A Game-Theoretic Framework for Interference Avoidance

Transcription

1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL A Game-Theoretic Framework for Interference Avoidance R. Menon, A. B. MacKenzie, Member, IEEE, J. Hicks, Member, IEEE, R. M. Buehrer, Senior Member, IEEE, and J. H. Reed, Fellow, IEEE Abstract Various iterative algorithms for interference avoidance IA in networks with co-located receivers, suitable for distributed implementation, have been proposed in the literature. In this paper, the IA problem is cast in a game-theoretic framework and is formulated as a potential game. This formulation accommodates previously proposed algorithms and, in addition, gives us a framework that enables the design of new distributed and convergent algorithms for IA including algorithms with nonidentical utility functions for the users. Two new convergence results for potential games are then derived. The rst result establishes the convergence of a class of potential games to the global solution while following best response iterations and when noise is added. The second result establishes the convergence of potential games to the Nash equilibria of the game while following random better response iterations. The rst result combined with the potential game formulation allows us to show that for a large class of network scenarios, arbitrarily small noise assures the convergence of best response IA algorithms, including the eigeniterations, to an arbitrarily small neighborhood of the globally optimal signature sequence set. The second result enables the design of reduced feedback mechanisms for IA that converge to desirable solutions. Index Terms Game theory, interference avoidance, iterative construction of signature sequences, reduced feedback schemes. I. INTRODUCTION INTERFERENCE avoidance IA refers to waveform adaptation techniques that reduce multiple-access interference in multi-user networks. In this paper, we consider IA techniques, suitable for a distributed implementation, in peer-to-peer multiple access networks with co-located receivers. A similar approach to IA has been extensively used in the literature e.g. [1] and [2]. It is believed that such a development could be a starting point for the design of IA techniques for truly distributed networks. The channel characteristics of networks with co-located receivers are similar to multiple access channels which consider multiple access communication to a centralized receiver. Consequently, a majority of the capacity results on multiple access channels is directly relevant to networks with co-located Paper approved by A. Anastasopoulos, the Editor for Iterative Detection, Estimation and Coding of the IEEE Communications Society. Manuscript received May 2, 2007; revised February 5, 2008 and June 28, This work was supported by the Of ce of Naval Research under grant N and Wireless@Virginia Tech af liates. R. Menon is with Tyco Electronics, 221 Jefferson Ridge Parkway, Lynchburg, VA USA menon.rx@gmail.com. A. B. MacKenzie, R. M. Buehrer, and J. H. Reed are with Wireless@Virginia Tech, Virginia Tech, Blacksburg, VA USA {mackenab, buehrer, reedjh}@vt.edu. J. Hicks is with The Aerospace Corporation, Chantilly, VA USA james.e.hicks@aero.org. Digital Object Identi er /TCOMM /09$25.00 c 2009 IEEE receivers. A characterization of maximum sum capacity and optimal sequences is given in [3] and [4]. It is shown that when no user is oversized a user is oversized if its received power constraint is large relative to the received power constraints of the other users, the optimal sequences satisfy the Welch Bound Equality WBE. However, if oversized users are present in the system, an optimum assignment of sequences is achieved by allocating orthogonal sequences to oversized users and WBE sequences to non-oversized users. It is shown in [5] that the sequences identi ed above are optimal even when user capacity de ned as the maximum number of users per unit processing gain such that the quality-of-service QoS requirement of each user is satis ed is used as the performance metric. In [1], it is shown that the optimal sequences that maximize sum capacity and identi ed above also minimize the generalized total squared correlation TSC of the sequence set. Based on this fact, an algorithm which iteratively decreases TSC is proposed for IA. In this algorithm called the MMSE iteration, each user sequentially replaces its signature sequence with its normalized Minimum Mean Square Error MMSE lter. When starting from random initial sequences, the algorithm is shown to empirically converge to the set of optimum sequences that minimize TSC and maximize sum capacity. In [6], the xed points of this algorithm are studied. It is shown that the sub-optimal xed points of the MMSE iteration are not stable and hence the introduction of small perturbations, by the addition of noise at the end of each iteration, guarantees almost sure convergence to the optimal sequence set. In [7], a general signal space formulation of the IA problem is described and an alternate distributed algorithm based on eigen-iterations is proposed. The xed points of this algorithm are studied in [2] and a technique called class warfare is used to ensure convergence to the optimal set of sequences. However, unlike the approach in [6], this technique is not amenable to a distributed implementation. The IA algorithms are extended to multi-carrier systems in [8] and to multi-cell systems in [9] and [10]. Other extensions of IA algorithms include adaptations in asynchronous CDMA systems [11] and multipath channels [12] [13]. Feedback is a signi cant issue in the implementation of distributed IA algorithms since the signature sequence can be calculated only at the receiver. Hence the real-valued sequence needs to be fed back to the user in each iteration. This could lead to a large increase in the network overhead. A reduced feedback mechanism is investigated in [14] where each user s sequence is restricted to a subspace of the sequence s original

2 1088 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL 2009 signal space. However, this scheme might not lead to the optimal sequences. A different approach is to quantize the realvalued signal [15] and [16], thereby reducing the feedback information. Distributed IA algorithms, as discussed above, attempt to independently shape the waveform of individual users such that the individual user s performance as well as the network performance is improved. However, note that the choice of each user s waveform depends upon the choice of waveforms by the remaining users. Game theory is a branch of applied mathematics that can be used to model such dependencies and interactions between independent decision makers with formalized preference structures. It provides tools to predict and analyze the outcome of these interactions. Thus, game theory can be used to investigate the convergence and steady states in distributed IA algorithms. A survey on the use of game theory to analyze wireless ad-hoc networks is presented in [17]. Game theory has been extensively used to investigate the impact of power control on network interference. [18], a seminal paper in the eld, develops power control algorithm for achieving target SINR based on game-theoretic concepts. Other papers on the subject include [19], [20], [21] and references within. A game-theoretic analysis of the Nash equilibria of joint power control and signature sequence adaptation in synchronous CDMA systems is presented in [22]. In this paper, we cast the IA problem as a potential game [23] [24] [25] [21], a game in which users can serve the greater good by furthering their own interests. We show that the negated TSC is a potential function for a large class of IA algorithms that includes some new algorithms where users do not need to have identical utility functions. Convergence properties, including two new results, for best and better response iterations wherein users choose sequences that maximize or increase their utility respectively of potential games are then derived. The rst result establishes the convergence of the best response iterations of a class of potential games to the global solution when noise is added in a manner similar to [6]. This result allows us to investigate the relevance of the convergence property in [6], that establishes the almost sure convergence to globally optimal sequence sets, to updates other than the MMSE-update for IA. This includes the eigen-iterations algorithm for IA, which was previously shown to converge to the global optima only using class warfare techniques which are not amenable to a distributed implementation. The second result establishes the convergence of potential games to the Nash equilibria of the game while following random better response iterations. This result coupled with the better response convergence properties of potential games is used to motivate IA schemes that reduce the adaptation overhead in the network. The rest of this paper is organized as follows. The system model for the network under consideration is described in Section II. Some game theory concepts used in this paper are brie y discussed in Section III. In Section IV, the IA problem is cast as a potential game and the NE of the game are delineated. Several utility functions including non-identical utility functions that allow this formulation are also identi ed. The best response and better response convergence of potential games are investigated