Signcryption: what, why and how Yevgeniy Dodis New York University.
Reference Cole, Dodis, Roughgarden (2006) How much can taxes help selfish routing? Journal of...
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Reference Cole, Dodis, Roughgarden (2006) How much can taxes help selfish routing? Journal of Computer and System Sciences 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 Paris Metro Pricing Expedited Service Braess’ Paradox General Networks Game-Theoretic Analysis of Network Quality-of-Service Pricing David R.M. Thompson Albert Xin Jiang Kevin Leyton-Brown [email protected] [email protected] [email protected] Introduction Network System Q: Does a network provide good quality of service? A: That depends on what its users want from it. Game Solver •Normally impractical: Nash equilibria are too expensive to compute (O( 2 2 n ) 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 •Can be treated as a black-box •Input: network •Output: usage pattern Game Theoretic Model Different users have different values for quality of service: •User’s experienced QoS (e.g. latency) influenced by other users’ actions (which cause congestion) •This interdependence means game theory applies. Definition: “Nash equilibrium”: a stable state where no user wants to change their action, given the actions of everyone else [Nash, 1950] Implications and Conclusions 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?) Definition: “Social welfare”: sum of all parties’ utilities (users and network providers) Definition: “Economic efficiency”: Latency U tility S M TP H TTP VoIP U serG roup 1 Action f(x) Action Action f(x) U serG roup 2 Action Action TCP/IP Back-off Network System 2 TCP/IP users, 1 shared link Converges to •Equal division of bandwidth •Limited congestion Game Theoretic Model Suppose user 1 hacks his TCP/IP back-off implementation: Converges to •Unequal share of bandwidth •More congestion Suppose both users hack: Converges to •Equal share of bandwidth •Even more congestion C O N G ESTIO N U ser1,H acked back-off:utility = 0 U ser2,N orm al back-off:utility = -4 C O N G ESTIO N U ser1,H acked back-off:utility = -3 U ser2,H acked back-off:utility = -3 Game Solver •Equivalent to “prisoner’s dilemma” •Only equilibrium is for both users to hack User 2 Normal Hacked User 1 Normal -1,-1 -4,0 Hacked 0,-4 -3,-3 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. Introduction First class Economy class Q: Why charge different prices for identical service? A: Because they’re expensive, first- class cars are less crowded. Same concept applied to highway traffic: Toronto 407’s toll is tuned to control congestion Network System Q: Can we use this idea to prevent internet congestion? [Odlyzko, 1997; Ros & Tuffin, 2004] Linear, additive model of latency: •Delay = ( Usage ) / Bandwidth ∑ •A “perfect” fair queue of unlimited length 1m b/s,$0 1m b/s,$1 Game Theoretic Model 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) 2 users: $1.00/s delay 18 users: $0.10/s delay 1m b/s,$0 1m b/s,$1 Game Solver •Iterate over a range of prices: $0.00 to $2.00 in $0.01 increments •AGG solver finds usage pattern given costs Implications and Conclusions •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 -41 -40 -39 -38 -37 -36 -35 -34 -33 0 0.5 1 1.5 2 2.5 C ost SocialWelfare Social W elfare Users's Share Network System 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 Game Theoretic Model Same utility and user model as Paris Metro pricing example 2 users: $1.00/s delay 18 users: $0.10/s delay 1m b/s,$0 1m b/s,$0 Expedited,$1 Game Solver •Iterate over a range of prices: $0.00 to $2.00 in $0.01 increments •AGG solver finds usage pattern given costs: 2 users: $1.00/s delay 18 users: $0.10/s delay 1m b/s,$0 1m b/s,$0 Expedited,$1 Implications and Conclusions •Economically efficient when cost > $0.72 (Cost of latency minimized) •Most profit goes to users. •No waste: Load always uniformly balanced -37 -35 -33 -31 -29 -27 -25 -23 -21 0 0.5 1 1.5 2 2.5 C ost Social W elfare S ocial W elfare Users's S hare Network System 