IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER Cogntve Herarchy Theory for Dstrbuted Resource Allocaton n the Internet of Thngs Nof Abuzanab, Wald Saad, Senor Member, IEEE, Choong Seon Hong, Senor Member, IEEE, and H. Vncent Poor, Fellow, IEEE Abstract In ths paper, the problem of dstrbuted resource allocaton s studed for an Internet of Thngs IoT) system, composed of a heterogeneous group of nodes compromsng both machne-type devces MTDs) and human-type devces HTDs). The problem s formulated as a noncooperatve game between the heterogeneous IoT devces whch seek to fnd the optmal tme allocaton so as to meet ther qualty-of-servce QoS) requrements n terms of energy, rate, and latency. Snce the strategy space of each devce s dependent on the actons of the other devces, the generalzed Nash equlbrum GNE) soluton s frst characterzed, and the condtons for unqueness of the GNE are derved. Then, to explctly capture the heterogenety of the devces, n terms of resource constrants and QoS needs, a novel and more realstc game-theoretc approach, based on the behavoral framework of cogntve herarchy CH) theory, s proposed. Ths approach s then shown to enable the IoT devces to reach a CH equlbrum CHE), a concept that takes nto account the varous levels of ratonalty correspondng to the heterogeneous computatonal capabltes and the nformaton accessble for each one of the MTDs and HTDs. Smulaton results show that the CHE soluton mantans a stable performance. In partcular, the proposed CHE soluton keeps the percentage of devces wth satsfed QoS constrants above 96% for IoT networks contanng up to devces wthout consderably degradng the overall system performance n terms of the total utlty. Smulaton results also show that the proposed CHE soluton brngs a twofold ncrease n the total rate of HTDs and deceases the total energy consumed by MTDs by 78% compared wth the equal tme polcy. Index Terms Cogntve herarchy theory, resource allocaton, Internet of Thngs, bounded ratonalty. I. M INTRODUCTION EETING the strngent qualty-of-servce QoS) requrements of a massve number of heterogeneous devces s the man challenge facng the successful deployment of Manuscrpt receved March 1, 2017; revsed June 28, 2017 and August 12, 2017; accepted August 12, Date of publcaton August 29, 2017; date of current verson December 8, Ths work was supported n part by the U.S. Offce of Naval Research under Grant N and n part by the U.S. Natonal Scence Foundaton under Grant OAC , Grant CNS , Grant ECCS , and Grant CNS Ths paper was presented n part at the IEEE Internatonal Symposum on Informaton Theory, Barcelona, Span, July 2016 [1]. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was C.-H. Lee. Correspondng author: Nof Abuzanab.) N. Abuzanas wth Wreless@VT, Department of Electrcal and Computer Engneerng, Vrgna Tech, Blacksburg, VA USA e-mal: nof@vt.edu). W. Saad s wth Wreless@VT, Department of Electrcal and Computer Engneerng, Vrgna Tech, Blacksburg, VA USA e-mal: walds@vt.edu). W. Saad was also wth the Department of Computer Scence and Engneerng, Kyung Hee Unversty, Yongn, South Korea. C. S. Hong s wth the Department of Computer Scence and Engneerng, Kyung Hee Unversty, Yongn, South Korea e-mal: cshong@khu.ac.kr). H. V. Poor s wth the Department of Electrcal Engneerng, Prnceton Unversty, Prnceton, NJ USA e-mal: poor@prnceton.edu). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TWC the Internet of Thngs IoT) [1] [4]. In partcular, the IoT ecosystem wll encompass both human type devces HTDs) and machne type devces MTDs). MTDs are expected to delver a wde range of applcatons rangng from healthcare to smart homes and transportaton and, as such, they wll exhbt a heterogeneous mx of QoS requrements [4]. In general, the three man mportant performance metrcs for MTDs are relablty, energy effcency, and latency. MTDs, such as those used for envronmental montorng, cannot be easly charged, and thus they must seek to mnmze ther energy consumpton. In contrast, MTDs that are used for crtcal applcatons such as alarm systems wll be prmarly seekng to relably delver ther packets under strngent delay constrants. HTDs, such as smartphones, typcally requre hgh-speed transmsson rates whle not exceedng a certan energy budget. Beyond ths heterogeneous nature of the IoT, ts massve scale wll sgnfcantly ncrease the competton for wreless resources. Ths requres desgnng very effcent resource allocaton schemes talored to the scale and heterogenety of the system. Moreover, n the IoT, a centralzed approach to resource allocaton can be prohbtve as t requres solvng an optmzaton problem wth a large number of varables and constrants correspondng to all of the IoT devces. Thus, there s a need to adopt dstrbuted and selforganzng IoT resource allocaton schemes [5]. In addton to beng dstrbuted, resource allocaton n the IoT must also explctly factor n the varous IoT devces constrants that stem from ther dfferent QoS needs and computatonal capabltes n terms of memory and processng powers. To address these challenges, there s a need for a jont desgn of resource allocaton and multple access for the IoT [6] [14], [16], [17]. In [17], random access was ntally proposed as a sutable multple access scheme for a system wth massve number of devces such as the IoT snce t does not requre coordnaton and can properly cope wth the bursty nature of the MTDs traffc. However, n a dense IoT system, random access can potentally lead to ncreased collsons, and thus, not all devces wll be able to meet ther QoS requrements [18]. For example, the performance of MTDs that requre ultra low latency or HTDs that requre hgh data rates can can be severely affected by collsons. Thus, there s a need to desgn a new multple access scheme for the IoT that can satsfy n a dstrbuted way the requrements of devces wth strct QoS constrants. A. Related Work There has been sgnfcant recent nterest n developng sutable resource allocaton schemes for the IoT such as n [8] [14], [16], and [17]. Centralzed schedulng schemes IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 7688 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 for IoT LTE networks are proposed n [8] [11]. In [8], a resource management scheme s proposed that dynamcally allocates tme resources between MTDs and HTDs based on current traffc condtons and QoS requrements. The works n [9] and [10] propose schemes that allocate the LTE resources to MTDs and HTDs based on a bpartte graph. In [11], the authors propose two seperate uplnk schedulng schemes for HTDs and MTDs n an LTE system based on channel condtons and delay requrements whle takng farness nto account. Other works such as n [12] and [13] adopted game-theoretc approaches for dstrbuted resource allocaton problems n the IoT. The authors n [12] study the problem of throughput maxmzaton of MTDs under random access. However, n [12], devces are consdered of equal capablty and QoS requrements. The work n [13] consders a heterogeneous system of MTDs n whch nodes can use dfferent routng and network codng schemes to optmze heterogeneous QoS requrements. An evolutonary game approach has been adopted n [14] to obtan an optmal transmsson strategy for a devce-todevce D2D) enabled LTE network where each MTD chooses to act noncooperatvely or cooperatvely wth other MTDs usng the same resources. In [15], the problem of uplnk user assocaton of IoT devces s studed n a dense small cell network usng a mean-feld game. However, these works typcally rely on the concept of a Nash equlbrum to solve the studed resource allocaton problems. Such a conventonal Nash equlbrum soluton may not be sutable to solve the dstrbuted resource allocaton problem when the optmzaton s done at the IoT devces end. Ths s because the Nash equlbrum assumes that players have equal and smlar capabltes, whle IoT devces are heterogeneous and have dfferent computatonal capabltes. Thus, the ratonalty of each IoT devce s bounded by ts computatonal capablty. Also, at the Nash equlbrum, n order to compute ts best resource strategy, each devce must know the true actons of other devces n the system, whch s not practcal for the IoT. Indeed, n practce, to gather nformaton on all the actons of the opposng devces, a sgnfcant amount of nformaton exchange and, thus, delay) s requred gven the massve number of devces n the IoT. Moreover, MTDs, such as small sensors, have lmted memory and are unable to store the actons of all other IoT devces. Further, meanfeld game solutons typcally rely on the exchangeablty property whch presumes that all devces are homogeneous and have a smlar mpact on the game. Ths assumpton s generally not approprate for heterogeneous IoT envronments. These lmtatons motvate the need to ntroduce new soluton approaches that are talored to the unque nature of the IoT, n terms of heterogenety, scale, and devce constrants. B. Contrbuton The man contrbuton of ths paper are summarzed as follows: We propose a dstrbuted self-organzng resource allocaton scheme that enables the IoT devces to fnd ther optmal allocaton of tme resources, whle explctly caterng for the ndvdual capablty of each devce and the lmted avalablty of nformaton. In partcular, we consder the problem of resource allocaton n the uplnk of an IoT network that s supportng a heterogeneous mx of IoT devces usng a tme dvson multple access TDMA) scheme. Thus, each IoT devce s self nterested n determnng the optmal tme fracton that meets ts own strct QoS guarantees n a dstrbuted manner. In partcular, the HTDs seek to maxmze ther data rates whle the MTDs must delver ther packets wthn strngent deadlnes. For both HTDs and MTDs, the proposed model accounts for energy effcency. Due to the dependence on the optmal tme fracton of each devce on the tme fractons chosen by the remanng devces, we formulate the problem as a noncooperatve game 1 n whch the IoT devces are the players that seek to guarantee ther QoS whle explctly factorng n ther heterogeneous requrements and devces capabltes. We characterze the generalzed Nash equlbrum GNE) soluton of the proposed game and show the condtons under whch the GNE s unque. Further, we present a learnng algorthm that allows the IoT devces to reach the GNE n a dstrbuted manner. We show that the computatonal complexty of the GNE learnng algorthm s polynomal n the number of IoT devces. As such, gven the massve scale of the IoT, t may not be feasble for IoT devces wth lmted computaton capabltes to fnd the GNE. We propose a novel and more realstc soluton method that extends the framework of cogntve herarchy CH) theory [19], a branch of behavoral game theory that assumes that players belong to dfferent dscrete levels of ratonalty, and that the players have dfferent belefs about the remanng players dependng on ther ratonalty level. The ratonale behnd usng the proposed CH approach s two-fold: a) The dfferent CH ratonalty levels are used to account for the heterogeneous computatonal capabltes of the IoT devces, whle the dfferent players belefs are dependent on the resources avalable at each IoT devce to obtan and store the nformaton about the remanng players, and, thus, the CH approach s a more realstc model than the classcal GNE that assumes that all players are fully ratonal and of equal capabltes; and b) In the proposed CH approach, each devce selects ts equlbrum strategy based on ts belefs about the remanng devces, and, thus, t does not ncur massve nformaton exchange to fnd the CH equlbrum as s the case of fndng the GNE soluton. These characterstcs make the CH framework sutable for ths problem. To model the IoT resource allocaton problem usng CH, we extend the conventonal framework to explctly take nto account the awareness of each devce of other devces at the same ratonalty level. Ths extenson s mportant n our problem as multple devces wth the same characterstcs and computatonal capabltes can exst n an IoT network. 1 Here, we note that, wthn the scope of ths work, devces are assumed to be noncooperatve, due to the overhead assocated wth cooperaton n a massve IoT.

