Application of Game Theory to Wireless Networking Tansu Alpcan Introduction Application of Game Theory to Wireless Power Control Games Networking Equilibrium Analysis Stability and Convergence Iterative Schemes Tansu Alpcan Simulations Conclusion Deutsche Telekom Laboratories { Congestion Control } 1 / 44
Application of Outline Game Theory to Wireless Networking Tansu Alpcan Introduction Introduction Power Control Games Power Control Games Equilibrium Equilibrium Analysis Analysis Stability and Convergence Stability and Convergence Iterative Schemes Simulations Iterative Update Schemes Conclusion { Congestion Simulations Control } Conclusion { Congestion Control } 2 / 44
Application of Objectives of this presentation Game Theory to Wireless Networking Tansu Alpcan ◮ Present a general game theoretic framework for Introduction distributed control under limited information Power Control exchange. Games ◮ Illustrate the game theoretic approach via a specific Equilibrium Analysis application: uplink power control in wideband Stability and Convergence wireless networks. Iterative Schemes ◮ Investigate existence and uniqueness of Nash Simulations equilibrium. Conclusion ◮ Convergence and stability analysis of { Congestion Control } continuous-time distributed algorithms. ◮ Study of relevant distributed iterative (update) algorithms and their convergence conditions to the equilibrium. 3 / 44
Application of Network Games Game Theory to Wireless Networking Tansu Alpcan ◮ Game theory (GT) involves multi-person decision making. Introduction ◮ Autonomous parts of the networked systems (such Power Control Games as mobiles, devices generating Internet traffic etc.) Equilibrium are modeled as players. Analysis Stability and ◮ Players interact and compete with each other on the Convergence same system for limited and shared resources: e.g. Iterative Schemes quality of service, bandwidth... Simulations Conclusion ◮ Players are associated with cost functions, which { Congestion they minimize by choosing a strategy from a well Control } defined strategy space. ◮ Nash equilibrium (NE) provides an appropriate solution concept, which is (approximately) optimal w.r.t. a global objective function. 4 / 44
Application of Why Game Theory Game Theory to Wireless Networking Tansu Alpcan Introduction ◮ The microprocessor revolution enabled production of Power Control Games systems with significant processing capacities → Equilibrium independent decision makers. Analysis ◮ These system are connected to each with a variety Stability and Convergence wired/wireless communication technologies resulting Iterative Schemes in networked systems → interaction between Simulations decision makers. Conclusion ◮ The systems share various resources (but often have { Congestion Control } only local information) → competition for available resources (resource allocation). 5 / 44
Application of Uplink Power Control in Wireless Networks Game Theory to Wireless Networking Tansu Alpcan ◮ Primary objective of (uplink) power control is to Introduction regulate the transmission power level of each mobile Power Control Games in order to obtain and maintain a satisfactory quality Equilibrium of service or Signal-to-interference ratio (SIR) level. Analysis Stability and ◮ In wideband systems such as CDMA, signals of the Convergence users interfere and affect each other’s service (SIR) Iterative Schemes level. Simulations ◮ In data networks, unlike in voice communication, SIR Conclusion { Congestion requirements vary from one user to another. Control } ◮ Emerging technologies such as cognitive radio empowers mobile users with independent decision making capabilities. 6 / 44
Application of A Multicell Wireless Network Game Theory to Wireless Networking Tansu Alpcan Introduction Power Control Games Equilibrium Analysis Stability and Convergence Iterative Schemes Simulations Conclusion { Congestion Control } 7 / 44
Application of Distributed Power Control Game Theory to Wireless Networking Tansu Alpcan Introduction Power Control Games Equilibrium Analysis Stability and Convergence Iterative Schemes Simulations Conclusion { Congestion Control } 8 / 44
