Spectrum Trading with Insurance in Cognitive Radio Networks

Transcription

1 212 Proceedngs IEEE INFOCOM Spectrum Tradng wth Insurance n Cogntve Rado Networks Hamng Jn, Gaofe Sun, Xnbng Wang Dept. of Electronc Engneerng Shangha Jao Tong Unversty, Chna Emal: {jnhamng, sgf hb, xwang8}@sjtu.edu.cn Qan Zhang Dept. of Comp. Sc. and Engn HK Unv. of Sc. and Tech., HongKong Emal: qanzh@cs.ust.hk Abstract Market based spectrum tradng has been extensvely studed to realze effcent spectrum utlzaton n cogntve rado networks CRNs). In ths paper, we utlze the concept of nsurance n spectrum tradng so as to mprove spectrum effcency n CRNs. We show that by addtonally purchasng a specfcally desgned nsurance contract from a PU, an SU can mprove ts utlty snce t wll be nsured aganst the potental accdent,.e., transmsson falure ncurred by excessvely low SINR. Therefore nsurance provdes SUs more ncentve to purchase PUs channels and spectrum utlzaton n CRNs can be mproved. In ths paper, the orgnal spectrum market ncludng multple PUs and multple SUs are modeled as a hybrd market consstng of a spectrum market and an nsurance market. In ths hybrd market PUs serve as spectrum sellers as well as nsurers and SUs act as spectrum buyers as well as nsureds. We further model the hybrd market game as a four-stage Bayesan game between PUs and SUs. We characterze the second-best Pareto optmal SBPO) market allocatons and players perfect Bayesan equlbrum PBE) strateges. Furthermore, through extensve smulaton, we have demonstrated that at the PBE, hgh rsk and low rsk SUs wll respectvely experence mprovement n ther utltes for approxmately 23.5% and 4.6%. I. INTRODUCTION Cogntve rado CR) has been regarded as a promsng paradgm to tackle the spectrum scarcty and neffcent spectrum usage n current wreless communcaton networks. CRNs are desgned based on the noton of dynamc spectrum sharng where cogntve rado users have the capablty to opportunstcally share the rado spectrum. Among mechansms for dynamc spectrum sharng, marketdrven spectrum tradng s one of the most commonly utlzed frameworks. In [12], Nyato et al. analyzed the spectrum tradng between multple PUs and SUs. They modeled SUs dynamc and evolutonary behavor as an evolutonary game and the competton among the PUs as a noncooperatve game. In [3], Km et al. studed the prce and qualty compettons of duopoly CRNs. The CRNs were modeled as a duopoly W-F 2. market n whch two wreless servce provders compete for customers and spectra. In [5] and [11], Duan et al. studed under spectrum supply uncertanty the optmal prcng and nvestment of a cogntve moble vrtual network operator C-MVNO). In short-term, a C-MVNO can obtan spectrum opportuntes dynamcally by sensng and leasng the spectrum from the spectrum owner. Ths paper models the spectrum negotaton between C-MVNO and SUs as a Stackelberg game and analyze ts equlbrum propertes. Moreover, the ssue of compettve prcng n CRNs was studed n [9] n whch Nyato et al. modeled the spectrum tradng between PUs and SUs as a Bertrand game and analyzed ts Nash equlbrum. Moreover, aucton based spectrum tradng has also been proved to be an effectve framework for dynamc spectrum sharng and lteratures abound n the realm of spectrum aucton. In [15], Zhou et al. proposed a truthful and computatonally-effcent spectrum aucton mechansm, VER- ITAS. Zhou et al. n [6], proposed TRUST, the frst soluton enablng truthful double spectrum auctons. In [16], Ja et al. showed that the prmary users revenue can be maxmzed by ntroducng a specfcally desgned spectrum aucton framework. In [1], Gao et al. proposed a multauctoneer progressve aucton framework enhancng spectrum effcency n CRNs wth multple PUs and SUs. In [4] and [1], Gopnathan et al. respectvely studed the revenue-maxmzng and strategy-proof aucton mechansm to acheve far and effcent spectrum allocaton. Furthermore, Deek et al. n [2] proposed Topaz, a truthful onlne spectrum aucton desgn to deal wth bd and tme-based cheatng n onlne auctons. In addton, only recently, contract-based dynamc spectrum sharng has also been proved to be a promsng mechansm for spectrum tradng. In [8], Gao et al. characterzed the ncentve compatble and ndvdually ratonal contractual forms for SUs to purchase PUs spectra wth the qualty-prce desgned for ther own types. Furthermore, n [7], Duan et al. modeled the CRNs as a labor market and ncorporated contract theory to resolve the problem of cooperatve relay between PUs and SUs. However, all the aforementoned spectrum tradng mechansms fal to consder the ncentves for SUs to purchase PUs channels n the scenaro where SUs face potental transmsson falures. Under such crcumstances spectrum effcency wll thus be undermned n that SUs may not have enough ncentves to purchase PUs channels. Therefore, n ths paper, we seek to mprove spectrum effcency n CRNs by ncorporatng nsurance mechansm nto spectrum tradng. Insurance theory has been wdely studed n the realm of economy. In [17], Clarke studed the duty of good fath n the law of desgnng nsurance contracts. Moreover, Schller n [18] proposed that a sem-separatng equlbrum may occur n nsurance markets where msrepresentaton takes the form of fraudulent clamng. In our framework, both PUs and SUs possess double denttes that s PUs serve as spectrum sellers as well as nsurers and SUs act as spectrum /12/$ IEEE 241