3 ABUZAINAB et al.: COGNITIVE HIERARCHY THEORY FOR DISTRIBUTED RESOURCE ALLOCATION 7689 We show that the computatonal complexty of the optmzaton done at each devce to fnd the cogntve herarchy equlbrum CHE) s much smaller than that for the GNE and s lnear n the number of CH levels. Smulaton results show that the CHE soluton mantans a stable performance. In partcular, the proposed CHE soluton keeps the percentage of devces wth satsfed QoS constrants above 96% for IoT networks wth up to devces whle not consderably degradng the overall system performance, n terms of the total utlty. Smulaton results also show that the proposed CHE soluton brngs a two fold ncrease n the total rate of HTDs and deceases the total energy consumed by MTDs by 78% compared to an equal tme polcy. The rest of the paper s organzed as follows. Secton II presents the system model and the heterogeneous tme allocaton HTA) problem. Secton III descrbes the noncooperatve game formulaton of the HTA problem and presents the GNE and the CHE solutons respectvely. Smulaton results are presented n Secton IV. Fnally, conclusons are drawn n Secton V. II. SYSTEM MODEL Consder the uplnk of an IoT system composed of a heterogeneous mx of machne type devces and human type devces. In ths model, we consder MTDs havng strct delay requrements and HTDs havng hgh data rate requrements that are served by a base staton BS) accordng to a TDMA scheme. In ths scheme, transmssons occur n tme perods of T seconds, and each IoT devce transmts durng a fracton τ of T. We denote by L the set of L IoT devces L = H M ) that ncludes the set H of HTDs and the set M of MTDs. All IoT devces n L transmt on the same frequency band of bandwdth W. Each devce transmts wth a fxed power P. The channel gans, h, between devces L and the BS are assumed to be ndependent block Raylegh fadng wth varances α 2. It s assumed that statstcal channel state nformaton CSI) s avalable at each devce,.e. each devce knows the channel statstcs but not the nstantaneous channel gan. Ths assumpton s sutable for the uplnk n the IoT as obtanng statstcal CSI requres less communcaton overhead than full CSI [20]. Addtve whte Gaussan nose of varance σ 2 s assumed to be present at each recever. Each devce L must determne the optmal tme fracton τ that meets ts qualty-of-servce requrements. Due to the heterogenety of devces, the QoS requrement of each devce wll depend on ts type. The detals of the QoS requrements of HTDs and MTDs are presented respectvely as follows. A. HTD QoS Requrements HTDs, such as smartphones, wll seek to maxmze ther expected transmsson rates whle not exceedng an energy budget, E for devce. The acheved rate s related to the receved sgnal-to-nose rato SNR) through the ergodc capacty formula and s gven by R = E h [W log 1 h 2 P )] σ 2 τ. 1) The energy spent by HTD durng tme fracton τ s: E[ξ ]=P τ T. 2) Thus, each HTD solves the followng optmzaton problem: [ max E h W log 1 h 2 P )] τ σ 2 τ, 3a) s.t. P τ E T, τ 1 τ j, 0 τ 1, j L where, agan, W s the channel bandwdth. B. MTD QoS Requrements 3b) Each MTD seeks to delver a packet of sze bts wthn a strct delay constrant. Due to fadng, the packet may not be delvered successfully wthn a sngle transmsson, and thus, the probablty p wth whch the packet transmtted by devce s successfully decoded at the BS s gven by the probablty that the receved SNR s greater than a requred threshold γ. The threshold γ s chosen so that bts can be transmtted successfully. Here, usng Shannon s capacty formula, we have = W log1 γ ). 4) T τ The probablty of successful decodng s therefore [ h 2 ] P p = Pr γ = e γ σ 2 α 2 P, 5) σ 2 where P s the transmt power of MTD. MTD uses retransmssons n order to relably delver the packet. Hence, the probablty dstrbuton of the packet success tme T of devce d follows a geometrc dstrbuton gven by Pr[T = k] =1 p ) k 1 p = 1 e γ σ 2 α 2 P ) k 1e γ σ 2 α 2 P. 6) where T s the number of tme slots needed to delver the packet successfully. Each packet of MTD should be delvered wthn a strct deadlne of d seconds. It has been shown n [21] that the MTDs traffc s bursty. Thus, the nter-arrval tmes of the MTD packets are assumed to be large. Hence, the queung delay can be gnored as t tends to be much smaller than the transmsson delay. Indeed, for characterzng latency, n such a scenaro, we allow each MTD to guarantee that ts tme to transmt does not exceed a certan threshold. Snce the tme to delver each packet successfully s random, the probablty that the successful transmsson tme exceeds a certan number of tme slots t must be very small.e. Pr[T t ] ɛ wth Pr[T t ]=1 p ) t = 1 e γ σ 2 ) α 2 t P, 7) and t = d T. For MTDs, there s a need to mantan a low energy consumpton to extend the overall lfetme of the devces. Note that the expected energy consumed by each MTD wll be E[E ]=P E[T ]= P τ T. 8) p

4 7690 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 Thus, n order to fnd ts optmal transmsson parameters, each MTD must solve the followng optmzaton problem: mn τ P τ T p, 9a) s.t. Pr[T t ] ɛ, τ 1 τ j, 0 τ 1, 9b) j = where the constrant n 9b) s based on the assumpton of statc allocatonoftme fractons,.e., eachdevce s assgned the fracton τ even after t succeeds n transmttng ts packet. Based on 4), the tme fracton τ s a decreasng functon of γ. Also, the expresson for Pr[T t ] as a functon of τ s complcated and the probablty p s a functon of γ. Hence for convenence, the optmzaton varable s changed from τ to γ yeldng P mn, 10a) γ W log1 γ )e γ σ 2 α 2 P s.t. 1 e γ σ 2 α 2 P ) t ɛ, T W log1 γ ), b j T W log1 γ j ). 10b) T W log1 γ ) 1 10c) j = By rewrtng the constrants n 10), the optmzaton problem becomes P mn, 11a) γ W log1 γ )e γ σ 2 α 2 P s.t. γ α2 P log1 t ɛ) σ 2, 11b) γ e T W1 b j j T W log1γ j ) ). 11c) Gven the scale and heterogenety of the IoT, a centralzed approach to fndng the optmal transmsson probabltes wll be challengng to mplement n practce. Ths s due to the fact that t wll requre solvng an optmzaton problem wth a very large number of varables and constrants. Thus, a centralzed soluton wll always lead to hgh latency whch can be sgnfcantly prohbtve for IoT servces that are extremely senstve to delay. Consequently, a self-organzng approach to resource allocaton s essental n the IoT as t allows the IoT devces to optmze ther optmal transmsson probabltes n a dstrbuted manner. Based on the optmzaton problems 3b) and 9b), the optmal tme fracton for each devce s clearly dependent on the tme fractons of the remanng devces, whch motvates the use of a game-theoretc approach [23] as detaled next. III. HETEROGENEOUS TIME ALLOCATION GAME We formulate a statc contnuous noncooperatve game defned by L,S ) L,U ) L ) wth the players beng the devces n L. The acton a of each HTD s to choose the tme fracton τ, and the acton of each MTD s to fnd the requred threshold γ whch corresponds to fndng ts tme fracton τ accordng to 4)). Thus, gven the constrants n 3b) and 11c) respectvely, the strategy space for each MTD s S = [ T W1 b j ] j M, j = T W log1γ e j ) ) j H τ j, and for each HTD s S =[0, mn{ TP E, 1 j H, j = τ j b j j M T W log1γ j ) }]. Consequently, the utlty functon of each MTD s U a ) = P W log1 γ )e γ σ 2 α 2 P, 12) whle the utlty functon of each HTD s [ U a ) = E h W log1 h 2 P ] σ 2 ) τ. 13) We can clearly see that the utlty of each HTD n 13) s ncreasng n τ. The concavty of the utlty of each MTD s asserted next. Lemma 1: The utlty of MTD n 12) s concave n γ. Proof: The proof of Lemma 1 s n Appendx A. Lemma 2: By equatng the dervatve of U γ ) to zero, the utlty functon of MTD attans ts maxmum at γ that satsfes 1 1 γ = σ 2 α 2 P log1 γ ). 14) In the proposed HTA game, the utlty of each devce based on 12) and 13) s dependent only on ts own strategy. However, the strategy space of each devce s dependent on the strategy vector a of the remanng devces. Hence, to fnd a sutable soluton for ths formulated game, we must study the GNE. The detals of the GNE soluton are explaned next. A. Generalzed Nash Equlbrum Soluton The GNE s a popular soluton concept used to solve gametheoretc scenaros n whch the acton spaces of the players are mutually dependent, as s the case n the formulated HTA game. Formally, the GNE for the proposed game can be defned as follows. Defnton 1: The GNE of the heterogeneous tme allocaton game s the vector of players actons a such that U a ) U a ), D, a S a ). In what follows, we show that a GNE for the HTA game always exsts. Subsequently, we show condtons under whch the GNE s unque and characterze the GNE set n the case when the GNE s not unque. Proposton 1: A GNE exsts for the heterogeneous tme allocaton game. Proof: The proof s n Appendx A. In order to fnd a GNE, the best response of each devce s derved as follows. Proposton 2: Gven a strategy vector a, the best response of HTD s gven by E a = mn, 1 τ j b ) j TP T W log1 γ j ) j H, j = j M 15) Proof: The proof of Proposton 2 s n Appendx A.