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) Network System Delay of a path is the sum of delays of link segments along the path l(x)= x l(x)= 20 l(x)= 20 l(x)= x s t u v Game Theoretic Model 20 users Each can choose any path from s to t At equilibrium: flow split between 2 paths Adding a link At equilibrium: all users choose path s,u,v,t All users are worse off l(x)= x l(x)= 20 l(x)= 20 l(x)= x s t l(x)= 0 u v Pricing Put price on link (u,v) When users have same values: (u,v) useless Q: What happens if users have different values? l(x)= 20 l(x)= 20 s t l(x)= 0 u v 18 users: $0.10/s delay 2 users: $1.00/s delay Price:$1 Game Solver •AGG solver finds usage pattern given costs: l(x)= 20 l(x)= 20 s t l(x)= 0 u v 18 users: $0.10/s delay 2 users: $1.00/s delay Price:$1 Implications •Economically efficient between $0.81 and $9.50 •Most profit goes to users •More efficient than without the link (u,v) -160 -150 -140 -130 -120 -110 -100 -90 0 0.12 0.24 0.36 0.48 0.6 0.72 0.84 0.96 1.08 1.2 P rice ofM iddle Edge S ocialW elfare Related Work 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) Game Representations: •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 Black Box Network System G am e- Theoretic M odel N ash Equilibrium Network U sage & U ser Satisfaction G am e- Theoretic Solver 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 Game Generator Object-Oriented Python API: •Takes Network object as input •Generates AGG file •Launches AGG solver •Interprets results •Restricted to parallel paths or Braess-structured networks, with perfect expedited service •Arbitrary latency functions for each link, L: f L (# of users) Real value → •Supports richer QoS measure than just latency: f L (# 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 → C O N G ESTIO N U ser1,N orm al back-off:utility = -1 U ser2,N orm al back-off:utility = -1
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Transcript of Reference Cole, Dodis, Roughgarden (2006) How much can taxes help selfish routing? Journal of...
ReferenceCole, Dodis, Roughgarden (2006) How much can taxes help selfish routing? Journal of Computer and System Sciences
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
Paris Metro Pricing Expedited Service Braess’ Paradox General Networks
Game-Theoretic Analysis of NetworkQuality-of-Service Pricing
David R.M. Thompson Albert Xin Jiang Kevin [email protected] [email protected] [email protected]
IntroductionNetwork System
Q: Does a network provide good quality of service?A: That depends on what its users want from it.
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
•Can be treated as a black-box•Input: network•Output: usage pattern
Game Theoretic ModelDifferent users have different values for quality of service:
•User’s experienced QoS (e.g. latency) influenced by other users’ actions (which cause congestion)•This interdependence means game theory applies.
Definition: “Nash equilibrium”: a stable state where no user wants to change their action, given the actions of everyone else [Nash, 1950]
Implications and Conclusions
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?)
Definition: “Social welfare”: sum of all parties’ utilities (users and network providers)Definition: “Economic efficiency”: maximizing social welfare
Latency
Uti
lity SMTP
HTTPVoIP
User Group 1 Action
f(x)
ActionAction
f(x)
User Group 2 Action Action
TCP/IP Back-offNetwork System
2 TCP/IP users, 1 shared link
Converges to•Equal division of bandwidth•Limited congestion
Game Theoretic ModelSuppose user 1 hacks his TCP/IP back-off implementation:
Converges to•Unequal share of bandwidth•More congestion
Suppose both users hack:
Converges to•Equal share of bandwidth•Even more congestion
CONGESTION
User 1, Hacked back-off: utility = 0
Use
r 2,
Nor
mal
bac
k-of
f: ut
ility
= -
4
CONGESTION
User 1, Hacked back-off: utility = -3
Use
r 2,
Hac
ked
back
-off:
util
ity =
-3
Game Solver
•Equivalent to “prisoner’s dilemma”•Only equilibrium is for both users to hack
User 2
Normal Hacked
User 1
Normal -1,-1 -4,0
Hacked 0,-4 -3,-3
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.
