Multi-user Power Allocation Games

Transcription

1 Supelec Multi-user Power Allocation Games Mérouane Debbah Alcatel-Lucent Chair on Flexible Radio December, 2007

2 Flexible Radio: bringing intelligence everywhere The design of wireless networks requires an increase for new capacity and higher performance. The development of these capabilities is limited severely by the scarcity of two of the principal resources in wireless networks, namely Energy Bandwidth. Recently, the community has turned to a third principal resource, the deployment of intelligence at all layers of the network in order to exploit increases in processing power afforded by Moore s Law type improvements in microelectronics. 1

3 Flexible Radio: a Shannon historic perspective 3 landmark papers from Alcatel-Lucent 1948 A Mathematical Theory of Communication, C. Shannon, Bell System Technical Journal, Vol. 27 (July and October 1948), pp and Communication Theory of Secrecy Systems, C. Shannon, Bell System Technical Journal, Vol. 28 (1949), pp Programming a Computer for Playing Chess, C. Shannon, Philosophical Magazine, Series 7, Vol. 41 (No. 314, March 1950), pp

4 Flexible Radio: a Shannon historic perspective 3

5 Flexible Radio: a Shannon historic perspective 4

6 Flexible Radio: a Shannon historic perspective 5

7 Flexible Radio: a Shannon historic perspective 6

8 Flexible Radio: a Shannon historic perspective Claude Shannon and his mouse Theseus, in one of the first experience on artificial intelligence in Shannon s mouse appears to have been the first learning device of this level. Like the learning mouse, flexible radio intends to build learning networks, just on a much larger, and more complex scale. 7

9 What is a Flexible Radio 50 years later? Joseph Mitola III, Cognitive Radio: An Integrated Agent Architecture for Software Defined Radio, Royal Institute of Technology (KTH) Stockholm, Sweden, 8 May, It is the ultimate point where devices (called the robot hereafter) are sufficiently computationally intelligent about radio resources to detect user communication needs as a function of use context, and to provide (without human intervention) radio resources and wireless services most appropriate to those needs. Definition of Mitola: A radio that employs model based reasoning to achieve a specified level of competence in radio-related domains. 8

10 Flexible Radio: the road ahead It is also called a Cognitive radio ( Cogito, ergo sum [I think, therefore I am]- René Descartes). 9

11 The big dilemma Three important aspects of the problem: Heterogeneity: systems are heterogeneous in transmit power, frequencies, range, QoS requirements, spectral efficiency and systems. Limited information: there may be limited or no communication between different systems and decisions have to be made based on such distributed information. Temporal requirements: systems change rapidly and the flexible radio needs to adapt fast. 10

12 Theoretical foundations for flexible radios. Statistical inference methods to build devices which would carry plausible reasoning (Maximum Entropy methods,..). Game theoretic techniques (based on rational players) to provide decentralized/adaptive ressource allocation schemes. Large system analysis based on random matrix theory/free probability/physics theory to reduce the dimensionality of the problem i.e find the parameters of interest in a network rather than optimizing through simulations with 1 billion parameters. Information theory to understand the fundamental limits achievable with intelligent devices. 11

13 Context What is the price of Anarchy? Efficient ressource management techniques are highly critical to ensure reliable data transfer between users. Usual techniques are network oriented. The networks centralizes the information and feebacks the decision to the users. However, information is distributed within the users. As mobility and the number of users in the system increases, signaling overhead can increase dramatically. With the advent of intelligent radios, we must rely on mobiles to take decisions on their own based on the local information they have. When different users interact with incomplete information, any decision contains a risk. Game theory provides an neat framework for analyzing the decision process. 12

14 Some notions on Game theory What is Game Theory? Basically, Game Theory is the mathematics of strategy. Game Theory aims to help us to understand situations in which decision makers interact. The primary theory is the Minimax Theorem which basically says that if all the players of a game play the best, most rational strategy, the resulting outcome of the game is predictable. 13

15 Limitations of Game theory Is it useful? Players are considered rational and should determine what is the best Game Theory aims to help us to understand situation in which decision makers interact. The primary theory is the Minimax Theorem which basically says that if all the players of a game play the best, most rational strategy, the resulting outcome of the game is predictable. 14

16 The Birth of Game Theory J. von Neuman and O. Morgenstern, Theory of Games and Economic Behavior, Princeton, NJ: Princeton University Press, John von Neumann,

