ADENSE wireless network is defined to be a wireless network

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1 1094 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 16, NO 5, OCTOBER 2008 The Analysis of Nash Equilibria of the One-Shot Random-Access Game for Wireless Networks and the Behavior of Selfish Nodes Hazer Inaltekin, Member, IEEE, and Stephen B Wicker, Senior Member, IEEE Abstract We address the fundamental question of whether or not there exist stable operating points in a network in which selfish nodes share a common channel, and if they exist, how the nodes behave at these stable operating points We begin with a wireless communication network in which nodes (agents), which might have different utility functions, contend for access on a common, wireless communication channel We characterize this distributed multiple-access problem in terms of a one-shot random-access game, and then analyze the behavior of the nodes using the tools of game theory We give necessary and sufficient conditions on nodes for the complete characterization of the Nash equilibria of this game for all 2 We show that all centrally controlled optimal solutions are a subset of this game theoretic solution, and almost all (wrt Lebesgue measure) transmission probability assignments chosen by a central authority are supported by the game theoretic solution We analyze the behavior of the network throughput at Nash equilibria as a function of the costs of the transmitters incurred by failed transmissions Finally, we conclude the paper with the asymptotic analysis of the system as the number of transmitters goes to infinity We show that the asymptotic distribution of the packet arrivals converges in distribution to a Poisson random variable, and the channel throughput converges to ( (1 + )) ln( (1 + )) with 0 being the cost of failed transmissions We also give the best possible bounds on the rates of convergence of the packet arrival distribution and the channel throughput Index Terms Channel throughput, game theory, Nash equilibrium, random access control, slotted ALOHA I INTRODUCTION ADENSE wireless network is defined to be a wireless network whose domain is fixed but the number of nodes lying inside the network domain approaches infinity Mainstream ongoing research on this type of networks can be categorized into two basic areas: 1) characterizing the upper and lower bounds on capacity as a function of and 2) developing scalable network protocols Papers such as [1] and [2], follow the first mainstream, while [3] and [4], follow the second one Manuscript received November 11, 2005; revised April 4, 2007 First published March 12, 2008; current version published October 15, 2008 Approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor R Mazumdar This work was supported in part by the National Science Foundation (NSF) TRUST Science and Technology Center and the NSF Nets-NOSS and ITR programs H Inaltekin is with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY USA and also with the Department of Electrical Engineering, Princeton University, Princeton, NJ USA ( hi27@ececornelledu; hinaltek@princetonedu) S B Wicker is with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY USA ( wicker@ececornelledu) Digital Object Identifier /TNET There are two types of bottlenecks associated with dense wireless networks The first bottleneck arises from their physical limitations When a node transmits, it causes interference to the other nodes in the network This interference, in turn, hinders the nodes in the vicinity of the transmitter from reception As a result, if the network domain is fixed, per-node throughput goes to zero as In [1], Gupta and Kumar showed that there exist two positive constants such that the wireless network capacity lies in the closed interval The works [1] and [2] determine the rate at which the network capacity goes to zero as a function of The second type of bottleneck derives from the need for scalable protocols that show little performance degradation as It is almost impossible to have a protocol which exhibits equally good performance in terms of a chosen metric for all values of Thus, it turns out that different protocols and rules should be imposed on nodes for different values of node density In addition, it is hard to guarantee that all the nodes in a dense wireless network run the same algorithm because users might be willing to modify their communication nodes in order to improve their network performance This might lead to a further degradation in the overall network performance The works [3] and [4] propose some scalable routing strategies, but they assume that all the nodes run the same algorithm In this work, we focus primarily on the second type of bottleneck We push all of the decision making mechanism to the individual nodes, and consider them as selfish agents who are trying to improve their network performance The only assumption we make about the nodes is that they are rational and intelligent entities A rational node is aware of its alternatives, forms expectations about unknown parameters, and chooses its action after some process of optimization to maximize its expected utility An intelligent node can analyze a conflict situation and takes its knowledge or expectations of other nodes behavior into account while making a decision The tool that we chose to analyze the wireless networks with selfish nodes is game theory Game theory is the formal analysis of strategic settings in which all the agents nvolved are intelligent, rational decision-makers The involved questions that this work addresses with the help of game theory are whether or not there exist stable operating points of the network at which all selfish nodes agree to operate, and how the selfish transmitters behave at these stable operating points for any Game theory has an important role to play in design of large networks in general, and that of sensor networks in particular (see [5]) Game theory and the related field of mechanism design bring key tools into play equilibrium concepts, utility functions, and signaling/bargaining that are critical to the design /$ IEEE

2 INALTEKIN AND WICKER: ANALYSIS OF NASH EQUILIBRIA OF THE ONE-SHOT RANDOM-ACCESS GAME 1095 of distributed network control algorithms At its core, such design begins with enabling local decisions based on (potentially) local and incomplete information Game theory enables solutions through the association of utility functions with constraints on local operation, such as power, connectivity requirements, and so forth The resultant decision-making process, aided and abetted by local signaling and data acquisition, may then lead to desirable global behavior, with verification of such behavior through equilibrium analyses We acknowledge that issues remain with this approach, including speed of convergence toward equilibria and the selection of equilibria, but we contend that the potential for this approach is clear, and is slowly but surely being actualized Results of the paper can be considered as characterizing the steady state local behavior of wireless networks containing selfish nodes and using a collision channel model in the medium access multiplexing (MAC) layer The collision channel model was extensively used in the past (eg, [6] [8]), and it is very appropriate to characterize the behavior of networks using no power control and containing nodes with single packet detection capabilities In a more realistic scenario, it is expected that signal-to-interference-plus-noise ratio determines whether a packet