive model of latency:
•Delay = ∑( Usage ) / Bandwidth
•A “perfect” fair queue of unlimited length
Converges to
•Unequal share of bandwidth
•More congestion
s
18 low priority users, 2 high priority users
Linear model of utility:
•Utility = –Delay × ValueForTime – LinkToll
•Utility measured in $ (cost-benefit trade-off of
QoS)
CONGESTION
v
At equilibrium: all users choose path s,u,v,t
All users are worse off
Pricing
Put price on link (u,v)
When users have same values: (u,v) useless
Q: What happens if users have different values?
1mb/s, $0
1mb/s, $0
18 users:
$0.10/s delay
Expedited, $1
u
2 users:
$1.00/s delay
l(x) = 20
Definition: “Social welfare”: sum of all parties’
utilities (users and network providers)
Definition: “Economic efficiency”: maximizing
social welfare
Game Theoretic Model
Future extensions to user model:
•Arbitrary source and destination nodes
•Uncertainty about the types of other agents (i.e.
Bayesian games)
Game Solver
•All proposed model extensions are possible
within existing AGG framework
•When utility, latency functions have simple
structure (e.g. path latency = sum of link
latencies, path bandwidth = min of link
bandwidths) even more optimization may be
possible
l(x) = 0
s
18 users:
$0.10/s delay
Implications and
Conclusions
1mb/s, $0
User 1, Hacked back-off: utility = -3
-1,-1
2 users:
$1.00/s delay
Related Work
•AGG solver finds usage pattern given costs:
-27
u
Social Welfare
-29
Users's Share
l(x) = 20
-31
l(x) = 0
Price: $1
-35
•Iterate over a range of prices: $0.00 to $2.00 in
$0.01 increments
•AGG solver finds usage pattern given costs
s
t
-37
0
0.5
1
1.5
2
2.5
Game Representations:
l(x) = 20
Cost
v
•Kearns et al (2001): Graphical Games
•exploits strict independence structure
•cannot compactly represent games here
•Monderer (2006): Player-specific congestion
games
•Can compactly represent games here
•Did not focus on computation of Nash
equilibria
2 users:
$1.00/s delay
Implications and
Conclusions
-3,-3
•Equivalent to “prisoner’s dilemma”
•Only equilibrium is for both users to hack
Implications and
Conclusions
The only equilibrium is the least economically
efficient state. Fortunately, TCP/IP hacks have a
cost to adopt and hackers have a disincentive to
share their work.
Social Welfare
-33
•Economically efficient when cost > $0.72 (Cost
of latency minimized)
•Most profit goes to users.
•No waste: Load always uniformly balanced
Implications
-90
-34
-100
-35
-110
-36
Social Welfare
-37
Users's Share
-38
Social Welfare
0,-4
-120
Reference
-130
-140
-39
Network Model:
•Ros & Tuffin (2004): game-theoretic analysis of
Paris-Metro Pricing
•Cole et al (2006): analyzes putting taxes on links
to reduce congestion
(neither paper modeled users with different
values for latency)
18 users:
$0.10/s delay
-33
Game Solver
-4,0
v
-25
2 users:
$1.00/s delay
Game Solver
Normal
l(x) = 20
Game Solver
-23
Hacked
t
-21
1mb/s, $1
Converges to
•Equal share of bandwidth
•Even more congestion
Hacked
The equilibrium of an AGG would allow us to
answer questions about the proposed network:
•What paths through the network would the
users choose?
•How much load would occur on each link?
•What is each user’s utility? (i.e. how happy
are they with the network?)
Network System
Price: $1
Action
Implications and
Conclusions
l(x) = x
l(x) = 20
18 users:
2 users:
$0.10/s delay
$1.00/s delay
Game Theoretic Model
Suppose both users hack:
t
User 1
Network
Usage
& User
Satisfaction
l(x) = 20
l(x) = x
•Restricted to parallel paths or Braess-structured
networks, with perfect expedited service
•Arbitrary latency functions for each link, L:
fL(# of users) → Real value
•Supports richer QoS measure than just latency:
fL(# of users) → Q
where Q is an arbitrary set (e.g. vectors of
features such as bandwidth, latency, probability
of packet loss)
•Arbitrary utility functions:
f U(Q) → Real value
Future extensions to network model:
•Arbitrary network topology
•Richer models of usage (e.g. bandwidth
consumption, burstiness)
•Richer models of tiered service (i.e. imperfectly
expedited service)
l(x) = 0
f(x)
•Can be treated as a black-box
•Input: network
•Output: usage pattern
l(x) = x
l(x) = 20
1mb/s, $0
Normal
User Group 2 Action
Object-Oriented Python API:
•Takes Network object as input
•Generates AGG file
•Launches AGG solver
•Interprets results
l(x) = 20
l(x) = x
Q: Can we use this idea to prevent internet
congestion? [Odlyzko, 1997; Ros & Tuffin, 2004]
User 2
f(x)
u
Expedited, $1
Network System
User 2, Hacked back-off: utility = -3
SMTP
HTTP
VoIP
Network System
Game Generator
Game Theoretic Model
1mb/s, $0
Suppose user 1 hacks his TCP/IP back-off
implementation:
Utility
GameTheoretic
Model
Same utility and user model as Paris Metro
pricing example
18 users:
$0.10/s delay
Game Theoretic Model
General Networks
Game Theoretic Model
Social Welfare
Game Theoretic Model
User 2, Normal back-off: utility = -1
Q: Does a network provide good quality of
service?
A: That depends on what its users want from it.
Braess’ Paradox
Network System
2 TCP/IP users, 1 shared link
Network
System
Expedited Service
Cole, Dodis, Roughgarden (2006) How much can taxes help selfish
routing? Journal of Computer and System Sciences
-150
-40
-160
0
-41
0
0.5
1
1.5
2
2.5
Cost
•Economically efficient between $0.72 and $1.10
(Cost of latency minimized)
•Most profit goes to network provider
•Significant waste: Costly link sits idle while users
wait in free link’s queue
0.12
0.24
0.36
0.48
0.6
0.72
0.84
0.96
1.08
1.2
Price of Middle Edge
•Economically efficient between $0.81 and $9.50
•Most profit goes to users
•More efficient than without the link (u,v)
Kearns, Littman, Singh (2001) Graphical Models for Game Theory,
UAI
Monderer (2007) Multipotential Games, IJCAI
Nash (1950) Equilibrium Points in N-person Games
Odlyzko (1997) A Modest Proposal for Preventing Internet Congestion
Ros, Tuffin (2004) A Mathematical Model of the Paris Metro Pricing
Scheme, Computer Networks