17 Static games in Strategic Games Strategic Games: A strategic game consists of: A set of players. for each player, a set of actions. for each player, preferences over the set of action profiles. Example 1: the players may be firms, the actions prices and the preferences the firm s profit. Example 2: the players may be candidates for political office, the actions campaign expenditures and the preferences a reflection of the candidates s probabilities of winning. We consider for the moment only static games (the players have only one move as a strategy). 16

18 Strategic Games: some notations Players: Let P be the set of players. The subscript i designates all the players belonging to P except i himself. Remark: These players are often designated as being opponents of i. In a two player games, player i has one opponent referred as j. 17

19 Strategic Games: some notations Utility (or payoff function): Let u i (s) be the benefit of player i given the strategy profile s. In the two players case, we have U = {u 1 (s), u 2 (s)}. Remark: The payoff function represents a decision maker s preferences in the sense that if he prefers a S to b S then u(a) > u(b). 18

20 Let us start: the prisoner s dilemma Context: Two suspects in a major crime are held in a separate cells. If they both stay quiet, each will be convicted of the minor offense and spend one year in prison. If one and only one of them finks, he will be freed and used as a witness against the other who will spend four years in prison. If they both fink, each will spend three years in prison. 19

21 Modeling the game Players: The two suspects Actions: Each player s set of actions is {Quite, F ink} Preferences: We need a function u 1 such as: u 1 (Fink, Quiet) > u 1 (Quiet, Quiet) > u 1 (Fink, Fink) > u 1 (Quite, Fink) For example, u 1 (Fink, Quiet) = 0. u 1 (Quiet, Quiet) = 1. u 1 (Fink, Fink) = 3. u 1 (Quite, Fink) = 4. 20

22 Games and matrices Suspect 2 Suspect 2 Quiet Fink Suspect 1 Quiet 1,1 4,0 Suspect 1 Fink 0,4 3,3 There are gains from cooperation (each player prefers that both players choose Quite than they both choose Fink). However, each player has an incentive to free ride (choose fink?) whatever the other player does. Question: How to solve it? In other words, how to predict the strategy of each player, considering the information the game offers and assuming that the players are rational? 21

23 Methods to solve static games Iterated dominance. Nash Equilibrium. Mixed Strategies. In all these techniques, we assume non-cooperative games. Cooperative games require agreements between the decision makers and might be more difficult to realize (additional signalization). 22

24 Best response framework If player 1 is quiet, then the best response of player 2 is to fink. If player 1 finks, then 2 is better off finking also. Definition Denote b i (s i ) the best response of player i to the opponent s strategy vector s i. The best response b i (s i ) of player i to the profile of strategies s i is a strategy s i such that: b i (s i ) = argmax si S i u i (s i, s i ) One can see that if two strategies are mutual best responses to each other, then no player would have a reason to deviate from the given strategy profile. This is the start of the Nash equilibrium 23

25 The Birth of the Nash Equilibrium J. F. Nash, Non-Cooperative Games, Doctoral Dissertation, Princeton University, John Forbes Nash, Jr.,

26 Nash Equilibrium Definition The pure strategy profile s constitutes a Nash equilibrium if, for each player i, u i (s i, s i ) u i(s i, s i ), s i S i Expressed differently, a Nash equilibrium embodies a steady stable social norm : if everyone else adheres to it, no individual wishes to deviate from it. 25

27 Cooperative, non-cooperative and coordination We only went through one type of equilibria in the case of non-cooperative games. Nash equilibirum may be very efficient and cooperation is sometimes better off (Pareto equilibrium). However, we don t necessarily need cooperation. Coordination is enough (Correlated equilibrium). Example: (Aumann and Schelling, Nobel prize): People can coordinate rather well without communicating. Correlated Equilibrium in Access Control for Wireless Communications, E. Altman, N. Bonneau and M. Debbah, Networking 2006, Coimbra, Portugal, 2006 Consider the game where two people are asked to select a positive integer. If they choose the same integer, both get an award, otherwise no award is given. In such a setting, the majority tends to select the number 1. This number is distinctive, since it is the smallest integer. 26