is successful or failed We are planning to consider such an extension by taking the location information of the nodes and the capture effect into account in our future work A Related Work Recently, there has been an intensified research interest in the game theoretical modeling of wireless networks such as [9] [11] The first reason is the excessive proliferation of mobile users and the resulting difficulties encountered in their regulation process The second is the distributed and scalable nature of the solutions obtained as a result of the game theoretic analysis The last but not least reason is the fairly general abstractions of game theory characterizing many real-life situations including issues naturally arising in wireless networking Saraydar et al [9] analyze the pricing strategies for power control in a multicell wireless data network They show that the solutions reached as a result of the distributed selfish optimization of the user utilities are Pareto efficient Cagalj et al [10] address the selfish behavior in the CSMA/CA MAC protocol They derive the conditions for the optimal functioning of the selfish transmitters in CSMA/CA They also propose an algorithm which guides selfish nodes to a Pareto-efficient Nash equilibrium Two closely related work are [11] and [12] MacKenzie and Wicker [11] model slotted ALOHA with selfish users by using repeated games They only consider the homogeneous case where all nodes are identical, and prove that there exists a symmetric Nash equilibrium Altman et al [12] give a similar game theoretic analysis of the slotted ALOHA However, they also assume that all the nodes are identical, and only consider the symmetric Nash equilibrium Furthermore, there is no formal existence proof of the Nash equilibrium in [12] They only perform some numerical analysis In contrast to them, in this work, we neither assume nodes are identical nor restrict ourselves only to symmetric Nash equilibrium We consider heterogeneous networks and completely characterize the Nash equilibria of our random-access game for heterogeneous networks, which has not been done up until now to the best of our knowledge Moreover, we give a very careful, thorough, and formal analysis of the channel throughput and the asymptotic distribution of packet arrivals including bounds on the rate of their convergence Such an analysis cannot be found in any other work either In addition, we also analyze the relation between centrally optimal and game theoretic solutions, and prove that all centrally optimal solutions are contained in our game theoretic solution B Key Results We will first establish the Nash equilibria of the one-shot random-access game, and then concentrate on its asymptotic properties as goes to infinity Here,, is the set of selfish transmitters with cardinality, and is the cost of unsuccessful transmission by transmitter Our main contributions can be summarized as follows 1) We give the necessary and sufficient condition on selfish nodes that enables us to completely characterize the Nash equilibria of the one-shot random-access game In particular, when nodes are identical (ie, ), there exists a fully mixed Nash equilibrium (FMNE), which turns out to be the unique focal equilibrium of the game 2) We show that all centrally controlled solutions are a subset of the game theoretic solution Moreover, almost all transmission probability assignments chosen by a central authority are supported by the game theoretic solution 3) For any Nash equilibrium at which only the nodes in contend for the access of the common wireless channel, we characterize the behavior of the network throughput at this Nash equilibrium for all fixed as a function of 4) In the homogeneous case (ie, for all ), we prove that the asymptotic distribution of the packet arrivals to the common wireless channel converges to a Poisson distributed random variable with mean for any sequence of Nash equilibria at which the number of transmitters contending for the access on common wireless channel goes to infinity Moreover, the best possible achievable convergence rate is given by 5) In the homogeneous case, we show that the channel throughput at any with converges to uniformly at a rate, which is also the best possible achievable convergence rate for any A practical implication of the result 2) is that it is always possible to drive a system of composing selfish agents to the desired operating point by manipulating their perceived utilities This manipulation can be in the form of providing monetary incentives and discouragements for the selfish nodes to increase or to decrease their costs per each failed packet Results 4) and 5) may look like counterintuitive in the first sight Due to total anarchy in the system and the complete incoordination among the transmitters, it might be expected that total number of packet arrivals should diverge to infinity and the channel throughput should converge to zero as the number of selfish agents competing for the channel access goes to infinity However, our analysis reveals that this is not true Selfish transmitters are intelligent enough that they are aware of the fact that sustaining unnecessarily high transmission probabilities will result in the collapse of the system, and nobody will be able to successfully deliver this packet over the wireless channel As a result, they

3 1096 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 16, NO 5, OCTOBER 2008 adjust their transmission probabilities so that they always obtain a positive channel throughput and a stable distribution for the total number of packet arrivals C Note on Notation When we write, we mean a sequence of Nash equilibria at which there are transmitters contending for the access on a common wireless channel With a slight abuse of notation, will also represent the set of these transmitters Apart from this, we use calligraphic letters to indicate a set We reserve to represent the number of transmitters involved in Again with a slight abuse of notation, will be the fully mixed Nash equilibrium FMNE of By FMNE, we mean the Nash equilibrium at which all nodes have positive probability of transmission denotes the set of transmitters playing is the 0 1 random variable showing the action chosen by the transmitter when is played 0 means backoff, and 1 means transmit and represent the total number of packet arrivals at and, respectively is the channel throughput (ie, probability of having only one packet arrival) at when nodes have costs of unsuccessful transmission and indicate the Poisson distribution with mean and 0 1 Bernoulli distribution with mean, respectively as well as the generic random variables with these distributions 1 All of the random variables of the paper are independent unless otherwise stated D Organization of the Paper The rest of the paper is organized as follows In Section II, we first describe our communication problem, and then explain our game theoretic solution to it We establish the Nash equilibria of the random-access game in Section III In this part, we first discuss the two-node case by establishing the Nash equilibria and characterizing the channel throughput at FMNE We then extend these results to the -node case In Section IV, we turn our attention to the asymptotic properties of the system as the number of selfish transmitters grow without bound In this section, we identify the asymptotic behavior of the total number of packet arrivals and the channel throughput Section V concludes the paper II PROBLEM DESCRIPTION We consider a wireless network consisting of a