5 ABUZAINAB et al.: COGNITIVE HIERARCHY THEORY FOR DISTRIBUTED RESOURCE ALLOCATION 7691 Proposton 3: The best response of each MTD s a,as shown at the bottom of ths page, where γ s gven n 14), and γ,ub = α2 P log1 t ɛ) s the upper bound on γ n 11b). σ Proof: The proof 2 of Proposton 3 s n Appendx A. Thus, based on the propertes of the best response, the GNE of both MTDs and HTDs s derved next. Frst, before characterzng the GNE of the proposed HTA game, we defne the followng key parameters: { The set M = M s.t. γ γ,ub }. The set of sets A = {A j L s.t. A H E T WP A\M M A M T W log1γ ) T W log1γ,ub ) 1}. Then, the GNE of the proposed HTA game can then be derved based on the followng theorem. Theorem 1: The GNE of the HTA game s dependent on the followng two condtons: M T W log1γ ) H E T WP If M M T W log1γ,ub ) 1, the GNE of the heterogeneous tme allocaton game s unque. Inths case, the GNE strategy for each MTD s a = γ M M.For M and a = α2 P log1 t ɛ) σ 2 each HTD, the GNE strategy s a If M M M T W log1γ ) T W log1γ,ub ) = E T WP. E H T WP > 1, the GNE s not unque and the GNE set N s gven by N = 1 j A N j where N j = {a 1 L S s.t. a = γ A j M, a E = T WP A j H, a = α2 P log1 t ɛ) A σ 2 j \ M M, and L A j a = 1 A j M E T WP A j M M ) A j H T W log1γ ) T W log1γ,ub ) s.t. a <γ L A j ) M, a < α2 P log1 t ɛ) σ 2 L A j ) M M ) and a < E T WP L A j ) H }. Proof: The proof of Theorem 1 s n Appendx B. Based on the best response functons n Propostons 2 and 3, t s clear that the game does not admt any domnant strateges. Thus, to fnd the GNE of the HTA game, we present a learnng algorthm based on the nonlnear Gauss-Sedel type method n [22]. Ths algorthm allows the IoT devces to fnd ther GNE strategy n a dstrbuted manner based on the best response dynamcs. The algorthm s defned n Algorthm 1, and ts complexty s derved next. Theorem 2: The complexty of Algorthm 1 s OL 2 ),and ths algorthm converges n at most three teratons. Algorthm 1 GNE Learnng Algorthm for the Heterogeneous Tme Allocaton Game 1 The BS chooses an ntal feasble vector of tme allocatons τ 0 and broadcasts t to ts assocated devces. 2 Each devce ntalzes the current sum of tme fractons as L τ,0. 3 repeat 4 foreach devce 1,..., L do 5 Devce computes ts best response. 6 Devce broadcasts ts newly computed best response to all devces n L. 7 All devces update ther current sum of tme fractons. 8 end 9 untl convergence to GNE; Proof: The proof of Theorem 2 s n Appendx B. Corollary 1: The complexty of the number of computatons done at each devce, to fnd the GNE, s OL). Proof: The proof of Corollary 1 s n Appendx B. The GNE assumes that players are fully ratonal whch mples that they have enough computatonal power to compute ther GNE and can gather precse nformaton on the actons of other devces. Indeed, to compute the GNE, each IoT devce wll need to record the actons of all other IoT devces. Ths can lead to a massve amount of nformaton exchange among the IoT devces at each teraton and also requres all IoT devces to have enough memory to store all of these actons. Ths may not be realstc n an IoT ecosystem n whch devces are heterogeneous and have dfferent computatonal capabltes n terms of memory and processng power. For nstance, IoT devces can range from small sensors and wearable devces that have very lmted computatonal capabltes to smartphones that have hgher computatonal capabltes. To account for such constrants, an alternatve approach to IoT resource allocaton s proposed next. Ths approach extends the powerful framework of cogntve herarchy theory [19], whch can properly accommodate such heterogeneous devce capabltes. B. Cogntve Herarchy Framework for IoT Resource Allocaton Cogntve herarchy theory [19] s a branch of behavoral game theory based on the concept of bounded ratonalty. In general, bounded ratonalty means that each player fnds the best strategy based on the nformaton that s accessble to ths player and the player s computatonal or cogntve capacty as well as the tme avalable for decson makng. γ,ub, f γ γ,ub a = γ, f e e T W1 j M, j = b j T W log1γ j ) j H τ j ) T W1 j M, j = otherwse b j T W log1γ j ) j H τ j ) γ γ,ub

6 7692 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 The CH framework consders a herarchy of players n whch players are dstrbuted nto dscrete levels of ratonalty. Here, players have dfferent belefs about other players dependng on ther ratonalty levels. The belefs formed by each player at each ratonalty level s dependent on the nformaton that the devce can obtan and store, whch s constraned by the devce s memory capacty and the resources avalable at the devce to obtan the nformaton such as energy). A player at level k selects ts strategy based on the strateges of players belongng to levels lower or equal to k. Players belongng to the lowest level 0 do not engage n any ratonal thnkng and, nstead, they select ther strateges randomly. We note that CH sgnfcantly dffers from classcal herarchcal games, such as Stackelberg games and ther varants [23], n whch all players are consdered to have the same ratonalty. Also, n a Stackelberg game, the herarchy levels are defned based solely on the roles of the players, where players are classfed as ether leaders or followers, rather than based on capabltes, as s done n CH. For our problem, devces are restrcted by ther computatonal capabltes and resources. A player at a hgher level of ratonalty can consder more levels when computng ts strategy. Hence, devces must be grouped nto multple levels of ntellgence dependng on ther capabltes. Thus, we propose a CH model that extends the model of [19] and whch has the followng characterstcs 1) The number of players n each level k s dstrbuted accordng to a Posson dstrbuton f wth rate τ. Snce MTDs generally have lower computatonal capabltes than HTDs, t s assumed that MTDs belong to lower ratonalty levels than HTDs. Hence, MTDs belong to level 0 up to some level l whle HTDs belong to ratonalty levels greater than l. Also, we assume that devces belongng to the same CH level are of the same type. 2) A player at level k knows the true proportons f 0), f 1),, f k 1) of players that are at lower ratonalty levels. Snce these proportons do not add to one, a player at level k computes the relatve frequency g k h) of players at a level h 0 h < k) f h) as g k h) =, and g k 1 kh) = 0, h k. Ths =0 f ) assumpton s known as the overconfdence assumpton, and t s a good model for games wth human players. However, n ths case, the players are IoT devces, and t s more lkely to have devces of the same computatonal capabltes. Thus, we relax ths assumpton so that a devce at level k also knows the proporton of devces at the same level k. Ths extenson to CH s challengng because the strategy of each devce wll depend on the strateges taken by devces at the same level thus requrng new solutons that go beyond the exstng lterature [12]. 