2 buyers as well as nsureds. Facng the nsurance contracts offered by PUs, any arbtrary SU determnes whether to smply purchase a channel or meanwhle sgns an nsurance contract wth the PU to obtan nsurance for the potental accdent,.e., transmsson falure ncurred by the excessvely low SINR. In ths paper, we model the market game as a four-stage Bayesan game and characterze the second-best Perato optmal allocatons SBPO) and the perfect Bayesan equlbrum PBE). Spectrum utlzaton s shown to be mproved by ntroducng nsurance mechansm nto spectrum tradng and both PUs and SUs wll experence ncrease n ther utltes at the PBE. The contrbutons of ths paper are demonstrated as follows: We show that by utlzng nsurance mechansm n spectrum tradng spectrum effcency n CRNs, where SUs face potental transmsson falures, can be mproved. Furthermore, we model the market game between PUs and SUs as a hybrd market and characterze the SBPO market allocatons, the PBE contractual forms and players PBE strateges. Moreover, we derve the condtons for the exstence of the PBE. We analyze the nformaton asymmetry between PUs and SUs about SUs rsk types and study the effect of PUs stochastc rsk verfcaton on SUs rsk type cheatng. Through extensve smulaton, we have demonstrated that at equlbrum, both hgh rsk and low rsk SUs wll respectvely experence an mprovement n ther utltes for approxmately 23.5% and 4.6%. The rest of ths paper s organzed as follows: n Secton II, we ntroduce the network model, hybrd market structure and elements of the PBE. In Secton III, we characterze the SBPO market allocatons. In Secton IV, we derve the condtons for the exstence of the PBE and characterze the PBE contractual forms and players PBE strateges. In Secton V, we provde the numercal and smulaton results and n Secton VI, we dscuss the conclusons and the future work. II. NETWORK MODEL, HYBRID MARKET STRUCTURE A. Network Model AND PBE ELEMENTS We consder a CR network ncludng one prmary operator PO), multple PUs and multple SUs. The PO can be an access pont or a base staton. Any SU s a dedcated transmtter-recever par, Tx -Rx and we denote the set of SUs as {Tx k -Rx k } n k=1. We assume that SU transmtters always have packets to send to ther dedcated recevers. Moreover, such transmsson can be carred out only when an SU has successfully purchased a channel from a PU. The network model s demonstrated n Fgure 1. SUs n ths network are heterogeneous n terms of ther dstances d from the PO. Wthn any arbtrary tme slot, we use fd) to denote the p.d.f. of the dstance between an SU and the PO. The specfc expresson of fd) s determned by SUs moblty model. For example, f SUs move accordng to the restrcted moblty model n [14], fd) s exponentally decayng. Then, we defne the type of an SU n Defnton 1. Insurance Contract 1 SU4 Insurance Contract 2 SU3 SU2 PU2 PU1 Successful Transmsson SU7 PO Faled Transmsson Fg. 1. Illustraton of Network Model: SUs 1 and 2 belong to type-1, SUs 3, 4 and 5 belong to type-2 and SUs 6 and 7 belong to type-3. Due to nterference from the PO, transmsson from SU Txes 1, 2 and 5 to ther RXes fal. PUs 1, 2 and 4 propose nsurance contract 1 and PUs 3 proposes nsurance contract 2. Defnton 1. We draw q concentrc crcles centered at the PO respectvely wth raduses r 1 < < r q, and we call SU a type-j SU f, n the current tme slot, SU locates n the j 1)th j 2) annulus,.e., r j 1 < d < r j and type-1 SU, f t s wthn the nnermost crcular regon wth r n beng the PO s maxmum nterference dstance. Hence, the probablty for an SU to be a type-j j 2) SU s β j = r j r j 1 fx)dx and type-1 SU β 1 = r j fx)dx. The SINR at SU recever wth SU transmtter and SU recever respectvely locatng at pont Xt and X r s: SU1 PU3 PU4 γ = P κ X t X r α I PO + I PU + σ2 We assume that the transmsson power of SU transmtter P, the path loss factor L X t, X r ) = κ X t X r α, the nose varance σ 2 and I PU are approxmately dentcal among all SUs. Hence, nterference s prmarly ncurred by the PO. Therefore, the dstance between SU and the PO dfferentates the SUs n that IPO s determned by d. Furthermore, we assume that SUs of the same type have approxmately the same SINR at ther SU recevers. Furthermore, we assume that SUs n ths network carry out packet-based transmsson. When, for any arbtrary type-j SU, γ j falls below a certan threshold γ th, the transmsson from the SU transmtter to the SU recever wll fal because, n ths scenaro, the partcular SU recever wll be unable to decode the packets transmtted by ts transmtter due to excessvely low SINR. Hence, the accdent probablty,.e., the probablty of faled transmsson for any arbtrary type-j SU can be denoted as ξ j = φγ j ) satsfyng that φ ) s strctly monotoncally decreasng wth range, 1). Apparently, we have < ξ j < ξ j 1 < 1 j 2). In ths paper, we consder the scenaro wth acknowledgement of successful transmsson that s f and only f an SU recever successfully receves and decodes the packets sent by ts dedcated transmtter, t sends a feedback sgnal, ACK SU5 SU6 1) 242