in Section V. In Section VI, it is shown that noisy best response iterations lead to almost sure convergence to the global optimum for a special class of Nash separable potential games and consequently for a large class of IA algorithms. Reduced feedback mechanisms based on better response iterations are investigated in Section VII. Finally Section VIII concludes the paper. II. SYSTEM MODEL We consider a network where multiple user-nodes communicate with receivers which are co-located. Interference caused to the transmission of a particular user-node due to other user transmissions is in uenced by the correlation between the waveforms of users, transmit power levels and the channel characteristics. We use a signal space characterization to represent the waveforms of nodes [7]. This signal space representation speci es the waveform of a node in orthogonal signal dimensions for instance, time, frequency, or spreading code and is referred to as the signature sequence of the node. Let N be the number of signal dimensions available for transmission and K be the number of transmitting nodes in the network. We denote the N-dimensional signature sequence associated with transmitting node k by s k. The signature sequences are real valued as opposed to bi-polar sequences. Without loss of generality, the signature sequences are assumed to have unit norm and hence { are constrained to the } N 1 dimensional sphere S = s k R N 1 : s k 2 =1. The received power level of the k th transmit node at its receiver is denoted by p k. The received power level is a function of the transmit power level of the k th node and the fading and path loss of the channel from the k th transmit node to its receiver. The transmit power levels of users are assumed to be xed by a process that is independent of the waveform adaptation process. The channel is assumed to be constant over all signal dimensions and also constant over the time required for the adaptation process. The received signal is assumed to incur no multipath. It is also assumed that the signature sequences are synchronized at the receiver. However, the framework can be easily extended to include multipath channels and asynchronous systems similar to the analysis in [12] and [11] respectively. The data symbol transmitted from the k th transmit node is denoted by b k. It is assumed that the symbols sent by each transmitter are independent, have zero mean and unit variance. The signal at the receiver is given by K r = pk s k b k + z = s pb + z. 1 k=1 Here, r R N 1 and the vector, z R N 1, models additive Gaussian noise with covariance matrix R zz = E [ zz ] T. If the noise process is white, R zz is a multiple of the identity matrix. Matrix s is a N K matrix with the signature sequences of different radios as columns, s =[s 1,..., s K ] such that s S, where K = {1, 2,..., K } is the set k K of transmitting nodes in the network. p is a K K diagonal matrix whose k th entry is the power level p k ; the square root in p is taken over each individual entry. b is a column vector with b =[b 1,...,b K ] T. The received cross-correlation matrix [ can now be written as R ] rr = E rr T = sps T + R zz. Let

3 MENON et al.: A GAME-THEORETIC FRAMEWORK FOR INTERFERENCE AVOIDANCE 1089 s k denote the signature matrix s without the k th column, i.e., s k =[s 1,..., s k 1,s k+1,..., s K ]. Let p k denote the diagonal matrix p without its k th row and column. Then the interference plus noise cross correlation matrix at the receiver for the k th user is given by R ii [k] =s k p k s T k + R zz = R rr s k p k s T k The k th user s receiver is assumed to know or compute the k th user s waveform s k and to be able to perfectly estimate R ii [k] and R rr. The receivers do not directly communicate with each other. However, they are assumed to be able to coordinate enough to allow their respective transmitters to update their waveforms in a round-robin fashion. In addition to the above, the following notations are also used in this paper. Given x 0 A, N ɛ x 0 = {x A : d x, x 0 <ɛ} is the neighborhood of x 0, where d x, x 0 is the Euclidean distance metric on A. Note that the Euclidean distance between x R N 1 and y R N 1 is N x n y n 2. Nɛ x 0=N ɛ x 0 {x 0 } is the deleted n=1 neighborhood of x 0. Given a set, B A, B denotes its closure and P B denotes its power set. III. GAME THEORY AND POTENTIAL GAMES Consider a normal form game [26] represented as the tuple, Γ= K, {A k } k K, {u k } k K. Here, K = {1, 2,..., K } is the set of players of the game. The set of actions available for player k is denoted by A k and the utility function associated with each player k by u k. If the set of all available actions for all players is represented by A = A k, then u k : A R. k K A Nash Equilibrium NE for a game is an action pro le from which no player can increase its utility by unilateral deviations. An action pro le, a A, is a NE if and only if u k a u k b k,a k k K,b k A k. Here, b k,a k = a1,...,a k 1,b k,a k+1,...,a K. Nash equilibria form the steady states of the game. Suppose that a normal form game is played repeatedly. At each stage of the game, players choose actions that improve their utility functions in a round-robin fashion. The criteria for a particular choice of action gives rise to the best and better response dynamics de ned below: 1 Best response dynamic: At each stage, player k deviates from a k A k to some action b k A k if u k b k,a k u k c k,a k, c k A k. Note that a NE is an action pro le, a A, such that a k is a best response for every player k K. 2 Better response dynamic: At each stage, player k deviates from a k A k if there exists an action b k A k such that u k b k,a k >u k a k,a k. Two games with the same sets K and A are said to be bestresponse-equivalent if x A and every player k K, the best response set of both games coincide. Two games are said to be better-response-equivalent if x A and every player k K, the better response sets of both games coincide. The convergence properties of best and better response dynamics are identical for games which are better-response-equivalent. The properties of only the best response dynamic are identical for games which are best-response-equivalent. A potential game [23] and [25] is a normal form game such that any changes in the utility function of a player in the game due to a unilateral deviation by the player is re ected in a global function referred to as the potential function. Potential games can be grouped into different types based on the relationship between the potential function and the utility functions of the players in the game. A function V : A R is called 1 an exact potential function if k K, a A and b k A k, u k a u k b k,a k =V a V b k,a k. 2 an ordinal potential function if k K, a A and b k A k, u k a u k b k,a k V a V b k,a k. 3 a best response potential function if k K and a A, arg max b k A k u k b k,a k = arg max b k A k V b k,a k. A game is an exact, ordinal or best response potential game if there exists an exact, ordinal or best response potential function respectively for the game. In addition, a game is a transformable ordinal potential game if there exists a strictly increasing transformation f k : u k A R, k K such that the game Γ= K, {A k } k K, {û k } k K with ûk a = f k u k a is an exact potential game. Consider a coordination game Γ = K, {Ak } k K, {V } k K, where each player s utility function is replaced by the potential function. Exact and ordinal potential games are better-response-equivalent to this coordination game while the best response potential game is best-response-equivalent to this coordination game note that better-response-equivalence implies best-responseequivalence. Hence, in general, improving each user s utility also increases the value of a global potential function making these games easy to analyze. In addition, if the potential function is also a global network performance measure, these games give a framework where users can serve the greater good by following their own best interest, i.e., can maximize a global utility by only trying to maximize their own utilities. Note that since a potential game is best-response-equivalent to its corresponding coordination game any result for the best response dynamic of a coordination game also applies to a potential game. A similar inference holds for better-responseequivalent games. Also, the maximizers local and global of the potential function are NE for the potential game. Note that not all NE are maximizers of the potential function. Saddle points [27] 1 of the potential function could also be NE of the game referred to as suboptimal NE. IV. POTENTIAL GAME FORMULATION FOR IA In this section, we interpret the IA problem as a game. The user-nodes in the network are the players of the game K = {1,..., K}. The transmit waveforms available to the user-nodes A k = S, k K are the action sets. An algorithm or game for IA can be formulated by allowing each user-node to iteratively adapt its waveform such that each adaptation improves better response dynamic or maximizes best response dynamic the utility function associated with the 1 At the saddle points of a function, the slope of the function with respect to each dimension is zero though the points are neither local minima nor maxima of the function. Hence at a saddle point, a change in any single users action while the actions of other users are kept constant does not improve its potential function and correspondingly its utility function.