3) In a classcal CH theory model [19], players at the lowest level 0 are assumed to make ther choces randomly accordng to a unform dstrbuton. In the context of IoT, level 0 devces correspond to MTDs wth very lmted resources such as sensors. However, a random choce of γ by a level 0 devce accordng to a unform dstrbuton mght result n occupyng a large fracton of the tme duraton, whch wll consderably degrade the performance of the remanng devces. Thus, we consder that level 0 devce wll choose ts acton randomly wthn [τ 0,LB, 1] accordng to a decreasng dstrbuton such as an exponental dstrbuton wth mean μ. The tme fracton τ 0,LB s the lower bound on the tme fracton of level 0 devces accordng to 9b). Here, we note that the Posson dstrbuton has been shown n [19] to accurately capture stuatons n whch, as the ratonalty level k grows larger, fewer players wll be at a hgher level than k. As a result, n our settng, ths assumpton holds because there s a lmt on the computatonal capablty of the devces n terms of memory and processng power). Indeed, devces wth hgher computatonal capablty are more expensve and hence are fewer n number. In a typcal IoT ecosystem, the number of HTDs s small compared to the numerous sensors pertanng to dfferent types of applcatons. Due to the aforementoned CH characterstcs, each player wll seek to fnd ts CH strategy usng an teratve process. Durng ths process, each player wll perform lmted steps of strategc thnkng dependng on ts ratonalty level. A devce at level k wll antcpate the strateges of devces at levels 0 to k 1. Gven that the strateges of lower level devces wll help a level k devce to make a more nformed and sutable decson about the IoT network, and, hence, the devce performs k steps of thnkng. In the HTA game, the strategy space of each devce s dependent on the actons of the remanng devces. Further, n CH, each devce has ts own belefs g k about the ratonalty and actons) of the remanng devces dependng on ts ratonalty level k. Consequently, the strategy space of each devce s dependent on the devce s belefs g k. The constrant [ ] L τ = 1 becomes E gk L τ = 1. We denote by S,gk the strategy space [ of devce based on ts belef g k.the ] expresson for E gk L τ of an MTD at a CH level k 1 k l 1) s gven by [ ] k E gk τ = g k h) b h T W log1 γ. 16) L h=0 j L j h)) [ ] The expresson for E gk L τ of an HTD at a CH level k l k) sgvenby [ ] l 1 E gk τ = g k h) b h T W log1 γ L h=0 j L j h)) k g k h) τ j h) 17) h=l1 j L where γ j h) s the CHE strategy of MTD j at level h and τ j h) s the CHE strategy of HTD j at level h. Snce devces belongng to the same CH level are of the same type, the only

7 ABUZAINAB et al.: COGNITIVE HIERARCHY THEORY FOR DISTRIBUTED RESOURCE ALLOCATION 7693 parameter that s dfferent s the channel qualty. Computng 16) and 17) requres evaluaton of the CHE strategy at each level h h k) for all the channel varance values of the devces n L, whch has a computatonal complexty lnear n the number of devces L. Gven that the number of devces L s very large, t s assumed that the BS quantzes the set of channel varances of all devces nto a dscrete set C of C values where C L. Then, t broadcasts the quantzed set to all of ts assocated devces. Hence, 16) and 17) can be rewrtten as [ ] k E gk τ = g k h) L h=0 q C [ ] E gk τ = L l 1 h=0 g k h) q C k N q N q h=l1 g k h) q C b h T W log1 γq 18) h)), b h T W log1 γ q h)) N q τq h), 19) where γq h) s the CHE of an MTD havng quantzed channel varance value q at CH level h, τq h) s the CHE of an HTD havng quantzed channel varance value q at CH level h, and N q s the number of devces havng quantzed channel varance value q. To compute 18) and 19), we can see that each devce at CH level k must know the expected equlbrum strateges of devces at the same level. Ths s challengng snce each devce fnds ts CHE strategy based on ts belefs and not by usng learnng based on the devces actons as n GNE. Snce devces belongng to the same level are of the same type, we assume that each devce beleves that all devces at the same ratonalty level wll choose the same acton. Next, snce the utlty of each player does not depend on the actons, we defne the cogntve herarchy equlbrum of the HTA game as follows. Defnton 2: Astrategyproflea s sad to consttute a cogntve herarchy equlbrum for the HTA game f and only f a k) = arg max U a ), L, 20) a S,gk where k s the CH level of player and S,gk s the strategy space of devce based on ts belef g k. To fnd the CHE, MTD at CH level k 1 solves the followng problem: P k b k mn, 21a) γ W log1 γ k))e γ k)σ2 α 2 P k s.t. 1 e γ k)σ2 ) tk α P k ɛ, 21b) b k T W log1 γ k)) 1 g k k) L 1 k 1 h=0 g k h) q C N q b ) h T W log1 γq h)). 21c) A level k HTD solves the followng optmzaton problem: max τ k) E h [W log1 h 2 P k σ 2 )]τ k), 22a) s.t. P k τ k) E T, 0 τ k) 1, τ k) 1 g k k) L 1 k 1 22b) l N q τq h) g k h) g k h) h=l1 q C h=0 b ) h N q T W log1γ q C q h)). 22c) Based on the optmzaton problems n 21) and 22), to fnd ts optmal CHE strategy, a devce at CH level k needs to compute the actons of the remanng devces for each CH level h 0 h < k) based on ts belef g k. Hence, n CH, each devce fnds ts CHE based on ts own belefs about other devces and not through an teratve learnng process as n GNE. Thus, the CHE s a stable soluton smlar to the stablty of a Nash equlbrum) snce each player, due to ts bounded ratonalty, has no ncentve to devate to another strategy once t computes ts CHE. Therefore, n terms of stablty, the CHE provdes the same stablty as the GNE, but under the more realstc model wth bounded ratonalty. Further, the CHE strategy of a devce at level k s a functon of the channel qualty between the BS and the devce. Thus, devces belongng to the same CH level mght have dfferent CHE strateges. In the followng proposton, we characterze the CHE of the HTA game. Proposton 4: The CHE of the HTA game s the vector a such that for MTD at level k 1theCHEstrategys a k), as shown at the top of the followng page, and the CHE strategy of HTD at level k s a k), as shown at the top of the followng page. Proof: The proof s n Appendx A. Gven ths characterzaton, we can determne the computatonal complexty of fndng the CHE at each CH level. Theorem 3: The complexty of the optmzaton performed at a devce of level k to fnd the CHE s Ok 1) C) where C s the sze of the quantzed channel varance set C. Proof: The proof s n Appendx B. Theorem 3 asserts that the computatonal complexty of the optmzaton performed by each devce at CH level k to fnd the CHE s lnear n the CH level, whereas the computatonal complexty of the GNE learnng algorthm s polynomal n the number of devces L. Sncek L, Theorem 3 valdates that the CH proposed approach s more sutable for IoT devces that are of heterogeneous computatonal capabltes and typcally resource constraned. In the proposed CH approach, all devces obtan ther CHE strategy based on] ther belefs. Ths s reflected n the [ constrant E gk L τ = 1. Hence, the resultng sum of tme fractons of all devces at CHE may not be equal to 1. However, when L τ,che 1, the tme fractons of all devces must be normalzed. The sum of tme fractons at

8 7694 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 γ,ub, f γ γ,ub, a k) = γ T W, fe g k k) b k ) T W 1 k 1 g e k k) L h=0 g k h) q C Nq b h T W log1γq h)) 1, otherwse a k) = mn { E TP, b k 1 k 1 h=0 g k h) q C Nq b h T W log1γ q h)) ) 1 1 k 1 g k k) L g kh) N qτ h=l1 q C q h) l g kh) N q h=0 q C 1 γ γ,ub, b )} h T W log1 γq h)). CHE s gven by τ,che = T W log1 a L M h )) a h ) 23) H where τ,che s the value of tme fracton of devce at CHE and h s the CH level of devce. Let γ gk = e b k ) T W 1 k 1 g k k) h=0 g k h) q C Nq b h T W log1γq h)) 1. The normalzed tme fracton for each MTD wll be ν = M M T W g k k) L T W log1γ,ub ) T W log1a h )) H a h ), f γ γ,ub, T W log1γ ) T W log1a h )) H a h ), f γ gk γ γ,ub, M 1 k 1 b k ) h=0 g kh) q C N b h q T W log1γq h)) T W log1a h )) H a h ) otherwse. 