28 Game theory and power allocation E. Altman, N. Bonneau, M. Debbah, and G. Caire, An evolutionary game perspective to ALOHA with power control, Proceedings of the 19th International Teletraffic Congress, Constrained Stochastic Games in Wireless Networks, GlobeCom 2007 Washington DC, USA, E. Altman, K. Avrachenkov, Nicolas Bonneau, Merouane Debbah, Rachid El-Azouzi, Daniel Sadoc Menasche, Best Paper Award Wardrop Equilibrium for CDMA systems, third workshop on Resource Allocation in Wireless NETworks, Limassol, Cyprus, 2007., N. Bonneau, M. Debbah, E. Altman and A. Hjrungnes. Non-Atomic Games for Multi-User Systems, submitted in Selected Areas in Communications Game Theory in Communication Systems, N. Bonneau, M. Debbah, E. Altman and A. Hjorungnes Continuum Equilibria for Routing in Dense Ad-hoc Networks, E. Altman, M. Debbah and Alonso Silva, 45th Annual Allerton Conference on Communication, Control and Computing,

29 Basic example: flat fading case We consider the uplink of a single cell network. K users are simultaneously communicating with a base station in the flat fading case At each time instant, the carrier of each user k is characterized by a fading realization h k. As a consequence, the received signal at the base station is given by: y = k h k x k + n where n is a zero mean gaussian noise with variance σ 2. 28

30 Basic example: flat fading case At the base station, the SINR of user k SINR k = P k h k σ 2 + K j=1,j k P. jh j The corresponding capacity of user k on a given carrier: C k = log 2 (1 + P k h k σ 2 + K j=1,j k P jh j ) 29

31 Utility function In order to place ourselves in a game theoretic setting, we have to define a utility for the users. Utility measures the gain of a user as a result of the strategy this user plays. It is natural to define utility as the ratio of the throughput to the transmit power. This is a relevant performance measure, as each mobile wants to use its (limited) battery power to transmit the maximum possible amount of information. 30

32 Utility function Therefore, the utility of user k can be written u k = C k P k. (1) This utility is expressed in bits per joule. In the non-cooperative game setting, each user wants to selfishly maximize his utility. A Nash equilibrium is obtained when no user can benefit by unilaterally deviating from his strategy. 31

33 Utility function From now on, we denote SINR k = β k, whichever filter is actually used. The throughput is given by C k = C(β k ). To obtain the maximum utility achievable by user k, we derive u k with respect to the power P k and equate to 0. We obtain P k β k P k C (β k ) C(β k ) = 0. However, in our case (and CDMA case using asymptotic random matrix theory), we have that P k β k P k = β k Thus, the problem reduces to finding β that satisfies β k C (β k ) C(β k ) = 0. 32

34 Utility function β k C (β k ) C(β k ) = 0. (2) The existence of a solution is guaranteed as long as the utility is a quasiconcave function of the SINR. In addition, we assume that the function C( ) takes value C(0) = 0, so that users cannot achieve an infinite utility by not transmitting. The uniqueness of the solution β is due to the fact that the SINR of each user is a strictly increasing function of its transmit power. 33

35 Utility function Define C k = R k.f(β k ) where R k is the transmission rate and f(β k ) is the efficiency function. The efficiency depends on the modulation, coding and packet size but should be increasing with the following constraints: f(0) = 0 and f( ) = 1. Example: For a packet which contains M bits, f(β k ) = (1 e β k) M. In this case, with M = 100, β = 6.48 = 8.1dB. 34

36 Solution From the definition of the SINR, we have: p 1 = 1 σ 2 β (1 + β ) h (β ) 2 p 2 = 1 σ 2 β (1 + β ) h (β ) 2 35

37 Multi-user CDMA Model: Frequency Selective Fading Model The N 1 received signal vector y at the base station has the form: y = H 1 w 1 P 1 s 1 + H 2 w 2 P 2 s H K w K P k s K + n s = (s 1,..., s K ) is the emitted symbol vector. w k is the N 1 k-th user code. H k is the N N channel matrix of user k. N is the spreading length. P k is the amplitude of user k. n is a N 1 white complex gaussian noise vector of variance σ 2. 36

38 Multi-user CDMA Model: Frequency Selective Fading Model The game theoretic power allocation strategy depends on the type of receiver user used but has the same form as the flat fading: Given the target SINR β, the exact power allocation is given by: Matched Filter P k = 1 σ 2 β E k 1 αβ for α < 1 β where E j = 1 N N m=1 h mj 2. MMSE P k = 1 E k σ 2 β 1 α β 1+β for α < β. 37