4 1090 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL 2009 user-node. Unless otherwise mentioned, the initial waveforms for all users are assumed to be randomly selected. We show that for particular choices of utility functions, the IA algorithm forms a potential game which could lead to sequences that maximize the sum-capacity of the network. As mentioned before in Section I, it has been shown in [1] that the optimal signature sequences/waveforms i.e. sequences that achieve the maximum sum capacity also maximize the negated generalized total squared correlation NTSC function de ned as V s = sps T + R zz 2 F. 2 Here, A F = i,j a ij 2 is the Frobenius matrix norm. Distributed algorithms are proposed in [1] and [7], where users iteratively replace their sequence with their corresponding MMSE lter sequence and the inferior eigenvector of the corresponding interference cross correlation matrix, R ii [k] respectively, in a round robin fashion. The MMSE-iterations and the eigen-iterations are shown to increase the NTSC function at each update. This property suggests the existence of utility functions that allow a potential function, given by the NTSC function, for IA. A. IA Algorithm as a Potential Game Consider a network game or an iterative algorithm where the utility function for the k th user is given by u k s k,s k = 2p k s T k R ii [k] s k. 3 Note that the utility function is the negative of a weighted measure of the interference seen at the receiver corresponding to the k th user 2. Expanding the NTSC function de ned in 2, we have V s = sps T + R zz 2 F = Rii [k]+p k s k s T 2 k F = R ii [k] 2 F p2 k 2p ks T k R 4 ii [k] s k. Here, the last step follows from the fact that s T k s k =1. The rst two terms are independent of the choice of signature sequence by the k th user and the last term is the same as the utility function given in 3. If user k replaces its sequence s k by a new sequence ŝ k, the change in potential function is given by V s V ŝ k,s k = R ii [k]+p k s k s T 2 k + R F ii [k]+p k ŝ k ŝ T k = 2p k s T k R ii [k] s k +2p k ŝ T k R ii [k]ŝ k = u k s k,s k u k ŝ k,s k. Therefore, the network game for IA, Γ IA = K, {S} k K, {u k s} k K, is an exact potential game with V s as its exact potential function. This shows that Γ IA is also best and better-response-equivalent to the game Γ IA,co = K, {S} k K, {V s} k K. Note that since R ii [k] is a symmetric positive semide nite matrix, the utility function is a negative weighted Rayleigh quotient of R ii [k]. This is 2 Substituting for R ii [k] in Equation 3, it can be seen that the utility is the weighted sum of the squared correlation of the k th user s sequence with the other user s sequences 2 F 5 maximized by the eigenvector corresponding to the minimum eigenvalue of R ii [k]. Hence the inferior eigenvector of R ii [k] is the best response for a user. Maximizers of the NTSC function V S are also the maximizers of the sum-capacity of the network. Therefore, the IA algorithm with 3 as the utility function could lead to sequences that also maximize the sum capacity of the network. Some other utility functions which allow a potential game formulation with V s as the potential function, and hence which could also be used to form IA algorithms that maximize the sum-capacity of the network, are described below. An algorithm for IA with the properties of a potential game can be constructed using any of these utility functions. The nature of these described utility functions depends upon two factors: receiver type and the user s end performance metric. 1 SINR/Correlator Game: A natural choice for the utility function is the SINR at the output of the correlation receiver given by p k u k s =SINR k s = s T k R. 6 ii [k] s k An ordinal transformation of this utility function is given by û k s = 2p2 k u k s = 2p ks T k R ii [k] s k. 7 This is the same as the utility function for the exact potential game, Γ IA, given in 3. Hence an IA game with this utility function is an ordinal potential game with the NTSC function as the ordinal potential function. This game is thus also betterresponse-equivalent to the coordination game Γ IA,co. 2 MSE/Correlator Game: The minimum mean squared error MMSE at the output of a correlation receiver is MMSE s, α =1 2α p k + α 2 s T k R rr s k. 8 Here, α = p k /s T k R rrs k. A choice for a utility function is as follows: u k s = MMSEs = 1 2 s T k R + rrs k s T k R rrs k = st k R rrs k p k s T k R rrs k p k = st k R ii [k] s k s T k R rrs k. The last step follows from the fact that R rr = R ii [k]+p k s k s T k and s T k s k = 1. Consider the transformation of the utility function given by p k 9 û k s = 2p2 k u k s u k s+1 = 2p ks T k R ii [k] s k. 10 This is an ordinal transformation since the function x/x + 1 is monotonically increasing for x> 1 and 1 <u k s < 0. It is thus seen that the utility function is an ordinal transformation of the utility function for the exact potential game Γ IA. Therefore, this game is also better-response-equivalent to the coordination game Γ IA,co. 3 SINR/MSINR Game: For the MSINR receiver, the linear lter, w k, for the k th user is given by w k =R ii [k] 1 s k. The utility function, given by the SINR at the output of the receiver, is then u k s =p k s T k R 1 ii [k] s k. 11

5 MENON et al.: A GAME-THEORETIC FRAMEWORK FOR INTERFERENCE AVOIDANCE 1091 The k th user s best response is the normalized inferior eigenvector of R ii [k]. Hence the IA game with this utility function is a best response potential game with the NTSC as the best response potential function. This game is thus best-responseequivalent to the coordination game Γ IA,co. 4 MSE/MSINR Game: The MSE at the output of the MSINR receiver is given by 1 u k s = MSE s = p k s T k R 1 ii [k] s k This utility function is again maximized when s k is the inferior eigenvector of R ii [k]. Hence the IA game with this utility function is also best-response-equivalent to the game Γ IA,co. 5 Hybrid Game: It is to be noted that the best response for a user is the inferior eigenvector of R ii [k] for any of the utilities considered in the games discussed above. Hence any combination of the utility functions also results in a best response potential game. This implies that if each user independently chooses its receiver type and performance criteria from the set of discussed utilities without regard to the choice of other users, the resulting game/algorithm is still a best response potential game and is best-response-equivalent to the game Γ IA,co. Note that as mentioned in Section II, the receivers are assumed to have access to the interference cross-correlation matrix this might be obtained by averaging received power measurements over multiple symbol periods. Hence, for each of the described games, the transmit sequence is assumed to be computed at the receiver and then fed-back to the transmitter. The MSINR receivers are also adapted with each transmit sequence adaptation. B. Nash Equilibria of the IA Game The Nash equilibria of a game are, by de nition, the xed points of the game under a best response dynamic. Let the xed points of the IA game, Γ IA = K, {S} k K, {u k s} k K, or equivalently the coordination game, Γ IA,co = K, {S} k K, {V s} k K, under a best response dynamic be given by F Φ,Γ. As mentioned before, the maximizers of the potential function are NE. Since the IA game has a potential function that is continuous, bounded and de ned over a compact set, the potential function is guaranteed to have at least one maximum. Consequently, the game has at least one NE. The following lemma gives a characterization of the Nash equilibria of the game. Lemma 1: Let s k K S. If s F Φ,Γ, then for all k {1,..., K}, s k is an eigenvector of R rr = sps