24) The normalzed tme fracton for each HTD s 25), as shown at the top of the next page. We defne γ ν to be the SNR threshold for devce when the tme fracton of devce s ν. In practce, for the frst two transmsson tme slots, the devces transmt accordng to ther computed CHE tme fractons. In our proposed scheme, devces transmt a certan predetermned order, e.g. devce 1, devce 2, untl devce L. Snce the IoT devces are densely deployed, devce 1 can hear and decode the packet transmtted by devce to the BS. Devce 1 can detect the sgnal transmtted by devce ether by carrer sensng or by energy detecton,.e., f the energy of the transmtted sgnal exceeds a predefned threshold, as done for example n CSMA protocols [24]. Also, devce 1 can decode the packet and determne from ts header f the packet was transmtted by devce. Thus, devce 1 starts transmttng as soon as devce completes ts transmsson. Then, each devce records the tmes t,1 and t,2 durng whch t accesses the channel n the frst and second tme slots, respectvely, and computes the tme duraton T as the dfference t,2 t,1. Fnally, devce normalzes ts tme fracton as ν = τ,che T t,2 t,1. In the proposed normalzaton process, each devce computes ts normalzed tme fracton based on the transmsson tmes of the remanng IoT devces. Thus, transmsson s not nterrupted durng the process of normalzaton. Further, there are no extra control messages requred to perform the normalzaton. Such a normalzaton scheme does not requre nformaton exchange among the devces, and, thus, t does not result n sgnalng overhead. Hence, the proposed normalzaton s sutable for dynamc networks where devces jon or leave the network. In the followng proposton, we compare the performance of the CHE and GNE strategy profles of the HTA game. We show condtons under whch the CHE strategy can consttute also a GNE strategy, and show that n the other cases.e. when the CHE strategy profle does not consttute a GNE) there always exsts at least a GNE that yelds better performance, n terms of the total system utlty. Naturally, such a performance mprovement stems from the fact that the GNE requres more nformaton and more computatons per devce. Theorem 4: Let d = d 1, d 2,...,d L ) be a strategy profle correspondng to the normalzed CHE tme fractons wth d = ν f devce s an HTD and d = γ ν f devce s an MTD. Then we have the followng: If L τ,che 1, the strategy vector d s a GNE strategy of the HTA game f for every level 0 devce, ts CHE strategy satsfes a 0) mn{γ,γ,ub}. If L τ,che < 1, the CHE strategy vector of the HTA game s a GNE f E M T W log1γ ) H T WP M M T W log1γ,ub ) 1, The CHE equlbrum strategy for each MTD s a k ) = γ f M and a k ) = γ,ub f M M k 0, and The CHE equlbrum strategy for each HTD s a k ) = E T WP. Otherwse, there exsts at least one GNE that yelds better performance than the CHE, n terms of the total system utlty. Proof: The proof of Theorem 4 s n Appendx B. Theorem 4 provdes condtons under whch the strategy profle d correspondng to the normalzed CHE tme fractons consttutes a GNE strategy. Hence, the devces do not lose n performance by havng lmted nformaton and computatonal capabltes. However, f these condtons are not met, the

9 ABUZAINAB et al.: COGNITIVE HIERARCHY THEORY FOR DISTRIBUTED RESOURCE ALLOCATION 7695 ν = { mn E 1 TP, g k k) L 1 k 1 h=l1 g kh) q C τ q h) l h=0 g k h) q C N q T W log1a h )) H a h ) )} b h T W log1γq h)) 25) CHE strategy does not consttute, n most cases, a GNE strategy, and the performance s degraded due to bounded ratonalty. C. Sgnalng Overhead The communcaton overhead s specfcally evaluated n terms of the total number of bts exchanged to reach the CHE and GNE solutons, respectvely. In the proposed CH scheme, the BS needs to broadcast one packet to ts assocated devces whenever devces enter or leave the network e.g., durng handshakng or devce regstraton). The broadcast packet ncludes the updated network nformaton necessary for the devces to compute the CHE. Thus, the broadcast packet wll nclude the number of devces N, the quantzed channel varance set C, and the proporton of devces f q havng quantzed channel value q C. The BS acqures the CSI of each devce upon regstraton to the network. Further, the packet ncludes the parameters values and QoS constrants of each type of devces. For each type h of MTDs, the packet ncludes the packet sze b h and the tme constrant t h. For each type h of HTDs, the packet ncludes the energy constrant E h. Thus, the sze of the broadcast packet depends on the number of bts requred to represent each of the aformentoned parameters. Concernng the number of devces N, the number of requred bts B N depends on N max, the maxmum number of devces that the BS can serve. Assumng that N max s , the requred number of bts wll be B N = 14. As for the channel nformaton and consderng a statc settng, the channel varance value α q s typcally less than one, and the number of requred bts B α s 4 bts assumng that the fractonal part of the quantzed varance value s represented by 1 dgt. Also, the proporton of devces havng a quantzed varance value α q can be represented by B f = 7 assumng that the fractonal part of each proporton s captured by 2 dgts. Next, for each type of MTD, the number of requred bts for the packet sze s B b = 10 snce MTDs usually have small packet szes < 1000 bts). Further, assumng that the maxmum delay s 1000 ms and that the tme slot duraton s 1 ms [25], [26], the number of bts B t requred to represent the tme constrant of each type of MTDs s B t = 10. For HTDs, assumng that the energy upper bound E h n μ J) s represented by two dgts, the number of bts requred wll be B E = 7. Hence, the number of bts to represent each type of MTD s B M = B b B t = 20 and for each type of HTD s B H = B E = 7. The sze of the packet n bytes) s M = 14C7N H 20N M 8 where N H and N H are the number of types of HTDs and MTDs respectvely. For a network wth two types of MTDs and one type of HTDs and when C = 5, the sze of the packet wll be 15 bytes. The typcal packet sze of an MTD wth tme crtcal applcaton [23] ranges from 40 to 1000 bytes. Thus, the proposed CH scheme does not ncur sgnfcant overhead. As for the GNE, the overhead stems manly from the actons exchanged among the devces to obtan the GNE. In Theorem 2, t s shown that, n the worst case, three rounds of communcaton are requred among the devces to converge to the GNE. Assumng that the fractonal part of the tme fracton s represented by 3 dgts, the number of bts requred to represent the tme fracton s 10. Thus, the total number of bts exchanged to reach the GNE s 30 L. When the number of devces s 1000, the total number of bts exchanged wll be Hence, the overhead of the GNE soluton s sgnfcantly hgher than that of the proposed CHE. IV. SIMULATION RESULTS AND ANALYSIS In our smulatons, we set the bandwdth W to 100 MHz, the tme perod T to 3 ms, the nose varance σ 2 to 90.8 dbm, and ɛ to The value of the tme perod s chosen to be small enough to be sutable for IoT devces that have ultra low latency requrements, yet adequate to accomodate the mnmum tme requrements for an IoT wth a massve number of devces. We consder three types of devces: the frst two are MTDs whle the thrd s an HTD. MTD type 1 devces represent MTDs wth strct latency requrements such as e-healthcare sensors, whle MTD type 2 devces represent MTDs wth relaxed delay constrants such as smart meters. Thus, the packet sze for the MTD types 1 and 2 are set as 128 and 50 bytes respectvely, and the latency constrant for MTD types 1 and 2 are set to be 5 ms and 1 s respectvely n lne wth the gudelnes of [25] [28]. The transmsson powers for MTD types 1 and 2 and for HTDs are 0.1, 0.1, and 0.5 W respectvely. The varance of the Raylegh fadng dstrbuton of all channels s α 2 = 0.1.TherateofthePosson dstrbuton of devces over the CH levels s τ = 1. Ths value mples that the proporton of devces n each CH decreases wth the CH levels, and the probablty that a devce belongs to a CH level hgher than 3 s neglgble. Thus, t s sutable for our system because the maxmum number of CH levels consdered s 3. Frst, we assess the performance of the GNE soluton of the HTA game by computng the prce of anarchy PoA) [23] as follows. A. Prce of Anarchy of the GNE Soluton The PoA s the rato between the optmal centralzed soluton that maxmzes total utlty, and the mnmum total utlty that can be possbly acheved by a GNE strategy. Thus, the PoA s a measure of how much the performance of the system degrades when the GNE s used. In our problem, the utlty of HTDs s expressed n terms of the acheved rate, whereas the utlty of each MTD s expressed n terms of the energy cost.