3 to ts transmtter. Thus, based on the presence or absence of the ACK sgnal, the partcular SU transmtter wll be able to decde whether the current transmsson s successful. In ths network, SU transmtters send dentcal number of packets denoted by P S n the case of successful transmsson. The prmary network conssts of m PUs and one PO. We assume that PUs always have dle spectra to trade wth SUs and the PO serves as the nterface of the spectrum tradng and nsurance contract sgnng between PUs and SUs. In ths paper, we assume that the PO possesses the locaton nformaton of SUs n the local area and charge c for each consultaton of ths nformaton ssued by a PU. The analyss of the strategc behavor of the PO s beyond the scope of ths paper. Our results n ths paper can be extended to the scenaro of largescale wreless sensor networks as specfed n [13]. B. Hybrd Market Structure The hybrd market conssts of a spectrum market and an nsurance market. PUs and SUs n ths market have double denttes that s PUs serve as spectrum sellers as well as nsurers and SUs act as spectrum buyers as well as nsureds. Apart from the prce of ts channels τ, any arbtrary PU offers a menu of nsurance contracts w.r.t. dfferent types of SUs.e., C = C 1,, C q). We assume that all PUs sell channels at dentcal prce τ = τ, {1,, m}. The channels sold by PUs expre after one tme slot and PUs can resell these channels f they are stll dle n the next tme slot. SUs can purchase only one channel and sgn one nsurance contract at the begnngng of every tme slot. In every tme slot, enlghtened by [18], [19] and [2] we model the market game as a four-stage Bayesan game. Stage I Insurance contract proposng) Each PU accesses the PO and proposes a menu of nsurance contracts C = C 1,, C q ). Stage II Channel purchasng and contract sgnng) SUs access the PO to obtan the channel prcng and nsurance contracts nformaton. If an SU decdes to sgn an nsurance contract, t meanwhle purchases a channel sold by the same PU. Otherwse, the SU stochastcally purchases a channel from one of the PUs. Stage III Data transmsson and clam flng) SUs carry out data transmsson from SU transmtters to SU recevers. If the transmsson fals, the SU then fles a clam for ndemnty to ts nsurer through the PO f t has sgned an nsurance contract wth the PU n stage II. Stage IV Audtng and ndemnty payng) When PU receves a clam for ndemnty, t conducts rsk verfcaton of the SU wth probablty p [, 1] by consultng the PO the locaton nformaton of the SU. The PU can then determne whether ths SU has cheated on ts rsk types and has selected an nsurance contract desgned for other SU types. If no rsk type cheatng s establshed, the PU pays the ndemnty to the SU. Otherwse, t rescnds the contract and pays no ndemnty. In Defnton 2, we defne the form of nsurance contracts. Defnton 2. An nsurance contract Cj proposed by PU to type-j SUs has the form of Cj = λ j, θ j ) n whch λ j represents the nsurance premum, θj the net payoff n the case of an accdent and therefore the total ndemnty λ j + θ j. For smplfcaton of analyss, n ths paper, we dwell on the scenaro where only two categores of SUs, namely typeh and type-l, respectvely denotng SUs wth hgh and low accdent probablty, exst n ths network. We dvde the regon nsde the PO nterference crcle nto two parts.e., the nner crcular regon, A 1 = { } X < X X PO r th and the outer annulus, A 2 = { } X r th X X PO r max, wth X denotng the poston of any arbtrary pont, X PO the poston of the PO, r th the threshold radus and r max the PO s maxmum nterference dstance. Snce SUs n regon A 1 suffer from hgher level of nterference from the PO than those n regon A 2, we denote the former type of SUs type-h SUs and the latter type-l SUs, respectvely wth accdent probablty ξ h = φγ h ) and ξ l = φγ l ) such that < ξ l < ξ h < 1. The probablty for an SU to be type-h and type-l are respectvely β h = A 1 fx) = r th fx)dx and β l = A 2 fx) = rmax r th fx)dx. Hence, approxmately we have β h + β l = 1. In ths paper, we consder an nsurance market wth nformaton asymmetry and costly rsk verfcaton. SUs have prvate knowledge of ther rsk types that are not avalable to PUs unless they carry out rsk verfcaton. PUs are charged for c by the PO for each rsk verfcaton. It s necessary for PUs to verfy SUs rsk levels because n nsurance markets one gst of nsurance contract desgn s to charge more premum for nsureds wth hgher rsk. Hence, type-h SUs have ncentve to msrepresent ther rsk types and sgn the contracts desgned for type-l nsureds to pay less nsurance premum. For example, when SUs are offered actuarally far full nsurance contracts.e., C h = {λ h, θ h } = {ξ h L, L ξ h L} for typeh SUs and C l = {λ l, θ l } = {ξ l L, L ξ l L} for type-l SUs, wth L denotng an SU s loss ncurred by an accdent, snce ξ l L < ξ h L, t s possble that a type-h SU chooses C l. For the same reason, PUs only carry out rsk verfcaton for alleged type-l clamants. For the costly nature of audton, there s an nherent trade-off between the avodance of payng ndemnty wth the payment of audton cost. Thus, each PU audts stochastcally type-l clamant wth probablty p [, 1]. C. Elements of the four-stage Bayesan game In ths subsecton, we defne the basc elements of ths four-stage Bayesan game ncludng players strateges, belefs, utlty functons and PBE condtons. In Defntons 3 and 4, we respectvely defne players strateges and belefs. Defnton 3. PUs strategy profle s defned as { C, p ) } wth C = C 1,, C m ) and p ) = p 1 ),, p m ) ) where C = Ch, C l ) s the contract menu proposed by PU and p C) s ts audton probablty gven contract profle C. Moreover, the strategy of a type-k SU s defned as π k ) = π kh ), πkl ),, πm kh ), πm kl )) n whch πkj C) denotes the probablty that gven C, a type-k SU sgns nsurance contract Cj f 1 and sgns no nsurance contract f = 243