T + R zz. Proof: It is shown in [7] that the xed points of the distributed IA algorithms under eigen-iterations are given by sequences that satisfy Lemma 1. Since the IA game given here is best-response-equivalent to the distributed algorithm in [7], the lemma follows. Theorem-2 from [6] can be adapted to show that the set of eigenvalue con gurations for sequence multisets, s, such that s F Φ,Γ is nite. It is also shown in [6], that all the local maxima of the NTSC function are also global maxima. Hence the xed points of the game consist of globally optimal sequence con gurations that lead to the global maximum value for the potential function given by the NTSC function and some sub-optimal points. The existence of such sub-optimal points with examples is shown in [2]. V. CONVERGENCE OF IA POTENTIAL GAMES This section investigates the convergence of the IA potential game to the NE, identi ed in the previous subsection, under best and better response dynamics. A special case of Zangwill s convergence theorem-c [28], stated below, is used to aid the convergence analysis. Theorem 1: Let the correspondence [29], Φ: X P X, determine an algorithm that given a point x 0 generates a sequence {x t } through the iteration x t+1 Φx t. Let a solution set i.e. the set of xed points of the algorithm, S X be given. Suppose 1 All points {x t } are in a compact set S X. 2 There is a continuous function V : X R such that a If x X is not a solution, then V x >Vx for any x Φx. b If x S, then either the algorithm terminates or for any x Φx, V x V x. 3 Given any convergent subsequence x t x, t T, if x is not { a solution, then there is a convergent subsequence V x t } Note T need not be t T contained in T, such that lim V x t >Vx. t T Then either the recursion x t+1 Φx t arrives at a solution, or the limit of any convergent subsequence of {x t } is a solution is in S. A. Best Response Convergence Theorem 1 can be directly used to show the convergence of the IA potential game, Γ IA,co, to the NE under a best response dynamic. In the context of this game, let Φ represent the map from the signature sequence space S K = S to k K the set of all best responses after one round-robin iteration Refer to Appendix B for a more detailed de nition. Since S is a compact set, S K is also a compact set. Hence all the points sequences generated by the adaptation process are in a compact set and condition 1 of Theorem 1 is satis ed. Let S be formed by the Nash equilibria of the game. By the de nition of a NE, the potential function satis es the properties required of function V. conditions 2.a and 2.b when best response iterations are used. Condition 3 is satis ed as follows: Φ is an upper-semi-continuous u.s.c correspondence Shown in Appendix B. An u.s.c correspondence in a compact space has a closed graph Proposition 9.8 in [29]. Hence the algorithm map, Φ, is closed and therefore, if there exists a sequence x t x, t T, then there exists a sequence x t Φ x, t T and x t Φ Φxt such that x Φx. Since x is not a solution, V x >Vx. Since all the conditions of Theorem 1 are satis ed, the IA potential game, Γ IA,co, converges to the Nash equilibria of the game under a best response dynamic. Note that in the current scenario, the solution set S equivalent to the set of Nash equilibria comprises of all the points to which the algorithm converges. This convergence property also holds for all IA

6 1092 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL 2009 games which are best-response-equivalent and better-responseequivalent to the IA potential game, Γ IA,co. Therefore, an IA algorithm that can be constructed using any of the utility functions described in Section IV converges to the NE for the network, when users follow a best response dynamic. B. Better Response Convergence Better response dynamic refers to an update procedure where the users choose actions that increase their utilities as opposed to maximizing their utilities in the best response dynamic. Theorem 1 can again be used to investigate the convergence of the IA potential game, Γ IA,co, with a better response dynamic. In the context of this game, let Φ represent a map from the signature sequence space S K to the set of all better responses after one round-robin iteration of an ordinal or exact potential game. As before, S K is a compact set and condition 1 is satis ed. Let S the set of all Nash equilibria. By the de nition of the NE and the better response dynamic, the potential function satis es the properties required of function V. in conditions 2.a and 2.b. However, a general better response can generate an in nite sequence of points, which are not a solution and which do not satisfy condition 3. For example, consider a better response scheme in which at the each iteration the scheme dictates players to choose actions, from the set of available better response actions, that improve the potential function in increasingly smaller step sizes. Then, the generated sequence converges to a point. However, the convergent point need not be a NE since there might be an improving action that improves the value of the potential function beyond the value at the convergent point and which cannot be chosen due to the constraints of the better response scheme. The solutions set of the scheme could thus be larger than the set of Nash equilibria i.e. the scheme could converge to sub-optimal xed points which are not NE. Hence additional properties such are required to establish the convergence of a better response scheme to xed points that are also NE. For example, a better response procedure with a nite minimum step size can avoid convergence to the above-mentioned suboptimal xed points which are not NE 3. Alternatively, a random better response procedure can be used. In this scheme, at any speci c stage of the game, a player selects an action with uniform probability from the set of available actions that increase its utility. In the context of this game, let Φ represent a map from the signature sequence space S K to a random better response after one round robin iteration of an ordinal or exact potential game a detailed de nition of a random better response iteration is given in Appendix A.. Again, as before, conditions 1, 2.a and 2.b are satis ed. The following theorem illustrates condition 3. Theorem 2: Consider an ordinal or exact potential game with potential function V and a compact action space. Given a point x 0, let the random better response algorithm generate a sequence {x t }, where x t is the output of the t th round-robin iteration. Then, if there exists a sequence {x n } n N such that 3 We call a xed point sub-optimal if it is not a NE. It should be noted however, that among the xed points that are NE, not all are necessarily optimal NE. x n x, Pr [x S ] = 1, where S is the set of all Nash Equilibria. Proof: Given in Appendix A. It is thus shown that a convergent sequence generated by a random better response dynamic almost surely converges to a NE S for an exact or ordinal potential game. Hence condition 3 is also satis ed for the IA potential game Γ IA,co and the game almost surely converges to the NE while following a random better response dynamic. Note that the convergence property of the random better response dynamic holds for IA games which are better-response-equivalent to Γ IA,co. Therefore, an IA algorithm that can be constructed using the utility functions described in Section IV except the utility functions of games 3 and 4 converges to the NE for the network, when users iteratively update their signature sequences by their respective random better responses. VI. NOISY BEST RESPONSE CONVERGENCE As seen in the previous section, under a best response dynamic, the IA potential