10 7696 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 Fg. 1. Prce of anarchy of HTDs vs. the number of devces. Fg. 3. Sum of CHE tme fractons vs. the number of devces for dfferent values of the mean of the dstrbuton of the level 0 devces tme fractons. Fg. 2. Prce of anarchy of MTDs vs. the number of devces. Hence, n order to have a better evaluaton of the performance of MTDs and HTDs, we compute one PoA for the MTDs and another PoA for HTDs. The PoA of MTDs s defned as the rato of the maxmum energy consumed that can be possbly acheved by a GNE and the mnmum possble energy consumed by MTDs. The PoA of HTDs, on the other hand, s computed as the rato of maxmum total rate of HTDs and the mnmum total rate that can be possbly acheved by a GNE soluton. Fg. 1 shows the PoA of HTDs versus the network sze. As shown n Fg. 1, the PoA of HTDs s one for the consdered network szes snce the GNE s unque n ths case. Fg. 2 shows the PoA of MTDs as a functon of the network sze. Agan, the PoA of MTDs s also one for the consdered network szes snce the GNE s unque. Thus, Fgs. 1 and 2 show that the GNE soluton provdes stable performance for both MTDs and HTDs. Next, we evaluate the CHE soluton as follows. B. CHE Soluton Evaluaton For the consdered smulaton values, we compute the CHE soluton when the network sze vares between 1000 and n steps of Snce level 0 devces choose ther tme fractons randomly, 1000 samples of level 0 devces tme fractons are generated, and the total sum of CHE tme fractons s computed. Then, the average sum CHE tme fracton as well as the average tme fracton of each MTD type 2 and each HTD are computed. We consder two cases: A frst case n whch the mean μ of the dstrbuton of tme fractons of level 0 devces s 2τ 0,LB where τ 0,LB s a lower Fg. 4. Normalzed CHE tme fracton of type 2 MTDs vs. the number of devces. bound on the tme fractons of level 0 devces and a second case wth μ = 3τ 0, LB. Fg. 3 shows the average sum of CHE tme fractons versus the network sze. As shown n Fg. 3, the average sum of CHE tme fractons ncreases wth the network sze where t exceeds one for network szes greater than When μ ncreases to 3τ 0,LB, the sum of CHE tme fractons ncreases for each consdered network. Ths s because as μ ncreases, level 0 devces have a hgher chance of transmttng wth larger tme fractons, whch eventually ncreases the sum of CHE tme fractons. Fg. 4 shows the normalzed CHE tme fractons for type 2 MTDs as the number of devces vares for both consdered values of the mean μ. Whenμ = 2τ 0,LB and when the number of devces s less than or equal to 8000, each type 2 MTD transmts wth ts optmal tme fracton whch s In ths case, the sum of CHE tme fractons of all devces s less than one as shown n Fg. 3. However, snce the sum of CHE tme fractons s greater than one for larger numbers of devces, the normalzed CHE tme fracton of each type 2 MTDs decreases untl t reaches when the number of devces s Fg. 4 also shows that when the number of devces s greater than 8000, the tme fracton allocated for type 2 MTDs decreases wth the mean μ due to the ncrease n the sum of CHE tme fractons. For smaller numbers of devces, the CHE tme fracton s the same for both values of the mean μ snce the sum of CHE tme fractons s less than one.

11 ABUZAINAB et al.: COGNITIVE HIERARCHY THEORY FOR DISTRIBUTED RESOURCE ALLOCATION 7697 Fg. 5. Normalzed CHE tme fracton of HTDs vs. the number of devces. Fg. 5 shows the normalzed CHE tme fractons for HTDs as the number of devces vares for two values of the mean μ. When the number of devces s less than or equal to 8000 and μ = 2τ 0,LB, each HTD transmts wth ts hghest possble tme fracton whch s For hgher number of devces, the sum of CHE tme fractons ncreases beyond one, and the normalzed CHE tme fracton for each HTD decreases untl t reaches Also, the normalzed CHE tme fracton of each HTD decreases wth the mean μ when the number of devces s greater than 8000, snce the sum of CHE tme fractons ncreases wth μ and s greater than one. C. CHE Performance Evaluaton The performance of the CHE soluton s assessed n terms of the average total utltes of MTDs and HTDs, and the average percentage of devces wth satsfed QoS. Also, we assess the average performance of the GNE soluton. The state-of-theart GNE soluton s used as a game-theoretc baselne snce the GNE assumes that the players are fully ratonal, whle the CHE soluton assumes that players belong to dscrete levels of bounded ratonalty. Further, for the consdered smulaton values, the GNE s unque, and, thus, performance of the GNE soluton serves as a bound on the performance of the CHE soluton. Ths s acheved by generatng 1000 random ntal vectors of the devces tme fractons. Then, for each generated random vector, the total utlty and the percentage of devces wth satsfed QoS of the resultng GNE are computed. Then, the average total utlty and the average percentage of devces are computed. We also consder a non game-theoretc baselne, the equal tme polcy that splts the tme duraton T equally among the IoT devces. Fg. 6 shows the percentage of devces wth satsfed QoS constrants resultng from the CHE, the average GNE, and the equal tme polcy as a functon of the number of devces. For the GNE, the percentage s mantaned at 100% for all consdered network szes. For the CHE and when μ = 2τ 0,LB, Fg. 6 shows that the percentage of devces wth satsfed QoS constrants s 100% when the number of devces s less than or equal to When the number of devces s greater than 8000, the percentage of devces wth satsfed QoS constrants decreases slghtly untl t reaches 97.35%. Ths decrease s manly due to the fact that the normalzed Fg. 6. Percentage of devces wth satsfed QoS vs. the number of devces for dfferent values of the mean of the dstrbuton of the level 0 devces tme fractons. tme fractons of some of the CH level 0 devces drops below the lower bound, whereas the tme fracton of type 2 MTDs and HTDs are above the lower bound as shown n Fgs. 4 and 5. When μ = 3τ 0,LB, the CHE mantans the same percentage of devces wth satsfed QoS constrants as the case when μ = 2τ 0,LB for network szes less than or equal to Then, the percentage of devces wth satsfed QoS drops to 96% as the network sze ncreases to As for the equal tme polcy, from Fg. 6, we can see that the percentage of devces wth satsfed QoS constrants s 100% for a network sze less than When the number of devces s greater than 8000, the percentage of devces wth satsfed QoS constrants decreases to 73%. Ths s because, n ths case, the tme fracton assgned to each devce drops below the lower bound on the tme fracton of each HTD. Thus, Fg. 6 shows that the CHE soluton can mantan a stable performance, n terms of the percentage of devces wth satsfed QoS, as good as the GNE soluton and always outperformng the equal polcy soluton. Next, we compute the mnmum, maxmum, and average total rate of HTDs and total energy of MTDs acheved by the GNE soluton, the average total rate of HTDs and total energy of MTDs acheved by the CHE soluton, and the total rate of HTDs and the total energy of MTDs of the equal tme polcy. Fg. 7 shows the total acheved rate of HTDs versus the network sze for the consdered value of the mean μ. For all network szes, the GNE soluton s unque and thus the mnmum, maxmum, and average total rates of HTDs acheved by GNE are the same, and they ncrease wth the network sze. For the CHE soluton, when μ = 2τ 0,LB,the total rate of the CHE soluton s the same as the total rate of the GNE soluton for network szes less than Ths s because the CHE tme fracton of each HTD s the optmal soluton, and the sum of CHE tme fractons s less than one as shown n Fgs. 3 and 5. For network szes larger than 8000, the total rate of the CHE becomes less than total rate of the GNE soluton snce the CHE tme fractons of all devces decrease due to normalzaton. When μ s ncreased to 3τ 0,LB, the total rate acheved by CHE decreases for network szes greater than 8000 snce the sum of CHE tme fractons ncreases and s greater than one. For the equal tme polcy and for

