1 Evolutionry Stbility of Reputtion Mngement System in Peer to Peer etworks Antriksh Goswmi, Ruchir Gupt rxiv:1703.08286v1 [cs.gt] 24 Mr 2017 Abstrct Ech prticipnt in peer-to-peer network prefers to free-ride on the contribution of other prticipnts. Reputtion bsed resource shring is wy to control the free riding. Insted of clssicl gme theory we use evolutionry gme theory to nlyse the reputtion bsed resource shring in peer to peer system. Clssicl gme-theoreticl pproch requires globl informtion of the popultion. However, the evolutionry gmes only ssumes light cognitive cpbilities of users, tht is, ech user imittes the behvior of other user with better pyoff. We find tht without ny extr benefit reputtion strtegy is not stble in the system. We lso find the frction of users who clculte the reputtion for controlling the free riding in equilibrium. In this work first we mde gme theoreticl model for the reputtion system nd then we clculte the threshold of the frction of users with which the reputtion strtegy is sustinble in the system. We found tht in simplistic conditions reputtion clcultion is not evolutionrily stble strtegy but if we impose some initil pyment to ll users nd then distribute tht pyment mong the users who re clculting reputtion theeputtion is evolutionry stble strtegy. Index Terms Gme Theory, Evolutionry Gme Theory, peerto-peer network, Reputtion system I. ITRODUCTIO PEER-TO-PEER systems re utonomous nd distributed dynmic resource-shring networks. Collectively, the resources of mny utonomous users builds n economic nd highly sclble pltform for dt-shring, storge nd distributed computing etc. In these systems, it is peremptory for peers to voluntrily contribute resources which includes storge, bndwidth nd dt content etc. However, instinctively, ech peer would prefer to free ride on the prt of other peers by consuming vilble resources nd services without contributing nything bck, nd thus void the corresponding costs. It ws reported tht nerly 70% of Gnutell users shre nothing with other users (these users simply free-ride on other users who shre informtion), nd nerly 50% of ll file serch responses come from the top 1% of informtion shring nodes [1]. In follow-up study (5 yr lter), it ws found tht 85% of users shre nothing [2], which implies the free-riding problem hd got worse in the intervening yers. Generlly, the lck of coopertion nd so free riding is mjor problem in these utonomous resource shring networks [3], [4], [5]. Designer of P2P system cn consider either of two wys for resource mngement in these systems: resource lloction in which the designer should decide whether nd wht percentge of good (with given predefined cpcity) ech peer should consume nd resource provision in which the designer s tsk is to entice independent prticipnt to provide resource (with its right shre). Both mechnisms in wy or other use the reputtion of the peers. Idelly, reputtion should be the mesure of coopertive behvior of node which is n bstrct quntity nd privte informtion of node. So, it is difficult to mesure the coopertive behvior of node nd we cn only mesure its implictions with some degree of uncertinty. However, it cn be estimted with certin ccurcy on the bsis of behvior observed by node. A number of mechnisms hve been proposed in literture [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] for clcultion of reputtion. A considerble mount of work hs lredy been done oesource lloction [19], [20] nd resource provisioning [26] using reputtion. Reputtion bsed resource lloction mechnisms ply crucil role to encourge coopertion mong utonomous nodes. Reserch till now revels tht reputtion clcultion is useful to entice the coopertion, but the evolutionry stbility of the reputtion system is yet to be investigted. In this work, we hve nlyzed the reputtion system for understnding the conditions of evolutionry stbility of the system. Reputtion systems lwys tend to give more benefit to those nodes which re contributing more to the system. But s clcultion of reputtion of the node requires some cost so, coopertors (C strtegy users) who re not clculting reputtion lwys gets more benefit when they interct with reputtion clcultors (R) s compre to R users