game Γ IA,co and therefore any of the IA algorithms that can be constructed using the utility functions in Section IV converge to the NE of the game given by the set of eigenvectors of R rr. However, as mentioned in the end of Subsection IV-B, all the NE are not globally optimal and the game might converge to sub-optimal NE. Empirical results suggest that when the initial sequence are chosen at random, the MMSE iterations a better response adaptation procedure and eigen iterations a best response adaptation procedure always converge to the optimal sequence con gurations [1] and [7] respectively. In [6], it is analytically shown that w.r.t the MMSE iteration, the sub-optimal points are unstable and arbitrary small perturbations can lead the game to a neighborhood of the globally optimal con gurations. A generalization of this result w.r.t best-response iterations including the eigen-iterations is established in this section. It is shown that for a special class of Nash Separable NS IA games, noisy best response iterations asymptotically converge to a neighborhood of an optimal sequence set without class warfare or any other form of coordination. Let Φ: A P A represent a correspondence that maps from an action pro le x A to the set of all possible best responses after one round robin iteration Refer to Appendix B for a more detailed de nition. In a noisy best response iteration NBRI, the best response of each player is perturbed by bounded noise, with bound δ> 0. For each χ A, where A is the action space of the game, let z χ be a random vector with arbitrary joint probability density function p z z; χ, δ. It is required that p z z; χ, δ be positive everywhere on N δ χ and zero outside N δ χ. For instance, p z z; χ, δ could be a uniform distribution on N δ χ. The δ NBRI is de ned below: Given a noise bound δ> 0 and x [0] A, for each round robin iteration, t a positive integer, 1 Choose χ [t] Φx [t] 2 x [t + 1] = z χ [t] Here, χ [t] is the sequence of chosen best responses. The index t N 0 indexes one round robin iteration. Let F Φ,Γ be the xed points for a best response dynamic of Γ or equivalently the Nash Equilibria of Γ. A continuous

7 MENON et al.: A GAME-THEORETIC FRAMEWORK FOR INTERFERENCE AVOIDANCE 1093 function V : A R, on a compact set A is called Nash Separable if: 1 there are no suboptimal local maxima on A. 2 its maximum V max is isolated from the image of other xed points i.e. ɛ m > 0, : N ɛ m V max V F Φ,Γ =Ø. 3 best response iterations are strictly improving in a neighborhood of the maximum i.e. ɛ > 0, with ɛ < ɛ m, Φ V 1 [V max ɛ, V max ] V 1 [V max ɛ, V max ]. A potential game is NS if its potential function is NS. Note that the NS condition results in regions of attraction a region such that if the game enters this region, δ NBRI cannot make it escape from this region around the global maxima of the function. Since the global maxima of a NS function are isolated from each other, the regions of attraction are also isolated from each other. Hence, once a game enters a region of attraction, the player converges to a neighborhood of the corresponding global maxima. The following theorem, a generalization of Theorem 8 in [6], shows that the δ NBRI almost surely leads a NS potential game to this region of attraction. The theorem thus illustrates that even if a NS game has suboptimal NE, arbitrarily small noise will asymptotically ensure the convergence of the game to a neighborhood of the global optima. Theorem 3: Consider a NS potential game with potential function V. Then ɛ > 0, δ 0 > 0, such that δ with 0 < δ < δ 0 and x [0] A, the δ NBRI with iterates {x [t]} obeys lim t inf V x [t] V max ɛ. a.s. Proof: Given in Appendix B. A levelable signal environment is one in which the optimum sequences whiten the spectrum of R rr. This, for example, occurs in an IA system where K N, noise is white and all signals are received with equal power. Another example is a network with K N, noise is white and no user is oversized [4]. Note that a user is called oversized [4] if p i > k K:p k <p i p k N k K:p k p i The following theorem shows that an IA-game in a levelable signal environment is NS. Consequently, the the δ NBRI almost surely converges to globally optimal con gurations in networks with levelable signal environments. Theorem 4: The IA game, Γ IA, or equivalently the IA coordination game, Γ IA,co, in a levelable signal environment is NS [30]. Proof: A levelable signal environment is one in which the optimum sequences whiten the spectrum of R rr. This, for example, occurs in an IA system where K = N and all signals are received with equal power in AWGN. In [6], it is shown that V s has no sub-optimal local maxima i.e. all local maxima are also global maxima. Also, F Φ,Γ is shown to be a nite set [6]. Hence V F Φ,Γ is also a nite set. This fact can be used to choose a 0 <ɛ m < min k K p2 k /2 such that s F Φ,Γ implies V s = V max on W ɛm = V 1 N ɛm V max. Hence conditions 1 and 2 for a NS game is satis ed. Given, 0 < ɛ < ɛ m and Wɛ = V 1 Nɛ V max, assume there is a non-improving sequence of best responses in a round-robin round. This occurs if s Wɛ and inf V Φ s = V s. Since s Wɛ, V s <V max. Note that s/ F Φ,Γ by choice of ɛ m. Hence, j K such that V Φ j s >Vs. Here, Φ j. is the best response of player j. Choose the smallest such j. This occurs when player j s response is blocked by a player k < j who changes its response. Consider the smallest such k. However, by choice of j, player k s signature was already a best response. Hence the two smallest eigenvalues of R ii [k] are identical and the inferior eigenspace of R ii [k] has a dimension of at least 2. Let λ R ii [k] = [λ 1,...,λ N ] where λ 1 λ 2... λ N. Also let λ m = λ N = λ N 1. Let s S K be an optimal signature set. Since the IA game is levelable, λ > 0 such that Rrr = s p s T + R zz = λi. Thus, V max V s = Rrr 2 F + sps T s p s T + Rrr 2 F = sps T s p s T 2 F +2λTr { sps T s p s T } = Rii [k]+p k s k s T k λi 2 F. 14 Using Fact in [31], V max V s p2 k n=n 2 n=1 λ n λ 2 +λ m + p k λ 2 +λ m λ > ɛ. This is a contradiction by initial choice of ɛ m. Hence there exists a neighborhood around the maximum of the potential function in which the best response iterations are strictly improving. VII. REDUCED FEEDBACK SCHEMES BASED ON BETTER RESPONSE CONVERGENCE Note that the implementation of the IA potential game Γ IA,co or equivalently any of the IA algorithms that can be constructed using the utility functions in Section IV under a best response dynamic requires knowledge of the covariance matrix, R ii [k] at the adapting node since the best response for a user is the inferior eigenvector of R ii [k]. However, R ii [k] is only available at the receiver. Hence, the signature sequence has to computed at the receiver and then fed back to the transmitter. Since the signature sequence is real-valued, this feedback introduces a signi cant overhead in the network. In this section, we exploit the better response convergence properties for exact and ordinal potential games, derived in Section V, to design two reduced feedback schemes for IA. These schemes are applicable to IA games or algorithms that are better-response-equivalent to the coordination game Γ IA,co. Note that we consider a synchronous network here. However, similar to the analysis in [11], the potential game formulation for IA and, consequently, the reduced feedback schemes can be easily adapted to more practical asynchronous networks albeit with more notational complexity. A. Random Better Response Consider the exact or ordinal potential IA games described in Section IV. In a random better response update procedure,

