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Outline

- Game Theory: A Premier
- Evolutionary Game
- Applications to Networks
- Potential Research Fields

Using as less math

as possible !

2

Game Theory: A Premier- What is “game” about?
- Game of Chicken
- driver who swerves away looses
- What should drivers do?
- To swerve or to stay?

2

Game Theory: A Premier- What is “game” about?
- Game of Chicken
- driver who swerves away looses

Driver 2

Drivers want to do opposite of one another

Driver 1

Game Theory: A Premier

- A Game consists of
- at least two players
- a set of strategiesfor each player
- a payofffor each strategy profile
- Basic assumption (rationality of players)
- Nash Equilibrium
- no player can improve its payoff by unilaterallychanging its strategy
- Pareto optimality, price of anarchy

Game Theory: A Premier

Classification 1:

- Non-Cooperative (Competitive) Games
- individualized play
- Cooperative Games
- play as a group
- Repeated, Stochastic and Evolutionary Games
- not one shot

Game Theory: A Premier

Classification 2:

Game Theory: A Premier

Internet Application

v

C(x) = x

C(x) = 1

C(x) = 0

s

t

C(x) = 1

C(x) = x

w

Selfish Routing game

Game Theory: A Premier

Internet Application

P2P Networks: Bittorrent, Xunlei, Pplive, PPStream, QQLive …

Outline

- Game Theory: A Premier
- Evolutionary Game
- Applications to Networks
- Potential Research Fields

Recap

- Classical game theory (CGT)
- Outcome depends on strongrationality assumption
- Each individual uses a strategy that is the "best response" to other players’ choice
- Question: what is the meaning of a symmetric NE , ,given a large number of players?

Follow the crowd!

Evolutionary game theory

- Evolutionary game theory (EGT)
- refinement of CGT
- game in a population
- dynamics of strategy adoption
- mutual learning among players

Evolutionary game theory differs from classical game theory by focusing more on the dynamics of strategy change as influenced not solely by the quality of various competing strategies, but by the effect of frequency with which the various competing strategies are found in the population.

Evolutionary game theory

- Evolutionary game theory (EGT)
- Usually two types of game: games against the field and games with pairwise contests

A game against the ﬁeld is one in which there is no speciﬁc “opponent”

for a given individual - their payoff depends on what everyone in the

population is doing. Ex: Choice of Gender

A pairwise contest game describes a situation in which a given

individual plays against an opponent that has been randomly selected

(by nature) from the population and the payoff depends just on what

both individual do. Ex: Hawk-Dove Game

Evolutionary game theory

- A profile of evolutionary game
- Payoff (fitness)

Given a set of pure strategy S. A population proﬁle is a vector x that gives a probability x(s) with which each strategy s Sis played in the population.

Consider a particular individual in the population with proﬁle x. If that individual uses a profile σ={,}, the individual’s payoff is denoted as . The payoff of this strategy for a pair-wise game is

Evolutionary game theory

- Evolutionary stable strategy (ESS)
- Theorem (ESS)

An evolutionarily stable strategy is a strategy which, if adopted by a population of players, cannot be invaded (or replaced) by any alternative strategy that is initially rare.

Evolutionary game theory

- Example (Hawk-Dove Game)
- H: aggressive; D: mild
- Population strategy
- Mixed strategy (H,D) of an individual
- Payoff matrix (v<c):
- Suppose the existence of an ESS

Evolutionary game theory

- Example (Hawk-Dove Game) ‘cont
- In the population, the payoff of a mutant is

Evolutionary game theory

- ESS
- no statement of dynamics
- monomorphic / polymorphic
- Replicator Dynamics
- individuals, called replicator, exist in several different types (e.g Hawk and Dove)
- each type of individual uses a pre-programmed strategy and pass it to its descendants
- individuals only use PURE strategy in a finite set
- the population state is ,where is fraction of individuals using strategy

Evolutionary game theory

- Replicator Dynamics
- Fixed point:
- Stability of fixed point:
- Stability proof:

Lyapunov stability vs asymptotic stability

Lyapunovfunction and Engenvalue approach

Evolutionary game theory

- ESSvs NE in associated two-player game
- An ESS is a (mixed) NE
- A NE might not be an ESS
- Asymmetric NE in monomorphic population
- Unstable NE

Evolutionary game theory

- Replicator dynamics and NE
- In a two-strategy game
- Any NE is a fixed point of replicator dynamics
- Not every fixed point corresponds to a NE
- Replicator dynamics and ESS
- ESS is an asymptotically stable fixed point
- Two strategy pair-wise contest
- More than two strategies

ESS Asymp. Stable f.p. sym. NEf.p.

ESS Asymp. Stable f.p. sym. NEf.p.