set of transmitter nodes and a set of receiver nodes Each transmitter node wants to communicate with a receiver node through the common wireless channel (Fig 1) For example, these transmitter nodes can be the mobile users of a cellular network requesting uplink reservation to communicate with a base station In this particular case, the set of receiver nodes only consists of the base station More generally, if there are some nodes which are close to one another in a wireless ad-hoc network and willing to communicate with another close-by node, our game theoretic approach will also model their behavior In this case, receiver nodes are close enough to one another so that only one of them can receive successfully at a time Therefore, results of this paper can be considered as, in some sense, characterizing the behavior of selfish nodes locally in a general wireless network 1 We assume there is a common probability space (; F ;P) on which all of the random variables are defined Fig 1 n transmitter nodes contending for communication with a receiver node over a common wireless channel We assume that there is always a packet waiting for transmission in the buffers of the transmitter nodes, which corresponds to a heavy traffic assumption The analysis that we present in this paper can also be extended to the case where packets arrive to the queues of the nodes probabilistically, which we reserve as a future work We say a transmission fails if there is more than one transmission The cost of unsuccessful transmission of node is 2 If a transmission is successful, the node which transmitted its packet successfully gets utility 1 As opposed to previous work [11], we let to be greater than 1 to embrace the cases where the cost of packet collision may be bigger than the utility of packet success For example, a selfish node s perceived utility resulting from a packet success may be smaller than the cost of packet failure under severe limitations on its remaining battery energy We model this situation by using a strategic game, which is defined formally as follows Definition 1: Heterogenous one-shot random-access game with transmitter nodes is the game such that, for all, where 1 means transmission and 0 means backoff, where is the cost of unsuccessful transmission for node, and the utility function for all is defined as if if and if and In Definition 3, is the norm for the vectors in, and defined as the sum of the absolute values of the components of a vector If there exists a uniform constant such that for all, then we will denote with, and call it homogenous one-shot random-access game When there is no confusion, we will drop the word transmitter, and refer to the players of the game simply as nodes for the rest of the paper III NASH EQUILIBRIA OF THE RANDOM-ACCESS GAME In this section, we present our results characterizing the Nash equilibria of By means of these results, behavior of the selfish nodes at the Nash equilibria of is understood More specifically, we provide closed-form formulas for the transmission probabilities of the selfish nodes at any given 2 c is node i s subjective evaluation of unsuccessful transmissions It is normalized with respect to the benefit that it gets from successful transmissions It may depend on node i s energy expenditure per transmission or remaining battery life

4 INALTEKIN AND WICKER: ANALYSIS OF NASH EQUILIBRIA OF THE ONE-SHOT RANDOM-ACCESS GAME 1097 (1) Fig 2 Two nodes playing the one-shot random-access game Nash equilibrium of the one-shot random-access game When we say Nash equilibrium, we mean a transmission probability vector such that for any, maximizes node s expected utility given Therefore, none of the nodes wants to deviate from It is a stable strategy profile Intuitively, it is expected that must depend on the cost vector, and this dependence is formally developed in Theorem 1 To motivate the discussion, we start with case We then generalize our analysis to any fixed A Homogenous One-Shot Random-Access Game With Two Nodes The two-node case gives useful insights about the behavior of selfish nodes We do not give the proofs of the results of this section because generalized proofs will be given later for any In addition, they can be found in [13] We first look at the homogenous one shot random-access game can be represented in a compact form as in Fig 2 It has three equilibrium points; two of which are in pure strategies, and the last one is an FMNE In pure strategy equilibria, either node 1 or node 2 transmits with probability one, and the other one always backsoff At FMNE, nodes randomize between transmit and backoff actions with probabilities, for When a game is played by rational and intelligent players, it is expected that they have to play one of the Nash equilibria However, if the game has multiple equilibria such as, predicting which one of them will be played becomes hard To understand such games, game theorists ask the question of what might cause the players in a game to expect each other to implement some specific equilibrium Schelling considered this question in detail in [14] He argues that anything that tends the attention of players to focus on one particular equilibrium may make all of them expect this equilibrium If there is such an equilibrium in a game with multiple equilibria, this equilibrium is called a focal equilibrium of the game If there is only one focal equilibrium, then we should expect to observe this equilibrium Even though has three equilibria, the fairness property of the FMNE makes it the focal equilibrium of On the other hand, pure strategy equilibria are unfair to either node 1 or node 2 A semblance of user equality makes it hard to justify the occurrence of pure strategy Nash equilibria The best metric for characterizing the quality of a MAC protocol is channel throughput Therefore, it is imperative to analyze the channel throughput in order to understand how much we pay for allowing nodes to act selfishly, ie, what is the price of anarchy In pure strategy equilibria, the channel throughput becomes 1 Probability of packet collisions is zero, and channel is reserved permanently to either node 1 or node 2 On the other hand, the channel throughput at FMNE is given by as because nodes are discouraged from transmission by the high cost of packet failures As a result, we hardly see any one of them transmitting, and therefore, channel throughput goes to zero On the other hand, as because both nodes become very willing to transmit when the cost of collision is very small As a result, we almost always see both of them transmitting Hence, channel throughput goes to zero In addition, it can be shown by means of differential calculus that attains its global maximum at, and is its turning point Now, consider the centrally optimal solutions 3 to the same communication problem Suppose there exists a central authority to which both nodes faithfully obey If the central authority is fair, then it assigns the same transmission probability, say, to both nodes 1 and 2 In this case, channel throughput is given by, and maximized at This is how selfish nodes behave at FMNE of when If the central authority is unfair, then it sets the node 1 s probability of transmission to 1 and that of node 2 to 0, or vice versa, to maximize the channel throughput This is how nodes behave at pure strategy Nash equilibria As a result, we conclude that centrally optimal solutions with respect to the channel throughput metric coincide with the Nash equilibria of One final note is that we do not bother to make concepts of being