E. Altman, C. Touati, R. El-Azouzi INRIA, Univ Avignon Networking Games ENS January 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

UNFAIR! Symmetric MIMD with synch losses: ratio of throughputs remains as the initial ratio since the rate of increase and decrease are the same UNFAIR! VERY 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! Sym: UNFAIR! Sym: null recurrent MC 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 competition AIMD-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

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