8 1094 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL 2009 a user randomly chooses a signature sequence. The receiver indicates if the signature sequence increases the SINR of the user. The user reverts back to the old sequence if it does not. This process is repeated iteratively for each user in a round robin manner until convergence is achieved. This random better response procedure is assured to converge to a NE of the ordinal or exact potential game by the arguments described in Section V-B. However, since the sequence space for the game is large, convergence could be very slow. Convergence speed can be increased by having a directed better response scheme as described in the next subsection. B. Gradient-Based Better Response In this update scheme, the receiver nds the q q {1,..., N} dimensions in which the gradient of the utility function has the largest magnitude. The variable q can be used to control the amount of feedback from the receiver to the transmitter. The receiver then nds the step size λ that maximizes the utility function in the direction speci ed by the q chosen dimensions with the largest magnitude referred to here as the ascent direction. For example, if the gradient of the utility function is a 4-dimensional vector given by d = [ ] T and q =2, the receiver nds the optimum step size along the ascent direction d q = [ ] T. Note that the scheme allows the non-zero dimensions in the ascent direction to change at each iteration according to the interference perceived at the user. To illustrate the scheme, consider the IA algorithm with the utility function speci ed by 3. The utility function is modi ed below to incorporate the constraint that the signature sequences have unit norm. u k s k,s k = st k R ii[k]s k s T k s. 16 k Note that this does not change the value of the utility function for a given unit norm signature sequence. The gradient of the utility function is given by s T k s k Rii [k]s k 2 s T k R ii[k]s k sk du k s k,s k = 2 ds k s T 2. k s k 17 Let d q be the ascent direction. The optimum step-size in this direction is given by the solution to the following optimization problem: max s k + λd q T R ii [k]s k + λd q λ 0 s k + λd q T. 18 s k + λd q The optimal step-size can be computed by a simple line search procedure. By the arguments in subsection V-B, the gradient-based better response algorithm converges. However, as mentioned before, the set of xed points of the algorithm might be larger than the set of NE. The xed points of the algorithm are characterized as follows: The utility function for each user is bounded. Hence, in each iteration, the gradient of the utility function, du ks k,s k ds k, is zero in the ascent dimension. Since dimensions with the largest gradients are chosen for the du ascent direction in each iteration, k s k,s k ds k is zero at the convergence point. However, from 17, it can be observed that du ks k,s k ds k is zero for any eigenvector of R ii [k] and not just for the inferior eigenvector of R ii [k]. Hence, the iterations could lead to xed points where the signature sequence of a user is not the inferior eigenvector of its interference-plusnoise cross-correlation matrix, i.e. xed points that are not a NE. An example scenario where this occurs is where all users start adapting from the same initial sequence none of the users adapt to a different sequence and the initial sequence set of identical sequences is the xed point. To steer the gradient-based iterations from these sub-optimal xed points and to aid convergence to the Nash equilibria of the game, a random better response spacer step is added. If the receiver notices that the convergent sequence is not the minimum eigenvector of R ii [k], it instructs the transmitter to randomly choose a signature sequence that improves utility or equivalently its SINR at the receiver. NE are the only xed points of a random better response as shown in Section V-B. Hence the gradient based algorithm with the spacer steps also theoretically converges to the NE. It has however been noticed empirically that, when starting from random initial sequences, users always converge to the optimal NE even without the spacer step. Hence, the algorithm should ideally be implemented such that users start from random initial sequences. A random spacer step can be used if this cannot be ensured. Fig. 1 illustrates the convergence of the gradient-based better response scheme with q = 1 using two example network scenarios. All the users are assumed to have an equal transmit power of 1 watt and to transmit in an AWGN channel with SNR of 10dB. Each iteration corresponds to a complete round-robin update by all the users in the system. The gure also includes plots for schemes where the feedback values are quantized to 6-bits. Note that the curves with just 6-bit quantization closely follow the curves with nonquantized feedback. It is seen that when 10users share 6 signal dimensions, the best response takes 2 iterations to converge and the better response scheme takes 4 iterations to converge. Let r be the number of bits used to quantize a real value r =6is the gure. Then the number of bits required to be fed back per transmit-receive pair is 2 6r = 12r feedback in the network is 12r 10 = 120r for the best response scheme. For the better response scheme, the descent direction and the step size need to be fedback. Therefore the number of bits to be fedback per transmit-receive pair is 4 q + 1 r =8r feedback in the network is 8r 10 = 80r. Similarly, when 20 users share 12 signal dimensions, the number of bits to be fed back per user transmit-receive pair is 2 12r = 24r feedback in the network is 24r 20 = 480r in the case of the best response adaptation scheme and 7 q+1 r = 14r feedback in the network is 14r 20 = 240r in the case of the better response adaptation scheme. Hence the better response scheme results in considerable reduction of the adaptation overhead in the network. It is to be noted that the speed of convergence of the better response scheme depends upon the number of available signal dimensions and the number of users. However, it has been observed that in general, the increase in the number of feedback bits due to the increase in the number of iterations required for convergence for the better response scheme is less than the increase in the number of feedback bits due to the increase in the number of signal dimensions for the best

9 MENON et al.: A GAME-THEORETIC FRAMEWORK FOR INTERFERENCE AVOIDANCE 1095 Potential Function Users 6 Dimensions 20 Users 12 Dimensions Best response Quantized 6 bits/feedback value best response Gradient based better response 1 dimension Quantized 6 bits/feedback value gradient based better response 1 dimension Number of Iterations Fig. 1. IA with gradient-based better response. Channel is additive white Gaussian noise with SNR = 10dB. Potential Function NTSC were then investigated and two new results regarding the convergence of noisy best response iterations and random better response iterations were established. The rst result coupled with the potential game formulation was used to show that for a large class of co-located receiver network scenarios, arbitrarily small noise assures the convergence of best response IA algorithms, including the eigen iterations investigated in [7], to an arbitrarily small neighborhood of the globally optimal signature sequence set. The second result proved that random better response iterations converged almost surely to the Nash equilibria in exact and ordinal potential games. This result is especially useful since it can be leveraged to construct practical distributed implementations of IA that require limited feedback in the network and that converge to desirable network solutions. Two speci c reduced feedback schemes that illustrate