fair or being unfair mathematically rigorous at this point When we say fair, we assume that the central authority assigns the same probability of transmission to both nodes When we say unfair, it favors one of the nodes, and can set other node s transmission probability to zero to maximize the channel throughput of the node that it favors Later, we will prove a more general result that almost all transmission probability assignments (except very irrational ones) chosen by the central authority are supported by the Nash equilibria of B Heterogenous One-Shot Random-Access Game With Two Nodes When the remaining battery lifetimes of the nodes are different, or the circuitry of their transmitters are different (so are their power expenditures per transmission), nodes may evaluate packet collisions differently, ie, Pictorially, can be depicted as in Fig 3 It has three Nash equilibria; two of which are in pure strategies and the last one is an FMNE In pure strategy equilibria, either node 1 or node 2 transmits with probability one, and the other one always backsoff At FMNE, nodes randomize between transmit and backoff actions with transmission probabilities, Note that for any transmission 3 When we say centrally optimal solution, we only consider probabilistic assignments of the common wireless channel to the transmitter nodes by a central authority In other words, we do not allow a multiaccess scheme like time-division multiple access (TDMA), which reserves the channel to each node for a certain amount of time in round-robin fashion In fact, if we model the same scenario by using repeated games instead of the strategic game G(n; c), the game theoretic solutions will also capture such centrally optimal solutions A repeated game approach with probabilistic arrivals of the packets and learning to play Nash equilibria will be subject to our future study

5 1098 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 16, NO 5, OCTOBER 2008 Fig 3 Two nodes playing the one-shot random-access game: c might be different than c Fig 4 Channel throughput as a function of c = (c ;c ) when two nodes contend for the access of the common wireless channel probability assignment selected by a central authority, we can find such that nodes transmission probabilities coincide with at one of the Nash equilibria The only assignments that cannot be supported by the game theoretic solution are very irrational ones such as (0,0), (1,1) etc The channel throughput at pure strategy equilibria becomes equal to one On the other hand, it can be given as follows at FMNE: In contrast to, does not have a local maximum or a local minimum is the saddle point of Similar to, as, where is the usual Euclidean norm In addition, it can be shown that as, ; for a fixed, as ; for a fixed, as Channel throughput at FMNE as a function of and is shown in Fig 4 C One-Shot Random-Access Game With Nodes Having obtained some intuition about the behavior of the selfish nodes, we are now ready to extend our observations in Sections III-A and B in a formal way to for a general Since is a finite strategic game, we know a priori that it has at least one Nash equilibrium in mixed strategies [20] However, we do not know exactly how nodes behave at this Nash equilibrium, ie, what their transmission probabilities are The following theorem, Theorem 1, is the central result of this section, and it allows us to say more about the behavior of the nodes at the Nash equilibria of In the first part of Theorem 1, we give a sufficient condition, which will be called regularity condition, on nodes for the complete characterization of the Nash equilibria of Regularity condition depends on a global property of the nodes, and (2) easy to check Intuitively, at any given Nash equilibrium, there should be two kinds of nodes: the ones having positive probability of transmission and the ones always backing off Once regularity condition is satisfied, for any given subset of (apart from ), there exists a Nash equilibrium at which only the nodes in contend for the access on the common wireless channel, and the rest of them always backoff Thus, if nodes satisfy the regularity condition, has equilibrium points: are in pure strategies, one is an FMNE, and the remaining mixed strategies The fairness property of the FMNE of makes it the focal equilibrium of the homogenous one-shot random-access game as it did in two-node case In the second part of Theorem 1, we give the necessary and sufficient condition (NSC) on nodes for the complete characterization of the Nash equilibria of It is subset dependent Basically, for any, if somebody asks for the existence of the Nash equilibrium at which only the nodes in randomize between their transmit and backoff actions, and the rest of them always backoff, then NSC must be satisfied Similarly, if NSC is satisfied, then there exists such a Nash equilibrium Let be such that for any, The theorem is as follows Theorem 1: Part I: Let regularity condition If the nodes satisfy the for all, then has Nash equilibria such that (i) of them are in pure strategies (ii) One of them is an FMNE, and it is given by such that for all (iii) For the rest of them, there is a group of players randomizing between transmit and backoff actions according to a nondegenerate probability distribution, and the rest of the players choose always backoff strategy 4 Moreover, if, then transmission probabilities for these nodes are given by for all Part II: Let and with Then, has pure-strategy Nash equilibria Moreover, any mixed-strategy profile such that nodes in mix between transmit and backoff actions according to a nondegenerate probability distribution, and nodes in backoff with probability 1 is a Nash equilibrium if and only if (3) for, and for all (with if ), where 4 When we say nodes randomizes between transmit and backoff actions according to a nondegenerate probability distribution, we mean they choose both of these actions with some positive probability

6 INALTEKIN AND WICKER: ANALYSIS OF NASH EQUILIBRIA OF THE ONE-SHOT RANDOM-ACCESS GAME 1099 Proof: Part I (i) For any, the strategy profile at which transmitter transmits with probability 1, and the rest backoff with probability 1 is a pure strategy Nash equilibrium because unilateral deviations from this strategy profile can only decrease the payoffs of the nodes Since there are different choices of, there are pure strategy equilibria (ii) Now suppose is an FMNE Let be the random vector in representing the actions chosen by the nodes Since nodes must be indifferent between their transmit and backoff actions at such a Nash (see [17, Ch 3]), we have for Let By using independence of s, we have for any (4) Now, if, then Otherwise,, where the last inequality follows from the regularity condition Therefore, we have an admissible solution It is easy to verify that is a Nash equilibrium Hence, it is the unique FMNE of (iii) Take any subset of nodes with Suppose they randomize between transmit and backoff actions according to a nondegenerate probability distribution, and the rest of them always backoff Let be the resulting mixed-strategy profile, and suppose it is a Nash equilibrium Since some of the nodes backoff with probability 1, we essentially have a reduced heterogenous one-shot random-access game, where only the nodes belonging to are contending for the access of the common wireless channel Then, the reduced mixed-strategy profile must be the FMNE of Let Computations similar to the ones in (ii) reveal that for any (5) (6) This is an admissible solution because From (4) (6), we conclude for all (7) We have equations with unknowns, We are interested in the solution of this set of equations restricted to We will solve them without considering the restriction, and then verify that this solution is indeed contained in To this end, we write all s,, in terms of as follows: where the last inequality follows from the regularity condition Therefore, is the unique FMNE of the reduced game Now, we will verify that all of the transmitters are happy at this strategy profile Any transmitter does not want to deviate because is the Nash equilibrium of Let Then (8) Using (7) and (8), we obtain Similarly, we obtain for all Since the expectation of is strictly negative on the event, backing off with probability 1 is in transmitter s best interest for any As a result, the above mixed-strategy profile is the unique Nash equilibrium corresponding to at which only the nodes belonging to mix between transmit and