4 such that m = π kh + π kl ) = 1 and k, j {h, l}. Therefore, the overall strategy profle s { C, p ), π h ), π l ) }. Defnton 4. The belef profle s denoted as η ) = η1 ),, η m ) ) n whch η C) s the probablty n PU s belef that a Cl clamant s n fact type-h gven the contract profle C. Moreover, η C) s calculated through Bayes law: η C) = β h ξ h π hl C) β h ξ h π hl C) + 1 β h)ξ l π ll C), s.t. π hlc) + π llc) >, {1,, m} In defnton 5 and 6, we defne players utlty functons. Defnton 5. 3)-7) hold for {,, m}. The expected utlty of an SU that purchases an nsurance contract desgned for ts own type s: R kk C k) =1 ξ k )ru S τ λ k ) + ξ kru F τ + θ k ), s.t. k {h, l} The expected utlty of a type-h SU that purchases C l s: R hl C l, p ) 2) 3) =ξ h 1 p )ru S τ + θ l ) + p ru F τ) ) + 1 ξ h )ru S τ λ l ) 4) The expected utlty of a type-l SU that purchases C h s: R lh C h ) =1 ξ l )ru S τ λ h) + ξ l ru F τ + θ h) The expected utlty of a type-l SU s: Λ l C, πl ) ) m = π lh C)R lh Ch ) + π ll C)R llcl )) 6) = The expected utlty of a type-h SU s: Λ h C, p ), πh ) ) m = πhh C)R hhch ) + π hl C)R hl C l, p C) )) 7) = In 3)-5), U S = P S denotes the successful transmsson reward whch we assume s lnearly related wth P S, the number of packets receved by the SU recever and U F denotes the transmsson reward n the case of an accdent whch s approxmately. Hence, an SU s loss ncurred by an accdent s L = U S τ) U F τ). Moreover, r ) satsfes that r ) > and r ) < and C h = C l =, ) represent the case n whch the SU remans unnsured. Defnton 6. 8)-12) hold for {1,, m}. The expected proft of PU for contract C h s: 5) G hk C h) = 1 ξ k )λ h ξ k θ h, k {h, l} 8) PU s expected proft for C l sgned by a type-l SU s: G ll C l, p ) = 1 ξ l )λ l ξ l θ l + p c) 9) PU s expected proft for C l sgned by a type-h SU s: G lh C l, p ) = 1 ξ h )λ l ξ h 1 p )θ l + p c ) 1) PU s expected utlty from an nsurance contract s: Φ C, p ), πh ), π l ) ) =β h π hh C)G hhc h ) + π hl C)G lh C l, p C) )) + β l π lhc)g hl C h) + π llc)g ll C l, p C) )) 11) PU s expected utlty gven that t has sold M channels and N nsurance contracts such that M N s: Ω C, p ), πh ), π l ), M, N ) =Mτ + NΦ C, p ), πh ), π l ) ) 12) The rest of ths paper seeks to characterze the equlbrum of ths four-stage Bayesan game. Such an equlbrum s a perfect Bayesan equlbrum formally defned n Defnton 7. Defnton 7. E = { C, p ), πh ), π l ), η )} s a perfect Bayesan equlbrum of the four-stage Bayesan game f t satsfes: Λ h C, p ), π h ) ) Λ h C, p ), π h ) ), π h ), C 13) Λ l C, πl ) ) Λ l C, π l ) ), π l ), C 14) The followng 15)-17) hold for {1,, m}: p C) η C)θl c ) p C) η C)θl c ), p ), C 15) Ω C, C ), p ), πh ), πl ), M, N ) Ω C, p ), πh ), π l ), M, N), C = Ch, ) 16) C l η C) = β h ξ h πhl C) β h ξ h πhl C) + 1 β h)ξ l πll C. 17) C), 13) holds for π h ) and C conveyng that πh ) s the optmal contract choce for type-h SUs gven PUs audtng strateges p ). Smlarly, holdng for π l ) and C, 14) reveals the optmalty of πl ) among type-l SUs contract selecton strateges. 15) means for C, p ) s the optmal audton strategy for PU gven ts belef η C). 16) says gven other PUs contract profle C = C1,, C 1, C +1,, C m ), C s the optmal contract offer for PU and 17) states that η ) can be calculated from πh ) and π l ) by Bayesan law. Hence, the PBE strategy profle { C, p ), πh ), π l )} satsfes sequental ratonalty gven η ). III. CHARACTERIZATION OF SECOND-BEST PARETO OPTIMAL SBPO) ALLOCATIONS In ths secton, we characterze the PBE of ths four-stage Bayesan game by delvng nto ts SBPO allocatons. For any strategy profle, { C, p ), π h ), π l ) }, a market allocaton s defned as {C, p, π h, π l } wth p = pc), π h = π h C) and π l = π l C). In ths paper, we defne Pareto optmalty n second-best sense because of the constrants ntroduced n C 1 later ths secton. Then, we ntroduce Lemmas 1 and 2 that shed lght upon several basc propertes of the PBE. 244

5 Lemma 1. For PBE E = { C, p ), πh ), π l ), η )}, such that πhl C) + π ll C) >, C, we have p C) < 1 and p C) η C)θ l c) =. Furthermore, f πhl C) = and πll C) >, then we have p C) =. Proof: Suppose that p C) = 1, we have R hl C l, 1) =ξ h ru F τ) + 1 ξ h )ru S τ λ l ) <ξ h ru F τ) + 1 ξ h )ru S τ) =R hh C h ) Hence, we also have πhl =. Then 17) gves η C) = and η C)θ l = < c. Therefore, p C) = accordng to 15), whch contradcts wth the assumpton that p C) = 1. Thus, p C) < 1. From 15), we have p C) η C)θ l c) =. Specfcally, for the case that πhl C) = and π ll C) >, we have η C) = and p C) = from 15) and 17). Remarks. Lemma 1 conveys that every PU carres out stochastc audton at PBE.e., p C) < 1. Ths s somewhat ntutve because when rsk verfcaton s carred out by PU wth probablty 1, type-h SUs wll be better off to reman unnsured or sgn nsurance contracts other than Cl. Furthermore, ths n turn ndcates that audton s unnecessary,.e., p C) =. Moreover, audton s conducted wth postve probablty only when η C) reaches a threshold c. θl Lemma 2. The proft of nsurance contract Cl for any PU at PBE E = { C, p ), πh ), π l ), η ) } can be represented by