Introduction
First class Economy classQ: Why charge different prices for identical service?A: Because they’re expensive, first-class cars are less crowded.
Same concept applied to highway traffic:Toronto 407’s toll is tuned to control congestion
Network SystemQ: Can we use this idea to prevent internet congestion? [Odlyzko, 1997; Ros & Tuffin, 2004]
Linear, additive model of latency:•Delay = ∑( Usage ) / Bandwidth•A “perfect” fair queue of unlimited length
1mb/s, $0
1mb/s, $1
Game Theoretic Model18 low priority users, 2 high priority usersLinear model of utility:•Utility = –Delay × ValueForTime – LinkToll•Utility measured in $ (cost-benefit trade-off of QoS)
2 users: $1.00/s delay
18 users: $0.10/s delay
1mb/s, $0
1mb/s, $1
Game Solver•Iterate over a range of prices: $0.00 to $2.00 in $0.01 increments•AGG solver finds usage pattern given costs
Implications and Conclusions
•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
-41
-40
-39
-38
-37
-36
-35
-34
-33
0 0.5 1 1.5 2 2.5
Cost
So
cial
Wel
fare
Social Welfare
Users's Share
Network SystemQ: 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
Game Theoretic ModelSame utility and user model as Paris Metro pricing example
2 users: $1.00/s delay
18 users: $0.10/s delay
1mb/s, $0
1mb/s, $0
Expedited, $1
Game Solver•Iterate over a range of prices: $0.00 to $2.00 in $0.01 increments•AGG solver finds usage pattern given costs:
2 users: $1.00/s delay
2 users: $1.00/s delay
18 users: $0.10/s delay
1mb/s, $0
1mb/s, $0
Expedited, $1
Implications and Conclusions
•Economically efficient when cost > $0.72 (Cost of latency minimized)•Most profit goes to users.•No waste: Load always uniformly balanced
-37
-35
-33
-31
-29
-27
-25
-23
-21
0 0.5 1 1.5 2 2.5
Cost
So
cial
Wel
fare
Social Welfare
Users's Share
Network SystemFuture 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)
Network System
Delay of a path is the sum of delays of link segments along the path
l(x) = x l(x) = 20
l(x) = 20 l(x) = x
s t
u
v
Game Theoretic Model20 users Each can choose any path from s to tAt equilibrium: flow split between 2 paths
Adding a link
At equilibrium: all users choose path s,u,v,tAll users are worse off
l(x) = x l(x) = 20
l(x) = 20 l(x) = x
s t
l(x) = 0
u
v
PricingPut price on link (u,v)When users have same values: (u,v) uselessQ: What happens if users have different values?
l(x) = 20
l(x) = 20
s t
l(x) = 0
u
v
18 users: $0.10/s delay
2 users: $1.00/s delay
Price: $1
Game Solver•AGG solver finds usage pattern given costs:
l(x) = 20
l(x) = 20
s t
l(x) = 0
u
v
18 users: $0.10/s delay
2 users: $1.00/s delay
Price: $1
Implications
•Economically efficient between $0.81 and $9.50•Most profit goes to users•More efficient than without the link (u,v)
-160
-150
-140
-130
-120
-110
-100
-90
0 0.12 0.24 0.36 0.48 0.6 0.72 0.84 0.96 1.08 1.2
Price of Middle Edge
So
cial
Wel
fare
Related WorkNetwork 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)
Game Representations:•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
Black Box
NetworkSystem
Game-Theoretic
Model
NashEquilibrium
NetworkUsage & User
Satisfaction
Game-Theoretic
Solver
Game Theoretic ModelFuture 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
Game GeneratorObject-Oriented Python API: •Takes Network object as input•Generates AGG file•Launches AGG solver•Interprets results
•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
CONGESTION
User 1, Normal back-off: utility = -1
Use
r 2,
Nor
mal
bac
k-of
f: ut
ility
= -
1