7 1100 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 16, NO 5, OCTOBER 2008 backoff actions, and the rest of them backoff with probability 1 We have mixed-strategy Nash equilibria of this form Similar arguments reveal that any other strategy profile, except for the ones found in (i), (ii), and (iii), is not a Nash equilibrium of Consequently, we have in total Nash equilibria for Part II: The regularity condition is not needed to prove (i) of part I Thus, for any, it continues to hold Let A mixed-strategy profile at which only the nodes in mix between transmit and backoff actions, and the rest of them backoff with probability 1 is a Nash equilibrium if and only if is the FMNE of the reduced game, and the nodes in have no incentive to deviate Computations similar to the ones in part I reveal that is the FMNE of if and only if Fig 5 Illustration of the c s satisfying the regularity condition when n =5 Given c, all c s located on the vertical line passing through c and lying between two curves y = c and y = (c =(1 + c )) =(1 0 (c =(1 + c )) ) satisfy the regularity condition for and only if in Note that this is an admissible solution if Nodes have no incentive to deviate if and only if for all For any,wehave Proof: First, let Then, one of the pure strategy Nash equilibria coincides with Second, take, and let be defined as above We need to find a cost vector such that has a mixed-strategy Nash equilibrium where only the nodes in mix between transmit and backoff actions, and Start with any, and let Then, by Theorem 1, we need to solve equations Thus, if and only if A pictorial illustration of s satisfying the regularity condition is depicted in Fig 5 One easy observation here is that when nodes are identical, they satisfy the regularity condition, and for all of the subsets of except the empty set, there corresponds a Nash equilibrium at which only the nodes belonging to this subset randomize between transmit and backoff actions The following corollary of Theorem 1 is particularly important because it shows that almost all (with respect to -dimensional Lebesgue measure) probability assignment vectors chosen by a central authority are supported by the game theoretic solution In other words, for almost all, we can find a such that nodes transmission probabilities coincide with at one of the Nash equilibria of The assignments that cannot be supported are very unreasonable ones such as or is such that there exists with and Furthermore, Corollary 1 gives the explicit construction of the cost vector for any given transmission probability assignment Corollary 1: Let, such that and and such that and for all Then, for any transmission probability vector chosen by the central authority, there exists a cost vector such that has a Nash equilibrium at which the mixed-strategy equilibrium profile is given by for all Moreover, if we let and with, the cost vector can be constructed as for, and is large enough that for with unknowns,, subject to restriction for all We will solve them without considering the restriction, and then verify that our solution is admissible Fix If for all, then If we plug this identity to the equation related with the unknown, it can be found that Similarly, it can be found that for all In addition or Now, we will construct an admissible solution If, then let as above If, choose large enough that Thus, for, we satisfy the necessary and sufficient condition in part II of Theorem 1 by the above choice of For, Practical ramification of this corollary is that it is always possible to drive the system composing of selfish transmitters to any desired operating point by manipulating their costs Such a manipulation can be in the form of providing monetary incentives for selfish nodes to decrease their costs Or, it can be in the form of pricing such as charging some money per use of wireless channel to increase the perceived cost of failed packets Now, we would like to analyze the behavior of the channel throughput at the Nash equilibria as a function of for a fixed We will only analyze the homogenous case However,

8 INALTEKIN AND WICKER: ANALYSIS OF NASH EQUILIBRIA OF THE ONE-SHOT RANDOM-ACCESS GAME 1101 similar results continue to hold even in the heterogenous case (see [15]) The homogenous case is particularly important for us because has a unique focal equilibrium, and the channel throughput can be defined on the positive real line In addition, it has some nice properties such as convexity, concavity, and existence of its global maximum For any set, will denote the channel throughput at the mixed-strategy Nash equilibrium where only the nodes in randomize between transmit and backoff actions, and the rest of them backoff with probability 1 will denote the channel throughput at the FMNE At the pure strategy Nash equilibria, the common wireless channel is probabilistically reserved to one of the nodes Therefore, packet collision probability is zero, and the channel throughput becomes equal to one On the other hand, it can be expressed at any other equilibrium point as follows: Theorem 2: The following hold at mixed-strategy Nash equilibria with (i) has a unique global maximum attained at Moreover,, which is the same maximum channel throughput obtained by the slotted ALOHA 5 (ii) There exists such that is strictly concave for and strictly convex for is the turning point of (iii) as (iv) as Proof: See Appendix I The properties of the channel throughput provided in Theorem 2 can also be observed in Fig 6 Again, the behavior of as and as result from the selfishness of the nodes If is very small, they want to transmit very frequently, and the channel throughput goes to zero If is very high, they hardly want to transmit to avoid packet collisions One corollary of part (i) of Theorem 2 is that if time is divided into time slots, and a large number of myopic, 6 selfish, and identical nodes play the game in the beginning of each time slot, then at any Nash equilibrium of with a large number of nodes contending for the access of common wireless channel, the statical average of the number of successfully transmitted packets at each time slot 5 Obviously, the necessary condition to be able to let jn j!1 is to let n!1 6 By myopic nodes, we mean the nodes which only consider their payoffs in the current time slot If they take not only their current payoffs but also previous and future utilities into account, then the resulting game will be a repeated game with the stage game G(n; c) repeated infinitely many times In general, the set of the Nash equilibria of a repeated game can be quite complex However, even in this case, we know that the infinite play paths