SUs contract choces πhl and πll wthout explctly ncorporatng PU s audton probablty p C). Proof: From Lemma 1, we know that p C) η C)θ l c ) =, p C) < 1 and PUs are ndfferent about whether to carry out rsk verfcaton or not. Hence, we have β h πhl C)G lh C l, p C)) + β l πll G ll C l, p C)) =β h πhlc)g lh Cl, ) + β l πll G ll Cl, ) =β h πhl C)G lh C l ) + β lπll G ll C l ) Therefore, we have proved Lemma 2. Then, n C 1 we ntroduce the constrants that SBPO allocatons have to satsfy n ths Bayesan game. Such constrants nclude nformaton asymmetry between SUs and PUs about SUs rsk types shown n 21) and 22) and no pre-commtment of PUs audton strateges demonstrated n 18) and 19). Furthermore, 2) requres that the expected proft that PU can get by proposng an nsurance contract should be no smaller than ω, because otherwse PUs wll have no ncentve to propose nsurance contracts. If an allocaton {C, p, π h, π l } satsfes constrant C 1, we call t a feasble allocaton. C 1 : A feasble allocaton {C, p, π h, π l } should satsfy the followng constrants for {1,, m} and C. βh ξ h πhl + 1 β h )ξ l πll ) c = βh ξ h πhlθ l, f p > 18) βh ξ h πhl + 1 β h )ξ l πll ) c βh ξ h πhlθ l, f p = 19) Φ C, π h, π l ) =1 β h ) π lhc)g hl C h) + π llc)g llc l) ) + β h π hh C)G hh C h ) + π hl C)G lh C l )) ω 2) Λ h C, p, π h ) = max { R hh C h ), R hlc l, p ) {1,, m} } 21) Λ l C, p, π l ) =max { R lh Ch ), R llcl 22) ) {1,, m}} Then n Defnton 8, we defne SBPO market allocatons. Defnton 8. An allocaton {C, p, π h, π l } that satsfes C 1 s an SBPO allocaton f no other feasble allocaton {C, p, π h, π l } exsts satsfyng that: Λ k C, p, π k ) Λ kc, p, π k ), k {h, l} 23) In the followng analyss, we characterze SBPO allocatons by consderng a seres of constraned optmzaton problems from O 1 to O 3. Frstly, we consder problem O 1. O 1 β h, c, ν, ω 1,, ω m ): {Ĉ, p, π h, π l } = arg maxλ l C, π l ), s.t. C 1 β h, c, ω 1,, ω m ) and Λ h Ĉ, p, π h) ν 24) The soluton of O 1 β h, c, ν, ω 1,, ω m ) sheds lght upon the characterzatons of the SBPO allocatons of the four-stage Bayesan game because any SBPO allocaton {C, p, π h, π l } s n fact an optmal soluton to O 1 β h, c, ν,,, ) satsfyng that ν = m =1 πhh C)R hhch ) + π hl C)R hl C l, p C) )). To analyze O 1, frstly we study the symmetrc varant of O 1.e., O 2 β h, c, ν, ω ) = O 1 β h, c, ν, ω 1,, ω m ) 1 m ω =ω. In O 2 PUs offer the same contract menu.e., C = C l, C h ) and audt SUs rsk types wth the same audtng probablty. Moreover, SUs are evenly dstrbuted among PUs. Then, n Theorems 1-4, we provde our results about the characterzatons of the SBPO allocatons of ths four-stage Bayesan game. We use { C, ṗ, π h, π l } to denote the optmal soluton for O 2 β h, c, ν, ω ). In ths paper, we restrct our analyss on the case where ṗ >. Smlar conclusons exst when ṗ =. Theorem 1. For { C, ṗ, π h, π l } such that ṗ >, SUs contract selecton strategy satsfes π ll = 1, π lh =, π hh = 1 π hl and 25), π hl = B θ l, β h, c) = 1 β h)ξ l c β h ξ h θ, 1) 25) l c) Proof: We denote the value functon of the constraned maxmzaton problem O 2 β h, c, ν, ω ) as ψ 2 β h, c, ν, ω ) and we denote O 2 β h, c, ν, ω ) under scenaros such that ṗ > and ṗ = as O 2 β h, c, ν, ω ) and O 2 β h, c, ν, ω ) respectvely wth value functon ψ 2 β h, c, ν, ω ) and ψ 2 β h, c, ν, ω ). Therefore, we have ψ 2 β h, c, ν, ω ) = max { ψ 2 β h, c, ν, ω ), ψ 2 β h, c, ν, ω ) }. If for a feasble allocaton Q, π ll < 1, there s another feasble allocaton Q of O 2 β h, c, ν, ω ) such that a type-l SU earns hgher utlty for allocaton Q than Q. Then, we consder allocatons feasble n O 2 β h, c, ν, ω ) wth no counterparts n Q that makes type-l SUs better off. Hence, we have π ll = 1 and π lh =. Snce ṗ >, 18) holds and by some manpulaton we can arrve at the concluson 25). Theorem 2. For { C, ṗ, π h, π l } such that ṗ >, type-h SUs mnmum utlty ν s less than the utlty granted by 245

6 the poolng contract C = ξl + ω, 1 ξ)l ω ), wth ξ = β h ξ h + 1 β h )ξ l and ν = ru S ξ ω ), that s R hh C h ) = R hl C l, ṗ) = ν < ν 26) Proof: Snce Theorem 1 reveals that C l s better off for type-l SU than C h, then we have 27), R ll C l ) R lh C h ) 27) Snce, π ll < 1 and π hl >, we have 28), R hl C l ) = R hl C l, ) > R hh C h ) 28) Moreover, wthout loss of generalty, we also assume 29), R hh C h ) ν 29) Suppose that ν ν, we have π hh R hh C h )+ π hl G lh C l ) < ξ ξ h )L + ω. From 2), we have G ll C l ) > ξ ξ l )L + ω and R ll C l ) < ν < R hl C l, ). Then we construct an allocaton Q 1 = {Ch 1, C1 l, p1, πhh 1, π1 hl, π1 lh, π1 ll } wth p1 =, πhh 1 = 1, π1 hl =, π1 lh =, π1 ll = 1, and C1 h, C1 l ) = C h, C l ). On one hand contract C l satsfes λ l + θ l = L and R llc l ) = R ll C l ) whch reveals that G ll C l ) > G ll C l ) and R hl C l ) > R hl C l ). On the other hand, contract C h satsfes that C h = C h when π hh > and G hh C h ) G lh C l ) and C h = C l when π hh =. Therefore, we have R ll C l ) R llc h ), R hhc h ) > R hl C l ) and R hhc h ) > ν. Then from 2), we have: β h G hh C h) + 1 β h )G llc l) >β h πhh G hh C h ) + π hl G lh C l ) ) + β l G ll C l ) =ω Therefore, we can deduce that allocaton Q 1 s domnated n O 2 β h, c, ν, ω ) because Q 1 s feasble n O 2β h, c, ν, ω ) wth PUs expected proft larger than ω and nsureds expected utlty satsfyng R ll C l ) = R ll C l ). Hence, we