this were also constructed and described in the paper. We considered a synchronous system in our analysis. However, similar to the analysis in [11], the game theoretic framework for IA including the reduced feedback schemes can easily be adapted to asynchronous systems. Until now the majority of the research on IA has focused on networks with centralized or co-located receivers. Direct extensions of IA techniques to networks with distributed receivers do not lead to convergent solutions due to the asymmetry of the mutual interference between users at different receivers. It is believed that game theory, in general, and potential games, in particular, could aid in the design of IA techniques for these de-centralized networks [32]. This paper provides a starting point for such a development Best response Gradient based better 1 dimension Gradient based better 3 dimension Gradient based better 6 dimension Number of Iterations Fig. 2. IA with gradient-based better response and different values for q. Network has 20 users sharing 12 signal dimensions. Channel is additive white Gaussian noise with SNR = 10dB. response scheme. Fig. 2 shows the convergence of the better response scheme for different values of q. It is seen that increasing q decreases the number of iterations required for convergence. However, increasing q also increases the feedback in each iteration. Hence, the optimal q for a given network size can be found by optimizing for the minimum amount of feedback in the network and the required convergence time for the network. VIII. CONCLUSIONS We analyzed the interference avoidance IA problem in a network with co-located receivers using a potential game model in this paper. This formulation allowed us to identify various utility functions for users based on performance metric and receiver type that allow the design of new distributed and convergent IA algorithms including algorithms where users have non-identical utility functions. Convergence properties of the best and better response iterations of potential games APPENDIX A SOME DEFINITIONS AND PROOF OF THEOREM 2 Since the potential function and utilities discussed in our paper could result in multiple best and better responses, all possible choices must be considered while establishing convergence results independent of the choices. To aid this approach, a discussion on set-valued functions or correspondences [26] is rst presented. A correspondence from metric space A to metric set S is a mapping, Φ: A P S, where P S denotes the power set of S. A correspondence is compact valued if the set, Φx, is compact for every x A. A correspondence is upper-semicontinuous u.s.c if for a given point x A, for every open neighborhood of Φx, Θ, there is an open neighborhood of x, U, such that ΦU Θ. The nite sum and composition of compact-valued u.s.c correspondences is compact-valued u.s.c. Also, the image of a compact set under a compact-valued u.s.c. correspondence is compact, and the pre-image of open sets are open. Consider an ordinal or exact potential game with potential function V. The set of better responses for player k, k K, is the correspondence C k : A P A k, C k x ={x k : x k A k & V x k,x k >Vx} 19 The random better response iteration for player k is the mapping Ψ k : A P A, Ψ k x =x k,x k, 20

10 1096 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL 2009 where x k is chosen with uniform probability from the set Ck x Actually, the theorem holds for any continuous probability distribution which has support Ck x.. The composite better response iteration is the composition of all player s better response iterations after one round-robin iteration, Ψx = Ψ K...Ψ 2 Ψ 1 x... Proof of Theorem 2 We will show that Pr [x / S ] = 0. Suppose that x/ S. Then there exists at least one player i with an action x i such that V x i,x i >Vx. Let ɛ = V x i,x i V x. By theorem 4.19 in [33], a continuous function in a compact metric space is also uniformly continuous. Since the parameter space we consider is compact, the continuous function V. is also a uniformly continuous function. Then, from the de nition of a uniformly continuous function, there exists a δ such that d V y, V z < ɛ for all y, z such that d y, z <δ. x n is the action-pro le at the end of the n th round-robin iteration. The round-robin iteration is de ned such that the j th player takes the j th turn to update. Let x n be the action pro le in the n + 1 th iteration after the i 1 th player has updated. Then x n = [ x n+1 1,..., x n+1 i 1,xn i,xn i+1 K],...,xn. At the n th iteration player i updates to an action x i if V x i, x i n >V x n i, x i n. Since x n x, there exists a N such that for all n > N, d x n,x < δ 2K. It follows that d x n+1,x < δ 2K. Consequently, d xn k,x k < δ 2K ; k {1,..., K} and d x n+1 k,x k < δ 2K ; k {1,..., K}. Therefore, d x n i,x i < δ 2. Now, for all n>n and all x i such that d x i,x i < δ 2, we have, d x i, x n i d x i, x n i, < δ 2 + δ 2 = δ., x i,x i x i, x n i + d x i, x n i, x i,x i 21 Therefore, d V x i, xn i,v x i,x i < ɛ. Thus V x i, xn i > V x. It follows that for all n > N, there exists a neighborhood around x i in which V x i, xn i >Vx. Let µ be the measure of the ball of radius δ 2 around x i and let µ tot be the full measure of player i s strategy space. Then for each n > N when i makes a choice, the player will choose to move to a point x with V x >Vx with probability > µ µ tot = p. Given the present position, each of the choices is independent of the others The system evolves as a Markov Chain.. Therefore as x n x, p = 22 n>n Pr V x n >Vx > n>n By using the second Borel-Cantelli lemma, Pr V x n >Vx = 1. This contradicts the fact that x n x. Hence the probability that x n x where x/ S is zero. Therefore, if x n x, Pr [x S ] = 1. APPENDIX B SOME DEFINITIONS AND PROOF OF THEOREM 3 The set of best responses for player, k K, is the correspondence Dk : A k P A k, Dk x k = arg max V x k,x k. 23 x k A k The best response iteration for player k is the correspondence Φ k x = Dk x k,x k. The composite best response iteration is the composition of all players best responses after one round-robin iteration, Φx =Φ K...Φ 2 Φ 1 x... For every k K, V is a continuous function on the compact set A k, Dk x k Ø k K, x k A k. The maximum theorem [29] states that for every k K, Dk is a compactvalued u.s.c. correspondence. Hence, so are {Φ k } and Φ. Proof of Theorem 3 Since V is a continuous function on a compact set, there exists V min = min V x and V max = max V x. By the fact x A x A that V is NS, an ɛ 0 can be chosen such that Nɛ 0 V max V F Φ,Γ =Ø. Without loss of generality, we can choose ɛ in the statement of this theorem to satisfy ɛ<ɛ 0. Let W + = V 1 [V min,v max ɛ], W 0 = V 1 [V max ɛ, V max ]. 24 Note that since W + and W 0 are the pre-image of a closed set with respect to a continuous function, they are both closed. Since they are also contained in a compact set A, they are compact. If W + =Ø, the theorem hold trivially. Hence in the rest of the proof we will consider W + Ø. Now we seek a noise bound δ 0 > 0, such that if x W 0, N δ0 Φ x W 0. This can be done by choosing δ 0 = min {d x, x :x W +,x ΦW 0 }. 25 This minimum exists since d is continuous on the compact set, W + ΦW 0. Note that since Φ is a compact valued u.s.c correspondence, ΦW 0 is compact. Also δ 0 0, since by condition 3 for Nash separability, ΦW 0 W 0 and by de nition, ΦW 0 W + =Ø; hence δ 0 is the minimum distance between two non-empty, non-overlapping compact sets. Now choose δ 0,δ 0. By choice of δ 0, once the δ NBRI enters W 0, it cannot escape. We now show that outside of W 0, the NBRI increases the potential function by a xed nite amount γ with nite probability. Towards this end, let the function, β : A R, be de ned as follows: { } β x = max V x 0 :x 0 N δ x. 26 Note that this maximum exists because V is a continuous function on a compact set. Moreover, by the maximum theorem [29], β is a continuous function. Hence V β attains the following minimum on the compact set W + : γ = 1 2 min {V x β x :x W +}. 27 Note that γ> 0, since if γ =0, then there is a x 0 with a neighborhood N δ x 0 such that V x <Vx 0, x N δ x 0