in which one of the Nash equilibria of G(n; c) is played at each play of the game lie in the Nash equilibria of the repeated game For such Nash equilibria with a large number of nodes contend for the access of the common wireless communication channel, the above observation holds even if nodes are not myopic (9) Fig 6 Change of channel throughput with c for different values of n over the probability space can be approximated by if is tuned to the optimum value Roughly speaking, this implies, by strong law of large numbers, that the time average of the number of successfully transmitted packets can be approximated by with probability 1 From here, we can conclude that if a large number of selfish, myopic nodes are involved in a medium access scheme similar to slotted ALOHA, the maximum channel throughput, viewed as the time average of the number of successfully transmitted packets, can be approximated by the maximum channel throughput of the slotted ALOHA with nonselfish nodes We also characterize the error term in this approximation in detail in Section IV As in the two-node case, the Nash equilibria of the homogenous game contain all of the centrally optimal solutions We only consider the centrally optimal solutions in the sense of probabilistic assignment of the channel The following corollary can also be stated after the Theorem 1 However, we have chosen to state it here since optimal solutions that we consider are in terms of the channel throughput metric Corollary 2: For all of the centrally optimal solutions with respect to the channel throughput metric, there exists a such that one of the Nash equilibria of the homogenous one-shot random-access game coincides with the centrally optimal solution Proof: Suppose there exists a central authority which tries to allocate the common wireless channel probabilistically to the nodes If it chooses to be fair among them, then it should assign the same probability of transmission, say, to all nodes So, the channel throughput becomes equal to Since [0,1] is a compact domain, and is a continuous function of, it attains its maximum at some point because On the other hand, selfish nodes set their probability of transmission to, which is a continuous function of, at the FMNE Therefore, by the intermediate value theorem, it takes all the values between 0 and 1 This implies that there exists a such that Hence, the centrally optimal solution, which is fair to all of the nodes, coincides with the FMNE of To complete the proof, one can repeat the similar arguments for the case where the central authority is unfair See [15] for details IV ASYMPTOTIC BEHAVIOR OF THE SYSTEM In previous sections, a careful and complete characterization of the behavior of the nodes and the channel throughput at any

9 1102 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 16, NO 5, OCTOBER 2008 Nash equilibrium of for any arbitrary is provided However, it is also crucial to know how nodes behave asymptotically at the Nash equilibria as in order to understand the asymptotic properties of dense wireless networks Therefore, the aim of the rest of the paper is to extend the results presented in Section III to the case where infinitely many players are involved in In this section, we restrict ourselves to the homogenous case, and concentrate on two important network parameters: channel throughput and the distribution of total number of packet arrivals to the channel In particular, we give closed-form expressions for the asymptotic channel throughput and the asymptotic packet arrival distribution as the number of nodes contending for the channel increases without a bound It is helpful to introduce some notation at this point Let for all and Then, we say if there exists a uniform constant such that for all and for all,wehave When goes to zero, this notation indicates how fast approaches to at all points A Asymptotic Distribution of Packet Arrivals Our first goal is to give a closed-form expression for the asymptotic distribution of the packet arrivals at Nash equilibria We show that for any,if, then the number of the packet arrivals converges in distribution to at a rate The metric that we use to measure the distance between any two distributions is total variation distance The formal definition is as follows Definition 2: Let and be two measures defined on the same measurable space, where is a countable set and is a -algebra of subsets of Then, the total variation distance between and on is defined as Without loss of generality, we set Some properties of total variation distance are that it defines a metric on the set of probability measures on, and the convergence in is equivalent to the weak convergence of probability measures We use to denote the weak convergence of probability measures We refer the reader to [21, Ch 2] for a more detailed discussion on The asymptotic behavior of the distribution of the packet arrivals is formally established in Theorem 3 In this theorem, we show that the distribution of the packet arrivals converges to at a rate for any with Furthermore, we also prove that this is the best possible achievable rate by showing Theorem 3: For any,if as, then and (10) Proof: See Appendix II For a sequence of fully mixed Nash equilibria, the error term in Theorem 3 could simply be given by One important corollary of this theorem is that it enables us to approximate the probability by the probability up to an error term Corollary 3: Let be fixed Then In addition, the error term becomes for At the first sight, these results might look like conflicting with our intuition It might be thought that a wireless communication system consisting of infinitely many selfish transmitters which are trying to individually maximize their own utilities should collapse because of the total anarchy in the system and complete incoordination among the nodes In particular, it might be expected that total number of packet arrivals to the wireless channel should grow without a bound because everybody is trying to get his packet through the wireless channel However, it turns out that this is not the case Selfish transmitters are intelligent enough that they adjust their transmission probabilities in a way stabilizing total number of packet arrivals to the channel Even though transmitters are selfish, they are also aware of the fact that maintaining unnecessarily high transmission probabilities as the number of competitors goes to infinity is in nobody s best interest B Asymptotic Channel Throughput In this part, we analyze the asymptotic behavior of the channel throughput as Specifically, for any given, we show that converges to uniformly if, and the rate of convergence is given by Moreover, we also prove that it is the best possible achievable rate by showing that is for any given Theorem 4: If as, then converges to uniformly, and the rate of the convergence is given by, this is the best possible achiev- Moreover, for any able rate ie, (11) Proof: See Appendix III In the light of Theorem 4, we conclude that the function approximates the channel throughput at any up to an error term, and this is the best possible error that can be achieved in the asymptotic characterization of the channel throughput One important point related to the upper bound is that Theorem 4 proves the existence of a uniform constant and such that for all,wehave For, the best possible achievable error term could simply be given by Corollary 4: Moreover, for any, this is the best possible achievable convergence rate In Fig 7, we plotted the sup norm between the channel throughput at and as a function of as well as the function is obtained