have a contradcton and n fact ν < ν. Also, from 21) we can arrve at the concluson that R hh C h ) = R hl C l, ṗ) = ν. Remarks. Theorem 1 conveys that at the optmal soluton to O 2 β h, c, ν, ω ) when ṗ >, type-h SUs randomze between contract C l and C h and selects C l wth probablty B θ l, β h, c). Moreover, the fact that ν < ν reveals that audton s an effectve way to stem type-h SUs from choosng contract C l because as ndcated by Theorem 2, type-h SUs mnmum utlty ν s less than that granted by the poolng contract. Proposton 1. For { C, ṗ, π h, π l } such that ṗ >, contract C l satsfes 3), C l = arg maxr ll C l ), s.t. C l β h B θ l, β h, c) G lh C l ) G hh C h ) ) + G hh C h ) 1 β h )G ll C l ) ω ) + 3) Proof: From Theorem 1, we get π ll = 1, π lh = and 25). We get 3) by substtutng π ll, π lh and π hl n 2). Next, we further characterze the optmal soluton of O 3 β h, c, R hh, ) = O 2 β h, c, ν, ω ) ω= ν = R, whch s also a hh crucal step towards the characterzaton of O 1. R hh denotes a type-h SU s utlty f t sgns an actuarally far full nsurance contract C h.e., Rhh = R hh C h ). Then we have Theorem 3 characterzng the optmal soluton to O 3. Theorem 3. There exst threshold values for PUs audton cost c < c < ) and the probablty that an SUs falls nto the category of type-h SU βh < β h < 1), such that the optmal soluton for O 3, { C, p, π h, π l } satsfes, When c < c < and βh < β h < 1, C = C h, Cl ), π hh = π ll = 1 and π hl = π lh =. When qβ h ) c < and < β h < βh λ, h + θ h = L, λ l + θ l < L, π hh = π ll = 1 and π hl = π lh =. When < c c and βh β h < 1 or < c qβ h ) and < β h < βh C, = C h, C l ), π hl = B θ l, β h, c), π hh = 1 π hl, π ll = 1 and π lh =. where Cl = { } ξ l L, 1 ξ l )L < L < L), curve c = qβ h ) < β h βh ) s shown n Fgure 2 and C l satsfes: C l = argmax R ll C l ), s.t. C l β h B θ l, β h, c)g lh C l) + 1 β h )G ll C 31) l). Proof: By substtutng C h = {ξ h L, L ξ h L} nto 8), we have G hh C h ) =. Together wth ω =, we get condton 31). Then, we denote the value functon of O 3 β h, c, R hh, ) as ψ 3 β h, c, R hh, ) = ψ 2 β h, c, ν, ω ) ω= ν = R. hh If p =, R ll C l ) s contngent merely on the value of β h and thus, n ths scenaro, we can denote that kβ h ) = R ll C l ). Accordng to the Rothschld-Stgltz RS) model [2], a threshold value βh < β h < 1) exsts satsfyng that kβ h) > R ll Cl ) and k β h ) < for β h < βh and kβ h) = R ll Cl ) for βh β h < 1. Snce C l depends only on cost c, we can denote that tc) = R ll C l ) such that t c) <. Accordng to the RS model n [2], a threshold value c < c < ) exsts such that tc) > R ll Cl ) f and only f when < c < c. Based on the aforementoned propertes of kβ h ) and tc), we have: When c < c < and βh < β h < 1, ψ 3 β h, c, R hh, ) = kβ h ) = R ll Cl ) > R ll C l ). When qβ h ) c < and < β h < βh, ψ 3 β h, c, R hh, ) = kβ h ) > max { R ll C l ), R ll Cl ) }. When < c c and βh β h < 1 or < c qβ h ) and < β h < βh, ψ 3 β h, c, R hh, ) = tc) = R ll C l ) > kβ h). Then, together wth Theorems 1 and 2, we get Theorem 3 concernng the optmal soluton of O 3. Remarks. In Theorem 3, we characterze the contract forms and SUs strateges at the soluton to O 3 β h, c, R hh, ). It reveals that threshold values c < c < ) and βh < βh < 1), together wth the functon c = qβ h) dvde the area S = { β h, c) < β h < 1 and < c < } n the c β h plane nto three regons shown n Fgure 2. In regon I and II, the contract forms are specfed n the frst two tems of Theorem 3 and SUs strongly prefer the contacts desgned for ther own rsk types. Hence, audton s not necessary,.e., p =. When β h, c) falls nto regon III, then for PUs the optmal contracts 246

7 Fg. 2. c * c Regon II c= q β h ) β * h Regon I Regon III Illustraton of c β h plane: c and βh are threshold values. are n the form of C = C h, C l ) and audton s conducted wth probablty p >. In ths case, type-l SUs also strongly prefer the contacts desgned for type-l SUs, whereas type-h SUs select C l wth probablty π hl = B θ l, β h, c). After we propose Theorems 1-3, we can characterze the optmal soluton to O 1 β h, c, ν, ω 1,, ω m ) defned n 24). Specfcally, we consder the scenaro where PUs obtan nonnegatve proft that s we consder O 1 β h, c, ν,,, ). Theorem 4. The optmal soluton to O 1 β h, c, ν,,, ) concdes wth the optmal soluton to O 2 β h, c, ν, ) characterzed n Theorems 1-3, that s for {1,, m}, we have Ĉ = C, p = ṗ, π jk = π jk m, j, k {h, l} under the condton that ν R hh where ν denotes type-h nsureds expected utlty.e., ν = Λ hc, p, π h ). Proof: The proof of Theorem 4 s n the appendx. IV. CHARACTERIZATION OF PERFECT BAYESIAN EQUILIBRIUM OF THE FOUR-STAGE BAYESIAN GAME In ths secton, we characterze the equlbrum of the fourstage Bayesan game and restrct our analyss to equlbrum allocatons that are SBPO. A contradcton wth the defnton of equlbrum wll occur f a non-sbpo allocaton s an equlbrum allocaton because any devant PU k wll then be able to offer a contract menu C k = Ch k, Ck l ) such that ths offer s better off to all SUs and PU k makes postve proft. We denote Λ e h and Λe l as the equlbrum expected utlty for type-h and type-l SUs. In C 2 and C 3, we ntroduce two addtonal constrants ncorporated n the followng analyss. C 2 : A PBE E = { C, p ), πh ), π l ), η ) } satsfes C 2 f for the contnuaton sub-game after PUs propose the