12 7698 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 Fg. 9. Prce of bounded ratonalty of MTDs vs. the number of devces. Fg. 7. Total rate of HTDs vs. the number of devces. the total energy consumed by MTDs stays fxed at mj for the consdered network szes. Fg. 8 shows that the energy consumed by the CHE soluton, when μ = 2τ 0,LB, s reduced by around 78% compared to the equal tme polcy. In order to assess the effcency of the CHE soluton, we defne a performance metrc smlar to the PoA, but talored to the CH approach. We call ths metrc the prce of bounded ratonalty PoB). Fg. 8. Total energy consumed by MTDs vs. the number of devces. all consdered network szes, the total rate stays fxed at Mbts/sec. Ths s because the tme fracton assgned to each devce decreases wth the network sze and becomes less than the HTD optmal value for the consdered network szes. Also, the value of the total rate s fxed snce we are consderng HTDs of the same type. Thus, Fg. 7 shows that the CHE soluton mantans the same performance as that at the GNE for network szes less than For network szes larger than 8000, the degradaton of the total rate usng the CHE soluton, compared to the GNE soluton, s only around 11%. Also, the CHE soluton can brng a two-fold ncrease n the total rate of HTDs compared to the equal tme polcy. Fg. 8 shows the total consumed energy as a functon of the network sze. Here, we note that, for the consdered networks, the GNE s unque and the total energy consumed by MTDs at ths GNE s ncreasng wth the network sze. For the CHE, when μ = 2τ 0,LB, the average total energy consumed by MTDs s hgher than the total energy of the GNE soluton for all consdered network szes. Ths s due to the fact that, because of ther lmted capabltes, level 0 devces choose ther tme fractons randomly whle n the GNE soluton level 0 devces choose the optmal tme fracton. When μ s ncreased to 3τ 0,LB, the total energy consumed ncreases for all consdered network szes snce level 0 devces transmt wth hgher tme fractons. For the equal tme polcy, D. Prce of Bounded Ratonalty The PoB s defned as the rato of the optmal total utlty and the total utlty acheved usng the CHE strategy. Due to the dfferent performance metrcs between MTDs and HTDs and smlar to the case of PoA, we defne one PoB for MTDs and another PoB for HTDs. The PoB of MTDs s defned as the rato of the total energy consumed by MTDs usng the CHE soluton and the mnmum total energy consumed by MTDs. The PoB of HTDs, on the other hand, s defned as the rato of the maxmum total rate of HTDs and the total rate of HTDs acheved by the CHE soluton. Fg. 9 shows the PoB of MTDs versus the IoT network sze for the consdered values of μ. In ths fgure, we can see that when μ = 2τ 0,LB, the PoB of MTDs s around 2.27 when the network sze s Then, the PoB ncreases wth the network sze untl t reaches 2.81 when the network sze s Ths s due to the fact that the number of level 0 devces that choose ther tme fractons randomly ncreases wth the network sze, whch results n a hgher energy consumpton compared to the energy consumed by the GNE soluton. For the case when μ ncreases to 3τ 0,LB, the PoB of MTDs ncreases for each consdered network sze. Ths ncrease s manly due to the fact that as μ ncreases, the probablty that a CH level 0 devce transmts wth a hgher tme fracton ncreases, whch ncreases the total energy consumed. Fg. 10 also shows that the PoB of MTDs ncreases at a low rate wth the network sze. Thus, the proposed CH approach can clearly mantan a stable performance of MTDs for larger network szes. Fg. 10 shows the PoB of HTDs versus the IoT network sze for the consdered values of the mean μ. When the network sze s less than 8000 and when μ = 2τ 0,LB,thePoBofHTDs s one. Ths s because the CHE tme fracton of each HTD s the tme fracton that maxmzes ts utlty. As the network sze ncreases beyond 8000, the PoB of HTDs ncreases untl t reaches 1.15 when the network sze s Ths ncrease

13 ABUZAINAB et al.: COGNITIVE HIERARCHY THEORY FOR DISTRIBUTED RESOURCE ALLOCATION 7699 we have log χ γ ) = log P logw) log log1 γ ) γ σ 2 α 2 P.Theterm log log1 γ ) s convex n γ snce log log1 γ ) s concave n γ. Also, the term γ σ 2 α 2 P s lnear n γ. Hence, the functon log χ γ ) s convex n γ. It follows that the energy functon χ γ ) s convex n γ as t s logconvex n γ. Then, the utlty of MTD s concave snce U γ ) = χ γ ). Fg. 10. Prce of bounded ratonalty of HTDs vs. the number of devces. n PoB s due to a decrease n the normalzed CHE tme fracton of each HTD as shown n Fg. 5. When μ ncreases to 3τ 0,LB, the PoB of HTDs ncreases for network szes larger than 8000 due to the decrease n the normalzed CHE tme fracton accordng to Fg. 5. Fg. 10 shows that the PoB of HTDs ncreases at a low rate wth the network sze. Thus, the proposed CH approach can mantan a stable performance of HTDs for larger network szes. V. CONCLUSION In ths paper, we have consdered the problem of dstrbuted uplnk tme allocaton n an IoT network n whch the IoT devces have heterogeneous qualty-of-servce requrements. We have formulated ths problem as a noncooperatve game that takes nto account the heterogeneous requrements of the IoT devces. In ths game, the players are the IoT devces, and ther actons are to choose the tme fractons necessary to ensure ther qualty-of-servce requrements. In the proposed game, the strategy set of each devce s dependent on the actons taken by the other devces. Hence, we have frst characterzed the set of GNEs. Moreover, we have proposed an algorthm to fnd the GNE of the devces and shown that the computatonal complexty s polynomal n the number of devces. Then, we have proposed a novel soluton usng cogntve herarchy theory to take nto account the heterogeneous computatonal capabltes of the IoT devces. Thus, the proposed CH soluton provdes a more realstc soluton than the GNE, whch assumes that all players have the same capabltes. We have characterzed the cogntve herarchy equlbrum and compared t analytcally to the GNE. Extensve smulatons have been conducted to thoroughly assess the varous performance tradeoffs of the proposed approach. Fnally, we note that, beyond the IoT applcaton treated here, the proposed cogntve herarchy framework can be generalzed to any wreless network n whch heterogenety and bounded ratonalty are key features. APPENDIX A A. Proof of Lemma 1 By applyng the log functon to the energy functon χ γ ) = P W log1 γ )e γ σ 2 α 2 P 26) B. Proof of Proposton 1 For each HTD, the utlty functon s lnear n ts acton a = τ and hence t s also concave. For each MTD, ts shown n Lemma 1 that the utlty s concave n ts acton a = γ. Also, for each devce the strategy space S a ) s nonempty, closed and convex. The result follows drectly from [22, Th. 4.1]. C. Proof of Proposton 2 The utlty of HTD s an ncreasng lnear functon of τ. Thus, the optmal value of τ s ts upper bound. From the constrants of 3b), we get 15). D. Proof of Proposton 3 As shown n Lemmas 1 and 2, the utlty of MTD s concave and attans ts maxmum at γ. Hence, gven any strategy vector a,mtd chooses a = γ f γ S a ). Otherwse, MTD chooses the upper bound of ts strategy set a = α2 P log1 t ɛ) f the upper bound s less than γ σ 2 snce the utlty s ncreasng over the strategy set n ths case. The last case s when the lower bound s greater than γ,nwhch MTD chooses the lower bound snce the utlty functon s decreasng over the strategy set. E. Proof of Proposton 4 Snce the utlty of MTD s concave n γ, the result follows usng a smlar argument as that n the proof of Proposton 3. For HTD, the utlty functon s ncreasng n ts strategy a. Thus, the optmal value s the upper bound of the strategy space of HTD. APPENDIX B A. Proof of Theorem 1 For the frst case,.e. when M T W log1γ ) E H T WP M M T W log1γ,ub ) 1, t s clear accordng to the best response equatons n Propostons 2 and 3 that for the acton profle a = γ M and a = α2 P log1 t ɛ) M M, and a E σ 2 = T WP for each HTD, no devce has the ncentve to change ts strategy snce a s the optmal soluton of the utlty of devce. Next, we show there exsts no other acton profle that consttutes a GNE. For any acton profle a other than a, a devce M wll always change ts strategy to the optmal strategy γ f a < γ snce γ s the optmal soluton of ts utlty and t yelds a lower tme

14 7700 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 fracton. Also, any devce M M has the ncentve to change ts strategy to the optmal strategy γ,ub f a <γ,ub resultng n a lower tme fracton. Smlarly, any HTD wll E reduce ts strategy to T WP f a > E E T WP snce T WP s the upper bound. Let D be the set of all such devces. For the newly formed acton profle a s.t. a = a f D and a = a f L D, D H E T WP L D) M D M T W log1a ) T W log1a ) E H T WP L D) H a < M T W log1γ ) M M T W log1γ,ub ) 1 snce Ua ) < Ua ) for a > a for each MTD n L D and a < a for each HTD n L D, and hence each devce n L D has the ncentve to change ts acton to a. For the second case,.e. H E M T W log1γ ) T WP M M T W log1γ,ub ) > 1 for any acton profle a / N, smlar to the frst case, any devce D D s the same set defned before) has the ncentve to change ts strategy to γ f D M, to γ,ub f D M M E ) and to T WP f D H. For the newly formed acton profle a s.t. a = a f D and a = a f L D, t can be easly verfed that each acton a N s a GNE. No devce n A j has an ncentve to change ts strategy snce a s the optmal value of ts utlty and all other devces n L A j cannot mprove ther utltes snce the sum of tme fractons correspondng to a s one and an mprovement n ther utltes wll requre a hgher allocated tme fracton. B. Proof of Theorem 2 Frst, we show that Algorthm 1 