when they interct with nother R users. Due to this R strtegy is not evolutionry stble strtegy if there is not extr benefit. In this work, we hve nlyzed pyment bsed mechnism which gives the required benefit to the reputtion strtegy so tht it could be evolutionry stble. In our work, for the ske of simplicity we considered only discrete vlue full contribution, full defection nd reputtion clcultion with full contribution s strtegic choice. Min findings of this pper re listed s follows: 1) The threshold of the number of reputtion clcultor R strtegy users which if keep fixed then coopertors lwys gets higher pyoff thn defectors nd so free riding cn be controlled. 2) The threshold of the number of R strtegy users which if keep fixed theeputtion strtegy users lwys gets higher pyoff thn defectors. 3) If we llow reputtion s the optionl strtegy then in generl conditions (without ny extr benefit to R nd C strtegy users) R strtegy is not n equilibrium strtegy. 4) If we impose some initil pyment nd distribute tht initil pyment mong the plyers who re clculting the reputtion theeputtion is evolutionry stble
2 strtegy for threshold of initil pyment. II. RELATED WORK In literture lot of study hs been done for estimtion of reputtion. Burgohin et.l. [6] tke the rtio of resource contributed by the node to the rtio of bsolute mesure of contribution, whether they does not discuss the mechnism to mesure of contributions of node by the receiving node. In [7] the receiving node computes the trust vlue of node on the bsis of received dt in the trnsctions with the sending node. Dutty et.l. [8] suggest tht ech node should provide rting to the other node on the bsis of service provided by the user nd then this rting is supervised by group of users. This scheme uses the reputtion in the form of rting. In [9] ech node clcultes reputtion of other node on the bsis of service received from the other nodes depending upon number of trnsctions done with those nodes, dely in the trnsctions nd the downlod speed. Andrde et.l. [10] clcultes the reputtion of node by tking the difference of resources received from nd provided to the node. In [11],[12],[13],[14] nd [15], node djusts the reputtion of other node on the bsis of qulity of trnsctions with tht node. Eigen-Trust [12] uses sum of positive nd negtive rtings, Peer-Trust [13] normlises the rting on ech trnsction wheres Power-Trust [14] uses Byesin pproch to clculte reputtion loclly. Some resource lloction nd resource provision schemes using generosity level of the peers hs been investigted. Feldmn et l.[16] estimted generosity of node s the rtio of the service provided by the node to the service received by the node. odes will be served s per their estimted generosity. Kung et l. [17] proposed selection of peer for lloction of resource ccording to its contribution to the network nd usge of resources. odes desirous to receive resources hve to contribute bove certin level to the network. Meo et l. [18] model the resource lloction problem s competition mong ll requesters on the bsis of resource request mount. Resource is llocted to the requesters who re demnding lest. In this work uthor sumed ll the requesters s generous mens: it does not wnt not to shre, but to shre s little s possible. Lter the term generosity level of the peer is replced by the reputtion of the peer nd some new resource lloction nd resource provision schemes worked on this. In [19] Stsiou et l. proposes the distributed reputtionbsed system. They propose the lgorithm which mximizes requesters stisfctions s well s mximizes the downlod cpcity of the user so s to its utility. In [20] Gupt et l. uses the probbilistic pproch to llocte the resources on the bsis of reputtion. They rgue tht by using this scheme nodes tht don t hve very good reputtion bout ech other, my lso serve ech other t lest some mount of resource with finite probbility. For voiding whitewshing in unstructured peer to peer to network Gupt et l in [21] proposes reputtion bsed resource lloction mechnism in which the initil reputtion is djusted ccording to the level of whitewshing in the network. In [22] Li et l. uses the decision function tht tkes shred nd subjective history of the previous