10 INALTEKIN AND WICKER: ANALYSIS OF NASH EQUILIBRIA OF THE ONE-SHOT RANDOM-ACCESS GAME 1103 Fig 7 Sup norm between (N ;c) and 0(c=(1 + c)) ln(c=(1 + c)) as a function of n by the curve fitting tool of Matlab It can be observed that fairly well approximates, which verifies our Theorem 4 Again, these results may look like counterintuitive at the first glance It might be thought that channel throughput should decrease to zero as the number of selfish transmitters contending for the channel access goes to infinity due to complete incoordination among them However, as proved in Theorem 4, it turns out that selfish nodes always achieve a positive channel throughput, and the channel throughput behaves like as the number of transmitters competing for the channel access goes to infinity Moreover, if is equal to, they also achieve the maximum possible channel throughput that can be obtained by using a centrally controlled slotted ALOHA protocol V CONCLUSION In this paper, we considered a wireless network consisting of selfish nodes, which are expected utility maximizers We modeled our network using the tools of game theory, and addressed the questions of whether there exist stable operating points of the network, how the selfish nodes behave at these stable operating points, and how the channel throughput changes with the cost vector Our main achievements are the necessary and sufficient condition on nodes for the complete characterization of the Nash equilibria of, relation between the centralized solutions and the game theoretic solutions of the communication problem under consideration and the characterization of the asymptotic properties of the system as the number of selfish transmitters increases to infinity We firstly give a sufficient condition, which is called the regularity condition, on nodes to have a Nash equilibrium associated with any subset of nodes such that only the nodes in contend for the access of common wireless channel, and the rest of them backoff with probability 1 at this Nash equilibrium In addition to regularity condition, we give, for any, the necessary and sufficient condition on nodes to have a Nash equilibrium at which only the nodes in contend for the access of the common wireless channel, and the rest of them always backoff The regularity condition depends on the global behavior of the nodes and easy to check, whereas the necessary and sufficient condition is subset dependent Once nodes satisfy the regularity condition, has equilibrium points, and one of them is an FMNE If nodes are identical, the FMNE of the game turns out to be the unique focal equilibrium because it is fair to all nodes Secondly, we characterize the behavior of at any equilibrium point, at which only the nodes in contend for the access of the channel for a fixed as a function of In the homogenous case, when goes to zero, behaves like This is because all of the nodes have a strong tendency to transmit as a result of low cost of packet collisions behaves like as because nodes are deterred from transmissions by high cost of packet collisions This observation squares with our intuition about the behavior of the selfish nodes The convexity and concavity properties of are also studied, and we determined the critical point at which the channel throughput attains its global maximum It is also noted that the maximum channel throughput goes to, which is the same maximum channel throughput obtained by the centrally controlled slotted ALOHA, as tends to infinity Important byproducts of the complete characterization of the Nash equilibria and the properties of the channel throughput are that all of the centrally optimal solutions in terms of the probabilistic assignment of the wireless channel are a subset of our game theoretic solutions, and almost all of the transmission probability assignments (apart from the very irrational ones) chosen by the central authority are supported by the game theoretic solution The probability assignments that are not supported are the ones like or This shows that it is possible to drive such a system composing of selfish transmitters to any desirable stable operating point by manipulating the costs of selfish transmitters Finally, we looked at the asymptotic properties of the system as the number of selfish transmitters approaches infinity We show that for any sequence of Nash equilibria, asymptotic distribution of the packet arrivals converges to a Poisson distribution with mean at a rate It is also shown that channel throughput converges to uniformly at a rate Moreover, we also prove that these are the best possible achievable convergence rates APPENDIX I PROPERTIES OF CHANNEL THROUGHPUT AT NASH EQUILIBRIA In this appendix, we provide the proof for Theorem 2 A Proof of Theorem 2 Proof: (i) We start with finding first and second derivatives of with respect to can be obtained

11 1104 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 16, NO 5, OCTOBER 2008 as Thus, Let when Similarly, the second derivative of can be obtained after some calculus as follows: for Thus, is strictly concave for and strictly convex for is the turning point for (iii) (iv) One line proof of this fact is possible but the below technique gives us a uniform bound over, which is uniform over all Let Then Thus, This implies that attains its local maximum at To see why this is also its unique global maximum, first notice that for and for Therefore, is strictly increasing for and strictly decreasing for As a result, it attains its strict global maximum at To compute we first note that Thus (ii) This is correct for (see [13]) Thus, it is enough to consider Convexity and concavity of depend on the sign of its second derivative Thus, we will show, defined as above, is a strictly increasing function of, and becomes equal to 2 for some To this end, first observe that APPENDIX II ASYMPTOTIC DISTRIBUTION OF THE PACKET ARRIVALS A Some Preliminary Results The following auxiliary results will be needed while proving Theorem 3 We skip the proofs of Lemmas 1, 2 and 3 Lemmas 1 and 2 are just simple facts Lemma 3 follows from two applications of L Hopital s rule Lemma 1: Let as Then, Lemma 2: Let Then Lemma 3: Let with as for all Then, for any sequence Also as increases, and strictly increase Therefore, is strictly increasing function of By intermediate value theorem for continuous functions, there exists such that Proposition 1: Let as Then, Proof: Let and In addition, by strict monotonicity of for, we have and

12 INALTEKIN AND WICKER: ANALYSIS OF NASH EQUILIBRIA OF THE ONE-SHOT RANDOM-ACCESS GAME 1105 where (a) follows from Lemma 1, and (b) follows from Lemma 2 Let and Then fixed,wehave for some One immediate corollary of this proposition is that if and are defined as in Proposition 1, we can estimate by up to an error term for any B Proof of Theorem 3 We need to show that there exist two positive constants and such that for all We begin with showing the existence of Let and for means identical in distribution For notational simplicity in the upcoming equations, we will also let, and, where Then by Lemma 3 Thus, there exists a constant such that for all For the proof of, we refer the reader to [16] APPENDIX III ASYMPTOTIC CHANNEL THROUGHPUT A Some Preliminary Results Again, we will first need to obtain some preliminary results We first introduce some more notation We say as if for all, there exist a constant such that for all large enough, where is the domain over which the functions are defined Let us define the functions and as follows: if if (13) (14) Lemma 4: For any and any sequence with, the following holds: as (15) Proof: If, then Hence, it is enough to consider the sequences with for all First let Then, by using Taylor formula for Let, and observe that the series in the above parenthesis converges Since as, there exists a natural number, which depends on the choice of the sequence, such that whenever As a result, one has (12) where means convolution of probability measures,, (a) follows from being a metric on the set of probability measures on, (b) follows from Lemmas 62 and 63 of [21] in Section 26, (c) follows from the Lemma 64 of [21] in Section 26, and (d) follows from Proposition 1 For a for all If, one can always write for some nonnegative function with We will design two bounds on the difference such that the first one will be good for the cases is bounded, whereas the second one will be good for the cases