Application of Game Theoretic Formulation Game Theory to Wireless Networking Tansu Alpcan ◮ Game theory provides a natural framework for power control in wireless systems, where mobiles (players) Introduction compete for service quality: e.g. cognitive radio. Power Control Games ◮ A mobile has no information on other player’s power Equilibrium level or preferences. Therefore, use of Analysis Stability and noncooperative game theory is appropriate. Convergence ◮ Existence of a unique Nash equilibrium (NE) point is Iterative Schemes established in this multicell power control game. Simulations Conclusion ◮ Convergence of continuous and discrete-time { Congestion synchronous and asynchronous update schemes as Control } well as of a stochastic update scheme is investigated. ◮ The power control game and the update algorithms are demonstrated through numerical simulations. 9 / 44
Application of Network Model Game Theory to Wireless Networking Tansu Alpcan ◮ The system consists of L := { 1 , . . . , ¯ L } cells, with M l Introduction users in cell l . Power Control Games ◮ Define 0 < h ij < 1 as the channel gain. Let Equilibrium Analysis secondary interference effects from neighboring cells Stability and be modeled as background noise, of variance σ 2 . Convergence ◮ The i th mobile transmits with an uplink power level of Iterative Schemes Simulations p i ≤ p i , max , which is received at the BS j as Conclusion x ij := h ij p i . Then, SIR obtained by mobile i is given { Congestion by Control } Lh ij p i γ ij := � k � = i h kj p k + σ 2 10/ 44
Application of Network Model Game Theory to Wireless Networking Tansu Alpcan Introduction Power Control Games Equilibrium Analysis Stability and Convergence Iterative Schemes Simulations Conclusion { Congestion Control } 11/ 44
Application of Cost Function Game Theory to Wireless Networking Tansu Alpcan ◮ Each mobile is associated with a cost function: Introduction J i ( x i , x − i , h i ) = P i ( x i ) − U i ( γ i ( x )) Power Control Games Equilibrium ◮ The benefit (utility) function, U i ( γ i ) quantifies the Analysis user demand for quality of service or SIR level. Stability and Convergence ◮ The “pricing” function, P i ( p i ) is imposed to limit the Iterative Schemes interference, and hence, improve the system Simulations performance. It can also be interpreted as a cost on Conclusion the battery usage. { Congestion Control } ◮ Terminology clarification: max Payoff = Benefit − “ Cost ′′ min Cost = − Utility + Price 12/ 44
Application of Nash Equilibrium (NE) Game Theory to Wireless Networking Tansu Alpcan Definition Introduction The Nash equilibrium is defined as a set of strategies Power Control Games (and corresponding set of costs), with the property that Equilibrium no player can benefit by modifying its own strategy while Analysis the other players keep theirs fixed. Stability and Convergence If x is the strategy vector of players and X is the strategy Iterative Schemes space such that x ∈ X ∀ x , then x ∗ is in NE when x ∗ i of Simulations any i th player satisfies Conclusion { Congestion x i J i ( x i , x ∗ Control } min − i ) , where J i is the cost function of the i th player and x ∗ − i is the equilibrium strategies of all other players. 13/ 44
Application of NE of a Generic Noncooperative Game Game Theory to Wireless Networking Tansu Alpcan Introduction Assumptions: Power Control A1 The strategy space X of a noncooperative game, Θ is Games convex, compact, and has a nonempty interior, X o � = ∅ . Equilibrium Analysis Stability and A2 The cost function of the i th player, J i ( x ) , is twice Convergence Iterative Schemes continuously differentiable in all its arguments and strictly Simulations convex in x i , i.e. ∂ 2 J i ( x ) /∂ x 2 i ≥ 0. Conclusion { Congestion Let ∇ be the pseudo-gradient operator: Control } ∇ x 1 J 1 ( x ) T · · · ∇ x M J M ( x ) T � T . � ∇ J := 14/ 44
Application of Game Theory to Wireless Networking Tansu Alpcan Let in addition G ( x ) be the Jacobian of ∇ J with respect to x : Introduction Power Control Games b 1 a 12 · · · a 1 M Equilibrium . . G ( x ) := ... . . Analysis . . Stability and a M 1 a M 2 · · · b M Convergence M × M Iterative Schemes where b i := ∂ 2 J i ( x ) a i , j := ∂ 2 J i ( x ) Simulations and ∂ x i ∂ x j . ∂ x 2 i Conclusion { Congestion We also define the symmetric matrix Control } G ( x ) := G ( x ) + G ( x ) T . 15/ 44
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