interctions in deciding whether to cooperte or defect with the requester. M et l. in [23] proposes wter filling squred bucket lgorithm. In which the width of the bucket is the contribution level of the user nd the height is the required demnd of the user. The lloction is given on the bsis of shorter height first. This mechnism ensures the mximiztion of individul nd socil utility. In [24] M et l. llocte the resources to the users on the bsis of their contribution level nd requested bndwidth. Yn et l. in [25] uses the contribution level s the rnking of the user nd llocte the resources on the bsis of the rnking of the user. All of these schemes considered reputtion clcultion s compulsory for ll the nodes nd on the bsis of reputtion they impose their lloction scheme. But we in this work nlyze reputtion clcultion s strtegy of the user nd found tht whether lot of resource lloction mechnism hs been given but eveeputtion clcultion is not n evolutionry stble strtegy. For mking reputtion s n evolutionry stble strtegy we devise mechnism for the utonomous peer-to-peer system so tht it could be n evolutionry stble strtegy. In [26] VPEF propose Evolutionry Gme Theory bsed mechnism, VPEF (Voluntry Principle nd round-bsed Entry Fee), to enforce coopertion in the society. In VPEF uthor modeled the interction mong the users s public goods gme, wheres we modeled the interction s two plyer strtegic gme becuse ll the reciproction in peer-to-peer network re pirwise interction. Sme s in VPEF we lso incorporte round bsed entry fees. VPEF highlights the role of selection of different strtegies wheres we highlight the role of stbility of strtegies ginst muttion. In [27] uthor, evolutionry gme theoreticlly nlysed the reputtion strtegy in mobile d hoc network using simultion nd s of their strtegic gme reputtion strtegy is not evolutionrily stble strtegy. They re minimizing the possibilities of invding of reputtion strtegy by lwys defect strtegy. As of the nonymous, utonomous nd dynmic nture of the peer-topeer network uthor in [28] proposes the mechnism in which some coopertors first behve like generous,nd then like hrsh ccording to peers current behviors. One of the wek-point of the bove scheme is whether the punisher will dominte the system, but neither punishment nor coopertion is evolutionry stble strtegy. III. MODELIG OF REPUTATIO SYSTEM AS A GAME In this pper, we hve used peer, user, node nd gent interchngebly. Peer-to-peer network hs been ssumed s pure i.e., without ny centrl server with totl number of peers. We lso ssume tht ny two peers in the network cn interct with ech other. A P2P system without ny punishment nd rewrd mechnism cn simply be modeled s fmous Prisoners Dilemm gme in which defection lwys strictly domintes coopertion strtegy. The coopertion strtegy cn only survive in the system when it cn dominte the defection strtegy. In the reputtion bsed resource lloction mechnisms, reputtion is clculted by the peers. Resources re llocted to the resource requesting peers bsed on their reputtion. Reputtion mngement is tool to punish the defectors but on the cost of reputtion clcultion. Peers prefer to sve this
3 dditionl cost involved ieputtion clcultion. Therefore, most of the coopertors does not clculte reputtion nd only cooperte. If the frction of reputtion clcultors in the popultion comes to lower thn threshold, this leds to the domintion of defectors nd consequent collpse of system. The threshold cn be clculted by modeling whole sitution s strtegic gme. Although, user cn interct with multiple other users t time but s ech interction is independent from other interctions, hence we cn model ll these interctions s pirwise interction gme between two users. We model this phenomen s symmetric simultneous gme where both plyers mke their moves simultneously. A. Generl Reputtion Gme Here peer s strtegy my be clssified into three types viz. Reputtion Clcultion with coopertion (R), Coopertion (C), Defection (D). Users plying R strtegy lwys provide requested services s per the reputtion of the requesters; Users plying C strtegy lwys provide the requested services to ll users; Users plying D strtegy lwys deny ny requested service. Therefore, reputtion system is