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E. Altman, C. Touati, R. El-Azouzi INRIA, Univ Avignon
Networking Games
ENSENSJanuary 2010
A Trip to Matrix Gameland
Chapter 1
Overview of Chap 1
1. TCP, competition between protocols: motivation for the game theoretic definition of equilibrium
2. Matrix Games and Nash Equilibrium, properties
3. Correlated equilibrium
4. Multi-access 2x2 matrix game
5. Coordinating games
6. Zero sum games, maple
Background: Early TCP
TCP – Transport Control Protocol, used for reliable data transfer and for flow control
Packets have serial numbers. The destination acknowledges received packets
A non-acknowledged packet is retransmitted Initially, data transfers over the Internet used flow
control with fixed window size K: transmission possible only when the number of packets not acknowledged is smaller than K
Problem: NETWORK COLLAPSE
Background: Modern TCP (Van Jacobson)
Adaptive window: keeps increasing linearly in time
When K acks arrive we transmit the window to K+1 and K+1 packets are sent
When a loss occurs we halve K
AIMD: Additive Increase Multiplicative Decrease
Other details are not modelled
Background: Contemporary TCP
Aggressive versions have been proposed to adapt faster
Scalable (Tom Kelly): when K acks are received we multiply K by a constant. MIMD – Multiplicative Increase Multiplicative Decrease
HSTCP (Sally Floyd) like AIMD but with increase and decrease parameters that increase with K
TCP versions are mostly open source (IETF standards) but also patents.
EVOLUTION OF TCP
First version aggressive Second version (Tahoe) the most gentle,
disappeared Third version Reno and its refinement are the
mostly deployed versions Vegas version was shown to perform better but
was not much deployed due to vulnerability against Vegas. Used in satellite communications
New aggressive versions appear (for grid and storage networks): Scalable, HSTCP…
How will future Internet look like?
Researchers have tried to determine which version of TCP will dominate
We can pose a more abstract question: will the Internet move towards an aggressive behavior of TCP, a friendly behavior? Or coexistence?
If coexistence, what proportions?
If there is a convergence to one of the above, we call this an Equilibrium.
Definitions of Equilibrium
A1(u) The isolation test: See how well the protocol performs if everyone uses the friendly protocol only. Then imagine the worlds with the aggressive TCP only. Compare the two worlds. The version u for which users are happier is the candidate for the future Internet.
A2(u) The Confrontation test Consider interactions between aggressive and piecefull sessions that share a common congested link. The future Internet is declared to belong to the transport protocol of version u if u performs better in the interaction with v.
A3. Game Theoretic Approach: We shall combine the approaches. If a version u does better than v under both then it will dominate the future Internet. It is called "dominating strategy". Otherwize both versions will co-exist. The fraction of each type will be such that the average performance of a protocol is the same under both u and v
Competition between MIMD
Competition between MIMD
Symmetric MIMD with synch losses: ratio of throughputs remains as the initial ratio since the rate of increase and decrease are the same UNFAIR!UNFAIR!
Asymmetric MIMD with synch losses: connection with lowest RTT gets all the bandwidth VERY UNFAIR!VERY UNFAIR!
Non synchronized losses, Asym: connection with lowest RTT gets all bandwidth VERY UNFAIR!VERY UNFAIR!
Sym: Sym: null recurrent MC UNFAIR!UNFAIR!
[EA, KA, B. Prabhu 2005]
Intra and Inter-version competition
MIMD-AIMD competition, [EA, KA, BP 2005] there is a threshold on the BDP below which AIMD has better throughput
AIMD-AIMD competitionAIMD-AIMD competition: “fair” sharing.
Vegas – Reno interaction
Reno is more aggressive than Vegas. Does better in the confrontation test but worse in the isolation
“The last issue, which was not addressed in this paper, concerns the deploying of TCP Vegas in the Internet. It may be argued that due to its conservative strategy, a TCP Vegas user will be severely disadvantaged compared to TCP Reno users, …. it is likely that TCP Vegas, which improves both the individual utility of the users and the global utility of the network, will gradually replace TCP” (Bonald)
Summary in a Symmetric matrix game
Matrix Game:Sc-NR:NR better in Isolation, Vegas- Better in confr.NR-Sc:Sc better in Isolation, NR- In confr.
Mixed Strategies
Assume that neither actions is dominant in the TPC game. The game approach predicts that both versions will coexist. The fraction of each is computed so that the average performance of a protocol is the same under both actions
We call this the Indifference Principle
Let this fraction of actions be p and 1-p. Take a=0
Applying the indifference principle
Equating these gives:
Nash Equilibrium
In both the cases of dominating strategy as well as the case of mixed strategy, we predicted the use of BEST RESPONSES at equilibrium – At equilibrium, each player uses an action that is best for him for the given actions of the others
Equilibrium is formally defined through this property.
Pure equilibrium
Equilibrium in Mixed strategies
Characterizing equilibrium
Computing mixed equilibrium
Multiple Access Game
Two mobiles, Collision Channel Only possibility for successful transmission:
only one transmitms Equilibrium:
Always transmit.
Pricing
Adding price E per transmission. 0<E<1. Mixed equilib
r=1-E Thp at equilib:
E(1-E)
maximized at E=1/2 Yields Thp=1/4 Utility at equilib: 0
Capture
Often collision does
not result in loss
Packet Err Prob:
Packet Loss Prob:
Coordination
A non symmetric equilibrium can achieve a global throughput of 1
Is this a correlated equilibrium?
Coordination Games
Coordination Games
Zero-sum Game
Lower Value
Upper Value
Saddle Point
Best response to q
Linear Program
Solution in Maple
Equivalent Games
Example: transformation into an optimization problem
Example: Transforming into a zero-sum game
A vous de jouer!
Concave Games and Constraintns
Chapter 2
Overview of Chap 2
1. Constrained Game
2. Concave Games
Exampl of general constraints