13 1106 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 16, NO 5, OCTOBER 2008 Our first bound is obtained as follows: as such that Construct another sequence as follows For all for all if Our second bound is obtained as follows: Define inductively as follows: (16) Let us first concentrate on the case where there are infinitely many s with the above property For example, is a sufficient condition for it Then, such that for all, the following holds: Then, as but for any, for infinitely many This contradicts with Lemma 4 Proposition 2: Proof: For all, find such that as for Then, is an open-cover for [0,1] Since [0,1] is a compact set, by Heine-Borel theorem, there exists a finite subcover of [0,1] Let and Takeany Then, for some Therefore, we have whenever Since this is true for any, for all If we make big enough, we satisfy this inequality for as well Thus, there exists a uniform constant such that for all B Proof of Theorem 4 We need to find two positive constants satisfying the following For all and and for a given, and (17) (18) Then, for all, we have for all large enough We being with the proof for the existence of throughput can be given by Channel We know by Proposition 2 that there exists a constant such that (19) where (a) follows from if, then Otherwise, If there are only finitely many s satisfying (16), then forms a bounded sequence Thus, we use our first bound to conclude that for all, for some It follows that as for all and for all sequences with as Lemma 5: For any, there exist,, and such that for all and That is, as for Proof: We prove this by contradiction Suppose not Then, a such that for all, and, a and a such that Then, we can find sequences and with and for all Since, it is also true that for all Then

14 INALTEKIN AND WICKER: ANALYSIS OF NASH EQUILIBRIA OF THE ONE-SHOT RANDOM-ACCESS GAME 1107 where (a) follows from the triangular inequality, and (b) follows from Proposition 2 Then, it is enough to show that Thus, there exists a satisfying (17) For the proof of, we refer the reader to [16] REFERENCES [1] P Gupta and P R Kumar, The capacity of wireless networks, IEEE Trans Inf Theory, vol 46, no 2, pp , Mar 2000 [2] M Grossglauser and D Tse, Mobility increases the capacity of wireless ad hoc networks, IEEE/ACM Trans Netw, vol 10, no 4, pp , Aug 2002 [3] J Eriksson, M Faloutsos, and S Krishnamurthy, Scalable ad hoc routing: The case of dynamic addressing, in Proc IEEE INFOCOM, Hong Kong, Mar 2004, pp [4] X Hong, K Xu, and M Gerla, Scalable routing protocols for mobile ad hoc networks, IEEE Netw Mag, vol 16, no 4, Jul Aug 2002 [5] A B MacKenzie and L A DaSilva, Game Theory for Wireless Engineers San Rafael, CA: Morgan and Claypool, 2006 [6] L Kleinrock and J A Silvester, Optimal transmission radii in packet radio networks or why six is a magic number, in Proc Nat Telecommun Conf, Birmingham, AL, Dec 1978 [7] H Takagi and L Kleinrock, Optimal transmission ranges for randomly distributed packet radio terminals, IEEE Trans Commun, vol COM-32, no 3, Mar 1984 [8] T C Hou and V O K Li, Transmission range control in multi-hop packet radio networks, IEEE Trans Commun, vol COM-34, no 1, Jan 1986 [9] C U Saraydar, N B Mandayam, and D J Goodman, Pricing and power control in a multicell wireless data network, IEEE J Sel Areas Commun, vol 19, no 10, pp , Oct 2001 [10] M Cagalj, S Ganeriwal, I Aad, and J P Hubaux, On selfish behavior in CSMA/CA networks, in Proc IEEE INFOCOM, Mar 2005, pp [11] A B MacKenzie and S B Wicker, Stability of multipacket slotted Aloha with selfish users and perfect information, in Proc IEEE IN- FOCOM, 2003, pp [12] E Altman, R El Azouzi, and T Jimenez, Slotted Aloha as a stochastic game with partial information, in WiOpt 03, Mar 2003 [13] H Inaltekin and S Wicker, A one shot random access game for wireless networks, in Symp Information Theory in Wirelesscom, 2005 [14] T C Schelling, The Strategy of Conflict Cambridge, MA: Harvard Univ Press, 1960 [15] H Inaltekin, Topics on wireless network design: Game theoretic MAC protocol design and interference analysis for wireless networks, PhD dissertation, Cornell Univ, Ithaca, NY, Aug 2006 [16] H Inaltekin and S B Wicker, The analysis of a game theoretic MAC protocol for wireless networks, in Proc IEEE Secon 06 [17] R B Myerson, Game Theory, Analysis of Conflict Cambridge, MA: Harvard Univ Press, 1997 [18] D Fudenberg and J Tirole, Game Theory Cambridge, MA: MIT Press, 1991 [19] P Billingsley, Probability and Measure Hoboken, NJ: Wiley, 1995 [20] J Nash, Equilibruim points in n-person games, in Proc Nat Acad Sci, 1950, vol 36, no 1 [21] R Durrett, Probability: Theory and Examples, 2nd ed Duxbury Press, MA: Duxbury, 1996 [22] W Rudin, Principles of Mathematical Analysis New York: Mc- Graw-Hill, 1976 Hazer Inaltekin (S 04 M 06) received the BS degree (with High Honors) in electrical and electronic engineering from Bogazici University, Istanbul, Turkey, in 2001, and the MS and PhD degrees in electrical and computer engineering from Cornell University, Ithaca, NY, in 2005 and 2006, respectively He was then with the Wireless Intelligent Systems Laboratory, Cornell University, as a Postdoctoral Research Scientist until August 2007 He is currently with the Department of Electrical Engineering, Princeton University, Princeton, NJ, as a Postdoctoral Research Scientist His research interests include information theory, game theory, wireless communications, wireless networks, and financial mathematics Stephen B Wicker (SM 92) received the BSEE degree (with High Honors) from the University of Virginia, Charlottesville, in 1982, the MSEE degree from Purdue University, West Lafayette, IN, in 1983, and the PhD degree in electrical engineering from the University of Southern California, Los Angeles, in 1987 He is currently a Professor of electrical and computer engineering at Cornell University, Ithaca, NY, and a member of the graduate fields of computer science and applied mathematics He is the author of Codes, Graphs, and Iterative Decoding (Kluwer, 2002), Turbo Coding (Kluwer, 1999), Error Control Systems for Digital Communication and Storage (Prentice-Hall, 1995), and Reed-Solomon Codes and Their Applications (IEEE Press, 1994) Dr Wicker was awarded the 1988 Cornell College of Engineering Michael Tien Teaching Award and the 2000 Cornell School of Electrical and Computer Engineering Teaching Award He has served as an Associate Editor for Coding Theory and Techniques for the IEEE TRANSACTIONS ON COMMUNICATIONS and is currently an Associate Editor for the ACM Transactions on Sensor Networks He has served two terms as a member of the Board of Governors of the IEEE Information Theory Society