Outline

- Game Theory: A Premier
- Evolutionary Game
- Applications to Networks
- Potential Research Fields

Peer-to-peer file sharing

Wireless networks

Peer-to-peer file sharing

- File Piece (e.g. chunk, block)
- A content is split in pieces
- Each piece can be independently downloaded
- Leecher
- A peer that is client and server
- In the context of content delivery
- Has a partial copy of the content
- Seed
- A peer that is only server
- In the context of content delivery
- Has a full copy of the content

Great improvement over customer-server mode

Ideal system: single chunk, fully cooperative

Big System: many peers, many chunks, stochastic system

Peer-to-peer file sharing

Seed

t=0

t=T

t=2T

Which peers shall I serve in each time slot?

time

Peer-to-peer file sharing

- If no good incentive strategy
- Slow service
- Even overwhelmed by requests
- Incentive model
- A strategyis the behavior (providing/rejecting a service) of a peer against other peers
- A policy is the set of rules of for incentivization
- A point is a utility measure of peers
- A system is robust : convergence and cooperation

Q. Zhao, J. Lui, D. Chiu“AMathematical Framework for Analyzing Adaptive Incentive Protocols in P2P Networks”, IEEE/ACM Trans. Networking, 2012

Peer-to-peer file sharing

- Incentive model (’cont)
- Strategy = type of peer
- Finite strategies
- {cooperator, defector, reciprocator}

Always serve

Always reject

Serve cooperators and reciprocators

with certain probabilities,

reject defectors

Peer-to-peer file sharing

- Incentive model (’cont)
- System description:
- Incentive scheme (esp. for reciprocators)

At the beginning of each time slot, each peer randomly selects another peer to request for service. The selected peer chooses to serve the request based on his current strategy. A peer obtains α points if its request is served and loses β (=1) points if it provides service to others.

- Prob. that a type i peer provides service to a type j peer

Peer-to-peer file sharing

- Utility model
- After a long way, the points gained by a type-i peer
- We can now study
- equilibrium state (given G)
- is the equilibrium stable?
- how to reach this equilibrium?
- how good is the incentive scheme

Type-i payoff

Network payoff

Is this enough?

Peer-to-peer file sharing

- Learning model in P2P networks
- Current best learning model

At the end of each slot, a peer chooses to switch to another strategy s’ with certain prob. To decide which strategy to choose, the peer learns from other peers.

Needs to compute the gains of all other peers !

Peer-to-peer file sharing

- Learning model in P2P networks
- Opportunistic learning model

At the end of each slot, each peer chooses another peer as its teacher with certain prob. If the teacher is of a different type and performs better, this peer adapts to the teacher’s strategy with another prob.

Simpler !

Peer-to-peer file sharing

- Now we can study
- Robustness of incentive scheme
- Mirror incentive policy
- reciprocators are tit-for-tat
- Proportional incentive policy
- A reciprocators always serves any other reciprocator
- Linear incentive policy

Prob. That reciprocators serve other types of peers!

Each scheme generates a different matrix G !

Peer-to-peer file sharing

- In relation to EG
- pair-wise contest population game
- peers players; chunk exchange2 players games

Opp. Learning

Curr. Best Learning

After some efforts

Replicator dynamics

Large-scale wireless networks

- Random multiple access (slotted ALOHA)
- A node transmits with prob. p in each slot
- Simultaneous transmission collisions

Large-scale wireless networks

- Power control game

(signal to noise interference ratio, SINR)

- Large power better throughput
- Large power more interference to other receivers

Large-scale wireless networks

- Sad facts:
- Selfishness is unsuccessful
- Optimal cooperation is hard in a large distributed networks (bargaining, Shapley value)
- Evolutionary game kicks in!

What if wireless nodes learn from each other

in local interactions?

H. Tembine, E. Altman, “Evolutionary Games in Wireless Networks”, IEEE Trans. Syst. Man Cyber. B, 2010

Large-scale wireless networks

- Challenges
- Standard EGT: a player interacts with all other players (or average population)
- Large-scale wireless networks:
- no longer strategic pair-wise competition
- random number of local players
- non-reciprocal interactions
- Finite strategies of a player

{transmit, stay quiet} in multiple access game

{high power, low power} in power control game

Non standard EGT

Standard EGT

Large-scale wireless networks

- WCDMA power control game
- SINR with distance r between transmitter and receiver of node i is given by

PL

PH

PH

Pi: the strategy of node i (i.e., PH or PL)

x : the proportion of the population choosing PH

g : channel gain, r0 is the radius-of-reception circle of receiver

α : the attenuation order with value between 3 and 6, σ : the noise power, and β: the inverse of processing gain

I(x): total interference from all nodes to the receiver of node i

Large-scale wireless networks

- WCDMA power control game
- Payoff of node i is as follows:

R: transmission range

wp : cost weight due to adopting power

Pi (e.g. energy consumption)

ζ(r) : probability density function given the density of receiver

Large-scale wireless networks

- WCDMA power control game
- Existence of uniqueness of ESS
- Replicator dynamics

This function is continuous and strictly monotonic, which is required for the proof of stability based on sufficient condition

Large-scale wireless networks

- Some other related works
- Extensions to EGT
- Applications

E. Altman, Y. Hayel. “Markov Decision Evolutionary Games”, IEEE Trans. Auto. Ctrl. 2010

X. Luo and H. Tembine. “Evolutionary Coalitional Games for Random Access Control”, IEEE Infocom 2013 (mini)

P. Coucheney, C. Touati. “Fair and Efficient User-Network Association Algorithm for Multi-Technology Wireless Networks”, IEEE Infocom 2009 (mini)

S. Shakkottai, E. Altman. “The Case for Non-cooperative Multihomingof Users to Access Points in IEEE 802.11 WLANs”, IEEE Infocom 2006

C. Jiang, K. Liu, “Distributed Adaptive Networks: A Graphical Evolutionary Game-Theoretic View”, IEEE Trans. Signal Processing, 2013

Large-scale wireless networks

- Summary
- P2P : practical problem EG theory
- WCDMA: EG theory practical problem
- Common Challenges:
- difficultto find important problem
- difficultto have theoretical contributions to EGT

Two different styles !

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