modeled s the strtegic gme. Plyers:- User1, User2 Strtegies:- Reputtion Clcultion with coopertion (R), Coopertion (C), Defection (D) Preferences U i (A i, A i ) = ( C l i (1 R l i ) + C l i R l i C l i ) d ( C l i (1 R l i ) + C l i R l i ) R l i α + ( C l i R l i ) β (1) where A i nd A i re the ctions of plyer i nd other thn plyer i respectively. C l i is the coopertion level of plyer i nd R l i is the reputtion clcultion level of plyer i respectively. For C (coopertion) strtegy : C l = 1 nd R l = 0. Becuse these users re lwys cooperting nd not clculting reputtion. Similrly for R (reputtion clcultion with coopertion) strtegy : C l = 1 nd R l = 1, for D (defection) strtegy : C l = 0 nd R l = 0 In the preference function the first term represents the benefit of shring, the benefit of shring resources cn only be obtined by first user when the second user is either coopertor (C) or when the first plyer is either coopertor or reputtion clcultor user (C nd R) nd second plyer is reputtion clcultor (R) user. The second term represents the cost of shring, the cost of shring will only be imposed when the plyer is either coopertor or he is reputtion clcultor nd second plyer is coopertor. Third term represents the cost of reputtion clcultion which is lwys incurred when the first user is reputtion clcultor (R) user. Fourth term is the benefit of reputtion increment. The pyoff mtrix of the gme is illustrted in tble I. In this mtrix, row corresponds to the possible ctions of peer A wheres, column corresponds to the possible ctions of peer B nd the vlues in ech box re the plyers pyoffs to the ction profile to which the box corresponds, with A s pyoff listed first. Ech first vlue ij of this tble symbolizes the pyoff of A with strtegy S i, when R (A) TABLE I: Simplistic Model R(B) C (B) D(B) d α + β, d α, d α + β d + β α,0 C (A) d + β,d α d,d,d D (A) 0, α d, 0,0 TABLE II: symbols used in modeling the reputtion gme Symbol d β α P R P C P D x i n d n c p Definition Benefit received by getting the service from the coopertor The cost incurred due to providing the service to the other plyer The benefit received due to improving the reputtion The cost incurred due to clcultion of reputtion Averge Pyoff of R strtegic Plyers Averge Pyoff of C strtegic Plyers Averge Pyoff of D strtegic Plyers The proportion of peers with strtegy i Totl number of D strtegy users Totl number of R strtegy users Totl number of C strtegy users Round bsed pyment pyed by users B opts for strtegy S j. Tke the first vlue 12 for instnce, the vlue d α is the pyoff of A with R strtegy when B opts C strtegy where nd α is the cost incurred due to providing the service to the other plyer nd the cost incurred due to clcultion of reputtioespectively. In this α < s the cost of reputtion clcultion is lwys less thn cost of shring, otherwise R strtegy users loss is more thn C strtegy users when they ply with D strtegy users nd so will lwys prefer only to cooperte without clcultion of reputtion. If user with R strtegy meets user with C strtegy, it will lwys grnt service to C strtegy user nd get service from the C strtegy user. Thus, it would obtin benefit d. However, to clculte the reputtion of the peers, the user with R strtegy hs to communicte to the other peers for informtion. So it hs to ber n extr cost α. Therefore, the totl pyoff of user with R strtegy in this trnsction is d α. In this d > s the benefit received by shred dt is lwys greter thn the cost of shring. ow consider the second vlue b 12 tht is the pyoff of A with C strtegy when B opts R strtegy. If user with C strtegy meets user with R strtegy it will lwys grnt service nd get service from the R strtegy user. Thus, it would obtin benefit d. However, due to its coopertive behvior its reputtion would lso increse, so it would get the extr benefit for reputtion increment β. Therefore, the totl pyoff of user with C strtegy in this trnsction is d + β. Anlysis:- In this gme if R strtegy user intercts with D strtegy user, then he gets pyoff α nd if C strtegy user intercts with D strtegy user, then he gets pyoff which re less thn 0. This shows tht users does not hve higher pyoff in unilterlly devition from profile (D, D). Therefore (D, D) is the pure strtegic strict sh equilibrium