converges n at most three teratons. We assume wthout loss of generalty that the ntal sum of tme fractons allocated to the devces s one. In the frst teraton of the algorthm, all MTDs that can mprove ther utltes by reducng ther current tme fractons wll change ther strateges. If there are no such MTDs that can mprove ther utltes n the frst round, the algorthm stops snce no devce wll change ts strategy. The worst-case scenaro occurs when durng the frst round the last MTD that changes ts strategy by reducng ts tme fracton s the last n the order. Ths s because by the end of the frst round, the sum of tme fractons of all devces s less than one. Hence, n the second round, t s only possble for the devces to mprove ther utltes by choosng hgher tme fractons. The algorthm termnates n the thrd round because no other devce can further mprove ts utlty. C. Proof of Corollary 1 Based on Theorem 2, the complexty of the computatons done at each devce, to fnd the GNE, s OL) snce Algorthm 1 converges n at most three teratons. Each devce solves at most two best response optmzaton problems to converge to ts GNE strategy. The complexty of fndng the best response based on Propostons 2 and 3 s OL). D. Proof of Theorem 3 To fnd the CHE, a devce at level k needs to evaluate CHE strateges of all devces assumng that they are level 1 to k based on ts belefs. Based on Propostons 4 and 5, fndng the strategy of each devce takes O1) tmes. Hence, to fnd the strategy of all devces of each level takes OC) tmes. Thus, to fnd the strategy of devces for all levels up to k takes Ok 1) C). E. Proof of Theorem 4 For the frst case, we start by nvestgatng the strategy profle d. Snce level 0 MTDs choose ther tme fractons randomly, the CHE strategy of level 0 MTD could possbly be γ ν < mn{γ,γ,ub}. In ths case, the normalzed tme fracton of a level 0 devce wll be greater than the tme fracton that maxmzes ts utlty. Hence, a level 0 MTD has the ncentve to change ts strategy to mn{γ,γ,ub}, and, hence, the CHE soluton wll not be a GNE. In the second case when γ ν > mn{γ,γ,ub}, the normalzed tme fracton of a level 0 MTD wll be less than the optmal tme fracton. In ths case, a level 0 MTD cannot ncrease ts tme fracton snce the sum of the CHE tme fractons s one. Now for any MTD belongng to a level hgher than 0, f γ γ,ub,thechestrategyofmtd wll be γ,ub and γ ν wll be greater than or equal to γ,ub. In ths case, the normalzed tme fracton ν wll be lower than the lower bound of the tme fracton whch s the value of the tme fracton that corresponds to γ,ub ). Hence, MTD has no ncentve to reduce further ts allocated tme fracton. For the whch e T W 1 k 1 g k k) b k h=0 g k h) q C k Nq b h case n ) T W log1γq h)) 1 γ γ,ub, 27) the CHE strategy of MTD s γ and hence γ ν wll be greater than or equal to γ. Thus, MTD has no ncentve to ncrease ts γ snce ths wll decrease the utlty. Also, MTD cannot ncrease ts tme fracton by reducng ts γ snce the sum of the normalzed tme fractons s one. When e b k T W 1 k 1 g k k) L h=0 g k h) q C k Nq ) b h T W log1γq h)) 1 >γ, γ ν wll be greater than γ usng an argument smlar to that n the prevous case, and, thus, MTD has no ncentve to change ts strategy d.nohtd can ncrease ts allocated tme fracton ν snce the sum of the normalzed tme fractons s one. For the second case, f M T W log1γ ) E H T WP M M T W log1γ,ub ) > 1, we know from Theorem 1 that the GNE s not unque and that the sum of tme fractons s one. Hence, the CHE strategy s not a GNE. Thus, for the CHE strategy d, there exsts a subset of devces that can mprove ther utltes untl the sum of tme fractons s one. Otherwse when M T W log1γ ) E H T WP 1, we know from Theorem 1 that M M T W log1γ,ub ) the GNE s unque and hence based on Theorem 1 the CHE strategy s a GNE accordng to Theorem 1 f the CHE strategy s for any MTD d k ) = γ f M and d k ) = γ,ub f

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Conf., Vctora, BC, Canada, May 2008, pp [25] IEEE p Machne to Machne M2M) Evaluaton Methodology Document EMD). IEEE Broadband Wreless Access Workng Group, IEEE Standard p-11/0014, [26] IEEE p Machne to Machne M2M) Communcaton Techncal Report. IEEE Broadband Wreless Access Workng Group, IEEE Standard p-10/0005, [27] M. Popovć, D. Drajć, and S. Krčo, The mpact of HSPA core network features on latency for M2M and OG-lke traffc patterns, n Proc. IEEE Telecommun. Forum Telfor TELFOR), Belgrade, Serba, Nov. 2013, pp [28] 5G empowerng vertcal ndustres, 5G-PPP, Whte Paper, Feb Nof Abuzanab receved the B.E. degree n computer and communcatons engneerng from the Amercan Unversty of Berut, Lebanon, and the M.S. and Ph.D. degrees from the Unversty of Maryland, College Park, MD, USA. She s currently a Post-Doctoral Fellow wth the Bradley Department of Electrcal and Computer Engneerng, Vrgna Tech VT). Pror to jonng VT, she was a Post- Doctoral Fellow wth the Natonal Insttute of Computer Scence and Automaton, France. Her research nterests nclude the Internet of Thngs IoT), resource allocaton n wreless networks, IoT, network codng, game theory, and wreless networks securty. Wald Saad S 07 M 10 SM 15) receved the Ph.D. degree from the Unversty of Oslo n He s currently an Assocate Professor wth the Department of Electrcal and Computer Engneerng, Vrgna Tech, where he leads the Network Scence, Wreless, and Securty Laboratory, wthn the Wreless@VT Research Group. Hs research nterests nclude wreless networks, game theory, cybersecurty, unmanned aeral vehcles, and cyberphyscal systems. He was a recpent of the NSF CAREER Award n 2013, the AFOSR summer faculty fellowshp n 2014, and the Young Investgator Award from the Offce of Naval Research n He was also a recpent of the 2015 Fred W. Ellersck Prze from the IEEE Communcatons Socety. He was the author/co-author of sx conference best paper awards at WOpt n 2009, ICIMP n 2010, the IEEE WCNC n 2012, the IEEE PIMRC n 2015, the IEEE SmartGrdComm n 2015, and EuCNC n In 2017, he was named College of Engneerng Faculty Fellow at Vrgna Tech. He currently serves as an Edtor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON COMMUNICATIONS,and the IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY.

16 7702 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 12, DECEMBER 2017 Choong Seon Hong S 95 M 97 SM 11) receved the B.S. and M.S. degrees n electronc engneerng from Kyung Hee Unversty, Seoul, South Korea, n 1983 and 1985, respectvely, and the Ph.D. degree from Keo Unversty, Mnato, Japan, n In 1988, he joned KT, where he was nvolved n broadband networks as a Member of Techncal Staff. In 1993, he joned Keo Unversty. He was wth the Telecommuncatons Network Laboratory, KT, as a Senor Member of Techncal Staff and the Drector of the Networkng Research Team untl Snce 1999, he has been a Professor wth the Department of Computer Engneerng, Kyung Hee Unversty. Hs research nterests nclude future Internet, ad hoc networks, network management, and network securty. He s a member of ACM, IEICE, IPSJ, KIISE, KICS, KIPS, and OSIA. He has served as the general char, a TPC char/member, or an organzng commttee member for nternatonal conferences, such as NOMS, IM, APNOMS, E2EMON, CCNC, ADSN, ICPP, DIM, WISA, BcN, TINA, SAINT, and ICOIN. In addton, he s currently an Assocate Edtor of the IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, theinternatonal Journal of Network Management, and the Journal of Communcatons and Networks, and an Assocate Techncal Edtor of the IEEE Communcatons Magazne. H. Vncent Poor S 72 M 77 SM 82 F 87) receved the Ph.D. degree n EECS from Prnceton Unversty n From 1977 to 1990, he was on the faculty of the Unversty of Illnos at Urbana Champagn. Snce 1990, he has been on the faculty at Prnceton, where he s currently the Mchael Henry Strater Unversty Professor of Electrcal Engneerng. From 2006 to 2016, he served as the Dean of Prnceton s School of Engneerng and Appled Scence. Hs research nterests are n the areas of nformaton theory and sgnal processng, and ther applcatons n wreless networks and related felds, such as smart grd and socal networks. Among hs publcatons n these areas s the book Informaton Theoretc Securty and Prvacy of Informaton Systems Cambrdge Unversty Press, 2017). Dr. Poor s a member of the Natonal Academy of Engneerng and the Natonal Academy of Scences, and s a foregn member of the Royal Socety. He s also a fellow of the Amercan Academy of Arts and Scences, the Natonal Academy of Inventors, and other natonal and nternatonal academes. He receved the Marcon and Armstrong Awards of the IEEE Communcatons Socety n 2007 and 2009, respectvely. Recent recognton of hs work ncludes the 2016 John Frtz Medal, the 2017 IEEE Alexander Graham Bell Medal, honorary professorshps at Pekng Unversty and Tsnghua Unversty, both conferred n 2017, and a D.Sc. honors causa from Syracuse Unversty n 2017.