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Game and Evolutionary Game in Communication Networks

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  1. Game and Evolutionary Game in Communication Networks YuedongXu 2013.12.04

  2. Outline • Game Theory: A Premier • Evolutionary Game • Applications to Networks • Potential Research Fields Using as less math as possible !

  3. 2 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?

  4. 2 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

  5. 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

  6. 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

  7. Game Theory: A Premier Classification 2:

  8. 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

  9. Game Theory: A Premier Internet Application P2P Networks: Bittorrent, Xunlei, Pplive, PPStream, QQLive …

  10. Game Theory: A Premier Internet Application Internet Ecosystem (Business Models)

  11. Game Theory: A Premier Internet Application Cloud Computing game

  12. Game Theory: A Premier Internet Application Online Social Networks

  13. Game Theory: A Premier Internet Application Network Security Game

  14. Game Theory: A Premier Wireless Application 802.11 multiple access game

  15. Game Theory: A Premier Wireless Application 3G/4G Power Control Game

  16. Game Theory: A Premier Wireless Application ? Green Blue ? Packet forwarding game

  17. Game Theory: A Premier Wireless Application Cognitive radio network game

  18. Game Theory: A Premier Wireless Application E Wireless jamming and eavesdrop games

  19. Outline • Game Theory: A Premier • Evolutionary Game • Applications to Networks • Potential Research Fields

  20. 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!

  21. 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.

  22. 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

  23. 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

  24. 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.

  25. 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

  26. Evolutionary game theory • Example (Hawk-Dove Game) ‘cont • In the population, the payoff of a mutant is

  27. Evolutionary game theory

  28. 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

  29. Evolutionary game theory • Replicator Dynamics • Fixed point: • Stability of fixed point: • Stability proof: Lyapunov stability vs asymptotic stability Lyapunovfunction and Engenvalue approach

  30. 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

  31. 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.

  32. Outline • Game Theory: A Premier • Evolutionary Game • Applications to Networks • Potential Research Fields Peer-to-peer file sharing Wireless networks

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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?

  39. 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 !

  40. 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 !

  41. 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 !

  42. Peer-to-peer file sharing

  43. Peer-to-peer file sharing • In relation to EG • pair-wise contest population game • peers players; chunk exchange2 players games Opp. Learning Curr. Best Learning After some efforts Replicator dynamics

  44. Large-scale wireless networks • Random multiple access (slotted ALOHA) • A node transmits with prob. p in each slot • Simultaneous transmission  collisions

  45. Large-scale wireless networks • Power control game (signal to noise interference ratio, SINR) • Large power  better throughput • Large power  more interference to other receivers

  46. Large-scale wireless networks

  47. 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

  48. 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

  49. 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

  50. 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