11 MENON et al.: A GAME-THEORETIC FRAMEWORK FOR INTERFERENCE AVOIDANCE 1097 by de nition 26. This shows that x 0 is a local maximum. However, since V is NS, all local maxima are global maxima and V x 0 =V max which contradicts the fact that x 0 W +. Now given x A and χ Φx, de ne B x, χ = Pr V z χ >Vx+γ z χ W B is a continuous function of x and χ since all primitives are continuous. Let B : A [0, 1] be de ned as B x = B x, Φx. This is a compact valued u.s.c. correspondence. We claim that B W + 0, 1]. Let x W +, χ Φx, and p = B x, χ. First consider χ W +. By the de nition of γ, choose χ N δ χ such that V χ > 2γ + V χ. By the continuity of V, choose δ > 0 such that V χ > γ + V x, χ N δ χ N δ χ. Since p z z; χ is non-zero almost everywhere on N δ χ N δ χ Ø, p Pr V z χ >Vx+γ Pr z χ N δ χ N δ χ > 0. Now suppose χ W 0. By the continuity of V, there exists a neighborhood N δ χ with δ > 0 such that N δ χ W 0 Ø. Since p z z; χ is nonzero everywhere on N δ χ W 0 Ø, p Pr z χ W 0 Pr z χ N δ χ > 0. Now since B x is a compact valued u.s.c., B W + is also compact. So, there exists a p = min B W + and by the previous discussion, p>0. We have thus shown until now that Pr V z χ >Vx+γ z χ W 0 29 p, x A, χ Φx. We now apply this to the NBRI. Let Q = Vmax ɛ V min γ and let V q = { [V max ɛ, V max] q =0 [V min + γ Q q,v max ɛ q =1 [V min + γ Q q,v min + γ Q q q Q 30 {V q } thus partitions V A. Let the state of the δ NBRI for iteration number t be de ned as σ [t] = {q : V x [t] V q }. 31 Here, x [t] A. At time t +1, let x t + 1 = z χ [t], where χ [t] Φx [t]. Let G t denote the event σ [t + 1] <σ [t] and E t denote the event σ [t] = 0. Since G t is determined only by knowledge of x [t] and χ [t], we have Pr G t µ1 l=1 G t l, E t µ2,x[t],χ[t] 32 = Pr G t x [t],χ[t] p, µ 1,µ 2 > 0. where the inequality follows from 29. Moreover, since 32 holds for arbitrary x [t] and χ [t], Pr G t µ1 l=1 G t l, E t µ2 p, µ1,µ 2 > The iterations numbers can be re-written using some m R and Q as Pr G Qm+q q 1 l=0 G Qm+l, E Qm p, 0 q Q Therefore the probability that there are Q consecutive iterations that decrease the state of the system is given by Pr Q 1 q=0 G Qm+q E Qm p Q. 35 However, if Q consecutive iterations decrease the state of the system starting from iteration number Qm, then the state of the system at iteration number Qm + 1 is given by σ [Qm + 1] = 0 implying E Qm+1. Thus Pr E Qm+1 E Qm p Q. 36 Also, we showed that if x [t] W 0 i.e. E Qm, x [t] W 0 ; t > Qm. Thus Pr E Qm+1 E Qm =1. Therefore, Pr E Qm+1 = Pr E Qm+1 E Qm Pr EQm + Pr E Qm+1 E Qm Pr EQm 1 1 Pr E Qm + p Q Pr E Qm. 37 The probability of event E Qm+1 is thus given by By induction, Pr E Qm+1 Pr EQm 1 p Q. 38 Pr E Qm+1 Pr E0 1 p Q m = 1 p Q m. 39 As m, 1 p Q m 0. Thus lim σ [t] = 0 and t lim inf V x [t] V max ɛ. 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MacKenzie, J. Neel, and J. Reed, A game theory perspective on interference avoidance, in Proc. IEEE Global Telecom. Conf., Nov./Dec. 2004, vol. 1, pp [31] D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton, NJ: Princeton University Press, [32] R. Menon, A. B. MacKenzie, R. M. Buehrer, and J. H. Reed, A gametheoretic framework for interference avoidance in ad-hoc networks, in Proc. IEEE Global Telecom. Conf., Nov [33] W. Rudin, Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, Rekha Menon received her B.E degree from REC, Trichy in 2000 and her M.S and Ph.D. degrees from Virginia Tech in 2003 and 2007 respectively. While at Virginia Tech, she was part of Wireless@Virginia Tech and worked on various projects involving game theory and software/cognitive radios. She also interned with the RF and Network Operations Division of Cingular Wireless in Hanover, MD. She is currently working as a DSP and systems engineer at Tyco Electronics in Lynchburg, VA. Her research interests include dynamic spectrum sharing, cognitive radio, Multiple Input Multiple Output MIMO systems, and Ultra Wideband. Allen B. MacKenzie has been an Assistant Professor in Virginia Tech s Bradley Department of Electrical and Computer Engineering since He joined Virginia Tech after receiving his Ph.D. from Cornell University and his B.Eng. from Vanderbilt University, both in Electrical Engineering. Dr. MacKenzie s research focuses on wireless communications systems and networks. His current research interests include cognitive radio and cognitive network algorithms, architectures, and protocols and the analysis of such systems and networks using game theory. In addition to the IEEE, Dr. MacKenzie is a member of the ASEE and the ACM. In 2006, he received the Dean s Award for Outstanding New Assistant Professor in the College of Engineering at Virginia Tech. James Hicks S 93-M 04 received the B.S.E.C.E. degree from George Mason University, Fairfax, Virginia, in He studied as a Bradley Fellow at Virginia Polytechnic Institute and State University in Blacksburg, Virginia, where he received the M.S.E.E., M.S. MATH, and Ph.D.E.E. degrees in 2000, 2004, and 2003, respectively. In 2004, he received the Motorola University Partners in Research UPR award. He has been an Engineering Specialist with The Aerospace Corporation, Chantilly, VA since His research interests are interference avoidance and interference rejection. R. Michael Buehrer joined Virginia Tech from Bell Labs as an Assistant Professor with the Bradley Department of Electrical Engineering in He is currently an Associate Professor and is a member of Wireless@Virginia Tech, a comprehensive research group focusing on wireless communications. His current research interests include dynamic spectrum sharing, cognitive radio, Multiple Input Multiple Output MIMO communications, intelligent antenna techniques, position location networks, Ultra Wideband, spread spectrum, interference avoidance, and propagation modeling. His work has been funded by the National Science Foundation, the Defense Advanced Research Projects Agency, the Of ce of Naval Research, and several industrial sponsors. Dr. Buehrer has co-authored over 30 journal and 75 conference papers and holds 11 patents in the area of wireless communications. He is currently a Senior Member of IEEE, and an Associate Editor for the IEEE TRANS- ACTIONS ON WIRELESS COMMUNICATIONS and the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. In 2003 he was named Outstanding New Assistant Professor by the Virginia Tech College of Engineering. Jeffrey H. Reed is the Willis G. Worcester Professor in the Bradley Department of Electrical and Computer Engineering. From June 2000 to June 2002, Dr. Reed served as Director of the Mobile and Portable Radio Research Group MPRG. He currently serves as Director of the newly formed umbrella wireless organization Wireless@Virginia Tech. Dr. Reed s area of expertise is in software radios, cognitive radio, wireless networks, and communications signal processing. His book, Software Radio: A Modern Approach to Radio Design was published by Prentice Hall in May His latest book, An Introduction to Ultra Wideband Communication Systems was published by Prentice Hall in April Dr. Reed received the College of Engineering Award for Excellence in Research in In 2005, Dr. Reed became a Fellow of the IEEE for contributions to software radio and communications signal processing and for leadership in engineering education.