4 nd consequently D is the evolutionry stble strtegy (ESS). U 1 (D, D) > U 1 (x, D) (2) where x is ny strtegy other thn D. Let us ssume tht the popultion frction of R, C nd D strtegies re x R, x C, nd (1 x R x C ) respectively. Therefore, the verge pyoff of ech strtegy is written s P R = d (x R + x C ) (x R + x C ) + x R β α (3) P C = d (x R + x C ) + x R β (4) P D = x C d. (5) From the bove equtions following observtions cn be mde. If the reputtion clcultion cost α is ssumed to be negligible, then the expected pyoff for R strtegy users will lwys be greter thn the C strtegy users till there re D strtegy users s in this cse x R + x C is less thn 1 nd it will be equl when there is no D strtegy user The pyoff received by R strtegy will be higher thn the D strtegy i.e., P R > P D when x R > (x R+x C )+α () i.e., when frction of R strtegy user is greter thn the rtio of totl expected cost incurred to R strtegy users by popultion nd individul benefit received by R strtegy user when plyed with R strtegy user. The pyoff received by C strtegy will be higher thn the D () strtegy i.e., P C > P D when x R > i.e., when the frction of R strtegy user is greter thn the rtio of totl expected cost pyed by C strtegy user nd individul benefit received by C strtegy user when plyed with R strtegy user. If frction of R strtegy users re lesser thn both the rtio mentioned bove, then the pyoff of D strtegy users becomes highest in the popultion nd therefore users imittes to D strtegy, becuse now P D > P R nd P D > P C. The pyoff to R strtegy users will be higher thn C nd D strtegy when P R > P C nd P R > P D i.e., x D > α i.e., when the frction of D strtegy users is greter thn the rtio of cost of reputtion clcultion nd cost of shring, nd lso when x R > (x R+x C )+α (). We hve lredy exmined pure strtegy equilibrium now let us exmine the mixed strtegy equilibriums of the gme. Existence of mixed strtegy sh Equilibrium For the mixed strtegy equilibrium first we will exmine the mixed strtegy with ny two strtegies, then we will tke the combintion of ll three strtegies. Let us consider the combintion of two strtegies. First tke C nd R strtegy. In this combintion C lwys domintes the R strtegy which is then dominted by the D strtegy. If we tke C nd D strtegy, then D lwys domintes the C strtegy. If we tke R nd D strtegy, then lthough in this combintion R nd D strtegy is in itself pure strtegy sh equilibrium but only D strtegy fulfills the condition for pure strtegy sh equilibrium in the presence of C strtegy. This two strtegy combintion provides nother mixed strtegy sh equilibrium with zero pyoff. The equilibrium cn be obtined by solving following equtions. x R (d α + β) + ( α) (1 x R ) = 0 (6) By this equlity we got α x R = (d + β) x C = 0 α x D = 1 (d + β) (7) (7b) (7c) The second mixed strtegy equilibrium 7 is only possible when the pyoff of C strtegy with the bove combintion is less thn or equl to the pyoff of R nd D strtegy users i.e., P C <= 0 i.e., (d β) 2 α. This equilibrium leds to zero pyoff so this is not useful from system designer perspective. ow we will nlyze the mixed strtegy equilibrium with ll three strtegy. For mixed strtegic equilibrium with ll three strtegies: x R (d α + β) + x C (d α) + (1 x R x C ) ( α) = x R (d + β) + x C (d ) + (1 x R x C ) ( ) = x R 0 + x C d + (1 x R x C ) 0 (8) By this equlity we got x R = d + β (d + β)( α) 2 x C = (d + β) (9) (9b) x D = 1 x R x C = α (9c) 1) Theoreticl Anlysis of Mixed Strtegy Equilibrium with All Three Strtegies: In this gme, (D,D) is strict sh equilibrium but this equilibrium stte leds to no shring from ll the peers nd results in collpse of the system. The only equilibrium stte which llow the survivl of the system is polymorphic mixed strtegy equilibrium depicted by eqution (9). In this section we nlyzed the mixed strtegy equilibrium eqution for vrying single prmeter vlue (viz. d,, α, β) while other prmeters remin fixed. By this nlysis of the mixed strtegy equilibrium in eqution 9 following things cn be observed:- With the increment in cost of the reputtion clcultion α, D strtegy users increses, C strtegy users decreses nd R strtegy users remins sme iesulting mixed