39 Multi-user CDMA Model: Frequency Selective Fading Model Optimum filter Asymptotically, as N, K, the power allocation is given by where β + is the solution of P k = 1 E k σ 2 β + β + α log 2 (1 + β +) α log 2 (e) 1 + β ++log 2 1 α β+ 1+β + for α < β + αβ + 1 α β+ 1+β + 1 β + (3) = α log 2 ( 1 + β ). (4) All these results are based on random matrix theory. 38

40 Simulations Comparison of utilities in the Nash equilibrium and uniform power allocations versus L for N = 256, K = x 105 Utility = Goodput/Power 10 5 MF MMSE Opt MFw MMSEw Optw Number of Multipaths L 39

41 Social optimum scenario for the flat fading case What is the users do not behave selfishly? On the contrary, in a social optimum scenario, the purpose for the users is to maximize the total utility of the system, i.e., the total throughput divided by the total power needed to attain this throughput: K u coop k=1 = C k K k=1 P. (5) k C k, as previously, will depend on the kind of receiver considered. 40

42 Social optimum scenario for flat fading (β Given the definition of P (k), for all k, C k) P C (β ) k P (k). Since P (k 0 ) = min k P (k), we (β k) further have C P C (β ) k P (k) C (β ) P (k 0 ). Thus we obtain C (β k ) C (β ) Summing over k = 1...K, we finally obtain P k P (k 0 ). K k=1 C (β k) K k=1 P k C (β ) P (k 0 ). This completes the proof. Only the user with the best channel transmits at a time. Hence, the optimal policy is time sharing. 41

43 Utility function with delay constraints Let X be the number of retransmissions required for a packet to be received without any errors. Any constraint on the transmission delay can be expressed on the number of retransmissions. P r(x = m) = f(β)(1 f(β)) m 1 The delay requirements of a particular user is given by the pair (D, q) such as: P r(x D) q which translates into D m=1 f(β)(1 f(β))m 1 q. 42

44 Utility function with delay constraints D f(β)(1 f(β)) m 1 q m=1 The delay constraint translates into a lower bound on the SINR β. Hence, the lower bound is given by: β lower k = f 1 (1 (1 q k ) 1 D k ) 43

45 Nash equilibrium of the game with delay constraints Each user will seek to maximize its utility while satisfying its SINR requirement. This can be captured by defining a delay-constrained utility for user k as u delay k u delay k = u k if β k β lower k = 0 if β k > β lower k Remember that β is solution of f(β) = βf (β). It can be shown that for some type of efficiency functions (the one we exhibited for example) that u k is an increasing function of p k if β k < β and for all β k > β, u k is a decreasing function of p k u delay k is maximized when the user transmits at a power level such as: β delay k = max(β lower k, β ) 44

46 Nash equilibrium of the game with delay constraints Result. The Nash equilibrium for the non-cooperative game is given by p delay k = min(p k, P max ) where p k is the transmit power that results in the SINR equal to β delay k solution of β delay k = max(β lower k, β ) 45

47 Nash equilibrium of the game with delay constraints Proof. β delay k β delay k is maximized when the transmit power is such as = max(β lower k, β ). If β lower k can not be achieved, the user must transmit at maximum power level to maximize his utility. Let p lower k be the power level for which the SINR is equal to β lower k. We have in any case p lower k P max (otherwise, the user is not admit it in the network) and because u delay k = 0 for p k < p lower k, there is no incentive for the user to transmit at a power level smaller than p lower k. Hence, the set of strategies are restricted to [p lower k, P max ], an interval in which the utility is continuous and quasiconcave and thus has a Nash equilibrium. 46

48 Nash equilibrium of the game with delay constraints Proof. As a result, β which is the solution of f(β) = βf (β) is unique and as a result β delay k is unique. 47

49 Conclusion We have only touched upon the basic applications of game theory and for only one application related to power allocation. Coalition Games in multiple access systems Coalition Formation for Distributed-User Cooperation in Wireless Networks, W. Saad, Z. Han, M. Debbah and A. Hjørungnes,, submitted to the IEEE Wireless Communications and networking Conference, Stackelberg equilibrium for cognitive radios R. Etkin, A. Parekh and D. Tse, Spectrum Sharing for Unlicensed bands, Proceedings of the Allerton Conference on Communication, Control and Computing, Monticello, IL, Sep , Jamming Games A. Kashyap, T. Basar and R. Srikant, Correlated Jamming on MIMO Gaussian Fading Channels, IEEE Transactions on Information Theory, 50(9): , September and still other applications are under study! 48

50 Last Slide THANK YOU! 49