contract profle C, {p, πh, π l } s a tremblng hand perfect Bayesan equlbrum ntroduced by Selten n [21]. C 3 : A PBE E = { C, p ), πh ), π l ), η )} satsfes C 3 f πl C) = π l C ) such that contract profle C and C dffers only n the contracts strctly domnated for type-l SUs. Then we have Lemma 3 to characterze basc propertes of PBE that satsfes constrant C 2. Lemma 3. We have p = for {1,, m} such that θl = c at any PBE E = { C, p ), πh ), π l ), η )} that satsfes C 2. Proof: We denote Π as the contnuaton subgame after the proposal of C and {p, πh, π l } s a perfect Bayesan 1 βh equlbrum of Π that satsfes C 2. We consder a sequence of perturbed contnuaton subgames {Π εn } n=1 whch converges to Π and the assocated PBE {p n, π hn, π ln, η n } satsfyng that {p n, π hn, π ln } converges to {p, πh, π l }. Then, from Bayes law, we have: ηn = β h ξ h πhln, 1), n. β h ξ h π hln +1 β h)ξ l π lln Maxmzng p ηnθ l c) w.r.t. p [ε n, 1 ε n ] means that f θl = c, n, we have p n = ε n. Therefore, the fact that lm ε n = reveals that p = lm p n =. n n Remarks. Lemma 3 reveals that under constrant C 2, the decrease n the audton probablty of PU s not able to serve as a threat to deter an devant PU to attract type-h SUs. Thus any PBE allocaton falls nto the category of SBPO allocaton, whch s formally proved n Lemma 4. Lemma 4. Any PBE allocaton satsfyng C 2 s SBPO and an optmal soluton to O 1 β h, c, Λ e h,,, ) wth Λe h R hh. Proof: Assume that E = { C, p ), πh ), π l ), η )} s a PBE and ts allocaton {C, p, πh, π l } satsfes C 2. {C, p, πh, π l } s feasble n O 1β h, c, Λ e h, Ωe j,, Ωe j ) wth Ω e denotng the equlbrum expected revenue of PU, for {1,, m}. Snce Ω e, we have {C, p, πh, π l } s also feasble n O 1 β h, c, Λ e h,,, ). Then we also prove ths lemma by contradcton. We assume that ths allocaton does not optmze O 1 β h, c, Λ e h,,, ). We consder a devaton made by PU j from Cj to C j. We denote Ω j as PU j s proft of the contnuaton equlbrum after ts devaton. Hence, Ω j and Ω e j satsfes 15) that s Ω j < Ω e j. In the scenaro where R hl Cj l ) < R hhc j h ), we have πhh C j, C j) = πll C j, C j) = 1 and p j C j, C j) =. Hence, Ω j Ω e j whch contradcts wth Ω j < Ω e j. Also smlar results hold n the case that R hl Cj l ) R hhc j h ). After Lemmas 3 and 4, we are ready to characterze the PBE of ths four-stage Bayesan game satsfyng C 2 and C 3. In Theorem 5, we propose the necessary and suffcent condtons for PBE to exst n ths four-stage Bayesan game. Theorem 5. Under C 2 and C 3 only two types of PBE exst: If and only f < c c and βh β h < 1 or < c kβ h ) and < β h < βh, PBE E 1 exsts at whch PUs propose contract menu C = C l, C h ), {1,, m} and carry out audton wth postve probablty. Type-l SUs strongly prefer C l whereas type-h SUs randomze between C l and C h. If and only f βh < β h < 1 and c < c <, PBE E 2 exsts where PUs propose contract menu C = Cl, C h ), {1,, m} and SUs only select contracts desgned for ther own rsk types. Thus, audton s not necessary,.e., p =. Proof: The proof of Theorem 5 s n the appendx. V. NUMERICAL AND SIMULATION RESULTS In ths secton, we provde our numercal and smulaton results. We demonstrate the exstence of PBE E 1 and E 2 n Fgures 3 and 4. Moreover, n Fgure 5, we demonstrate the comparson of the utlty of an SU wth and wthout nsurance. Wthout loss of generalty we set the smulaton parameters as follows: U S = 6, U F = 2, τ = 1, ξ h =.7 and ξ l =

8 Illustraton of exstence of PBE 1 6 Illustraton of exstence of PBE 2 8 Comparson of an SU s utlty wth and wthout Insurance Y State dependent wealth of an SU Non accdent) Indfference curve of a type h SU Indfference curve of a type l SU Auxlary lne Far odds lne of a type h SU Far odds lne of a type l SU Bsector of the 1st quadrant C D B Y State dependent wealth of an SU Non accdent) Indfference curve of a type h SU Indfference curve of a type l SU Far odds lne of a type h SU Far odds lne of a type l SU Bsector of 1st quadrant C E B The Utlty of an SU % Utlty of an SU wth Insurance Utlty of an SU wthout Insurance 4.7% 4.67% 24.6% A A X State dependent wealth of an SU Accdent) X State dependent wealth of an SU Accdent) Type h SU PBE1 Type l SU PBE1 Type h SU PBE2 Type l SU PBE2 Fg. 3. Illustraton of exstence of PBE 1. Fg. 4. Illustraton of exstence of PBE 2. Fg. 5. Comparson of an SU s utlty wth and wthout nsurance. In Fgure 3, the horzontal axs represents the statedependent wealth of an SU when an accdent does not occur to t and the vertcal axs s the state-dependent wealth of an SU when an accdent happens to ths partcular SU. Pont A59, 1) represents the combnaton of the state-dependent wealth of an SU when t does not purchase any nsurance contract. The dashed lne AC and AB respectvely represents the far odds lne of a type-h SU and a type-l SU. For a type-j SU, ts far, j {l, h}. We notce that for type-h SU one of ts ndfference curves s tangent to ts far-odds lne at pont C18.4, 18.4) whch s the ntersecton of lne AC and the bsector of 1st quadrant. Thus, we have that at PBE E 1 the contract desgned for type-h SU s C h = {4.6, 17.4} = {ξ h L, L ξ h L}. From 28) and 3), we can calculate that contract C l = λ l, θ l ) odds lne goes through pont A59, 1) wth slope ξj 1 ξ j breaks even when λ l =.21θ2 l.49θ l.4c. Next, we set c = 1 and draw an ancllary curve x = 59.21y 1)2.49y 1) 4 shown as the dashed blue curve. As n Fgure 3, we can fnd an ndfference curve for type-l SU tangent to the ancllary curve at pont D37.41, 41.13). At PBE E 1, the nsurance contract desgned for type-l SUs s C l = 21.59, 4.13). Therefore, we have shown the exstence of PBE E 1 n Fgure 3. Smlarly, n Fgure 4, we demonstrate the exstence of PBE E 2. For PBE E 2, we do not need to fnd the pont where one of the type-l SUs ndfference curves s tangent to the ancllary curve n Fgure 3. Snce lne AE, the far odds lne of a type-l SU ntersects wth the ndfference curve of type-h SU at pont B54.56, 11.36), we can conclude that that PBE E 2 exsts and the PBE contracts desgned for type-l SUs and type-h SUs are respectvely Cl = {4.44, 1.36} and C h = {4.6, 17.4}. Moreover, based on the aforementoned results about the contract forms and SUs contract choces at PBE E 1 and PBE E 2. We can calculate the expected utlty of an SU at both PBEs and compare them wth the expected utlty of an SU when nsurance s not ntroduced nto spectrum tradng n CRNs. In Fgure 5, we show the comparson of an SU s expected utlty wth and wthout nsurance. From Fgure 5, we can conclude that the SUs experence ncrease n ther utltes f they choose to purchase nsurance contracts at both PBEs. VI. CONCLUSIONS AND FUTURE WORK In ths paper, we utlze nsurance theory n spectrum tradng n CRNs and model the market game as a four-stage Bayesan game. We show that at PBE, both PUs and SUs wll experence ncrease n ther utltes. Ths ndcates by sgnng nsurance contracts, SUs wll have more ncentves to buy the channels and PUs can make non-negatve proft from every contract t offers. Thus, spectrum utlzaton n CRNs can be mproved. In future work, we am to analyze the scenaro where more than two types of SUs exst and analyze the related equlbrum. 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From the defntons of ψ 2 and ψ 2, we get ψ 2 β h, c, ν, ) ψ 1β h, c, ν, ) ψ 2 β h, c, ν, ) Snce ψ 2 decreases f and only f ν < Rhh = R hh, ) and ν > R hh, we have βh β h. We denote that N = π hh + π hl + π lh + π ll whch s the proporton of SUs buyng contracts from PU. Hence, we have β h = m = β h N and m = N = 1. We get the concluson that βh = β h. When ν < R hh We can get smlar results usng smlar analyss as before. Thus, n the case where ν R hh, we have βh = β h and ψ 1 β h, c, ν, ) = ψ 2 β h, c, ν, ) = π lhr ll Ĉ h) + π llr ll Ĉ l) Therefore, for {1,, m} such that N >, {Ĉ, p, π hh, π hl, π ll, π lh } s feasble n O 2β h, c, ν, ) and t reveals that the optmal soluton to O 1 β h, c, ν,,, ) does concde wth the one to O 2 β h, c, ν, ) when ν R hh. B. Proof of Theorem 5 Proof: Suppose ν = { ν ψ 2 β h, c, ν, ω ) R ll Cl )}. If Rhh ν < ν, we have ψ 1 β h, c, ν, ) = ψ 2 β h, c, ν, ). Hence, ψ 1 β h, c, ν, ) > R ll Cl ). Then, we conclude that f Rhh ν < ν, each SU sgns an nsurance contract at the optmal soluton to O 1 β h, c, ν, ). From Lemma 4, we have Λ e h R hh. Under C 3, we have Λ e h = R hh. We get at PBE that satsfes C 2 and C 3, each SU purchases an nsurance contract. We prove the frst case of Theorem 5. Snce we consder PBE under C 2, Lemmas 3 and 4 reveal ψ 3 β h, c, R hh, ) = R ll C l ) s a necessary condton for PBE E 1 to exst. We seek to prove t s also a suffcent condton. Hence, we assume ψ 3 β h, c, R hh, ) = R ll C l ) and {C, p, πh, π l } s the PBE allocaton characterzed n ths case. We consder the case where PU devates from C by offerng C = Ch, C l ) nstead of C = C h, C l ). In the contnuaton subgame after the proposal of C, C ) C, p = p C, C ), π h = πh C, C ), π l = πl C, C ) and η = η C, C ) can be chosen such that 13)-15) and 17) hold. Then we have to prove that 16) also holds. We denote the expected proft of PU obtaned from an nsurance contract after the proposal of C, C as Ω = Ω C, C ), p C, C ), πh C, C ), πl C, C ) ) and we assume Ω e =, so 17) can be wrtten as Ω. When max { R ll Cl ), R lhch )} R ll C l ) We can construct the contnuaton equlbrum such that πlh + π ll = whch reveals that Ω. When max { R ll Cl ), R lhch )} > R ll C l ) We can choose πlh and π ll such that π lh + π ll = 1 and πlh R lhch ) + π ll R llcl ) = max{ R ll Cl ), R lhch )}. If πlh R lhch )+π ll R llcl ) <, we have Ω <. Otherwse, we consder two cases: R hh Ch ) R hh and R hh Ch ) < R hh. When R hh C h ) R hh We construct the contnuaton equlbrum after C, C ) s proposed. Addtonally, f R hl C l ) > R hhc h ), θ l > c and Bθ l, β h, c)π ll 1, we can choose p, 1) such that R hh C h ) = R hlc l, p ), π hl = Bθ l, β h, c)π ll and π hh = 1 π hl. Otherwse, we choose p = and π lh and π ll such that π lh + π ll = 1 and π hh R hhc h ) + π hl R hl C l ) = max{ R hh C h ), R hl C l )}. Hence, the strateges we have just constructed satsfes 13)-15) and 17).e., t s a contnuaton equlbrum. Moreover, the allocaton Q 2 = {C h, C l, p, π hh, π hl, π lh, π ll } satsfes 18), 19), 21) and 22). If 2) s also satsfed, we have that Q 2 s feasble n ψ 3 β h, c, R hh, ) and max { R ll C l ), R lhc h )} ψ 3 β h, c, R hh, ) = R ll C l ). Therefore, we arrve at a contradcton and Ω <. When R hh C h ) < R hh Smlar methods can be used to tackle ths case. Thus, we have fnshed the proof of the frst tem of Theorem 5. Its the second tem can be proved n smlar manner as before. Hence, we get the conclusons n Theorem