strtegy equilibrium With the increment in cost of shring, R strtegy users increses, D strtegy users decreses. With the increment in benefit of shring d, C strtegy users increses, R strtegy users decreses nd D strtegy users remins sme iesulting mixed strtegy equilibrium. With the increment in benefit of reputtion increment β the frction of R strtegy users decreses, frction of C strtegy users decreses nd frction of D strtegy users remins constnt. The resoning behind first observtion is tht s cost of reputtion clcultion α increses, then the pyoff to R strtegy users decreses nd therefore the R strtegy becomes less lucrtive to the users so they switches to C strtegy nd D strtegy users. As R strtegy users decreses nd
5 () Fig. 1: Frction of popultion in mixed sh equilibrium for the network () vs benefit of shred dt d with other prmeters{ = 3, α = 3, β = 4, p = 0.5} vs cost of reputtion clcultion α with other prmeters {d = 8, = 3, β = 4, p = 0.5} vs cost of shring with other prmeters {d = 8, α = 2, β = 4, p = 0.5} C strtegy users increses, then the pyoff to C strtegy users lso decreses nd pyoff to D strtegy users increses. Due to this the C strtegy users lso switches to D strtegy users. As D strtegy users increses the pyoff to C strtegy users decreses more nd comes to lower thn R strtegy. Due to this C strtegy users now switches to R strtegy till the pyoff of ll the strtegy equlizes. Due to this in new equilibrium D strtegy users increses, C strtegy users decreses nd R strtegy users remins sme s shown in figure 1b. The resoning behind second observtion is tht s cost of shring increses, then the pyoff of R nd C strtegy users decreses. Due to this most of the R nd C strtegy users switches to D strtegy users. As C strtegy users decreses the pyoff to D strtegy lso decreses nd so D strtegy users lso switches to R strtegy. As R strtegy users increses the pyoff to C strtegy users increses nd so D strtegy users lso switches to C strtegy. Due to this in finl equilibrium R strtegy users increses, D strtegy users decreses. The resoning behind third observtion is, t one equilibrium stte when the expected pyoff of ll the three strtegies re sme, then s benefit of shred dt increses the expected pyoff of R user nd C user increses eqully wheres the expected pyoff of D user increses x R frction lesser thn R nd C strtegy users. Due to this D strtegy users switches to C strtegy users nd R strtegy users eqully. But s the frction x R nd x C increses, then the expected pyoff of reputtion R users increses lesser thn expected pyoff of C strtegy users nd s frction of C strtegy users increses the expected pyoff to D strtegy users lso increses. So R strtegy users switches to coopertion nd defection users. As x R decreses the pyoff to C strtegy users lso decreses so C strtegy users switches to D strtegy. This drift process comes with next equilibrium stte in which R strtegy users decreses, C strtegy users increses nd D strtegy users remins sme s shown in figure 1. The resoning behind fourth observtion is sme s the benefit of shred resources. In this s the benefit of reputtion increment β would increse the expected pyoff of reputtion nd C strtegy users would increse wheres the expected pyoff of D strtegy users remin sme. Due to this D strtegy users switches to R nd C strtegy users. As the C strtegy users increses the pyoff of D strtegy users gin increses. As the frction of reputtion nd coopertion increses the expected pyoff of R users increses lesser thn C strtegy. So R strtegy users switches to D nd C strtegy users nd in next equilibrium D strtegy users remin unchnged nd frction of R strtegy users decreses nd frction of C strtegy users increses. B. Reputtion system with round bsed initil pyment distributed to coopertive users In the previous subsection, we hve nlyzed the reputtion system in which there is no initil pyment required for the peers. ow in this subsection we will nlyze the reputtion system with round bsed initil pyment imposed to the peers, which is distributed mong the coopertion nd reputtion with coopertion peers (C nd R strtegy). The gme cn be defined s follows. Plyers:- User1, User2 Strtegies:- Reputtion with coopertion (R), Coopertion (C), Defection (D) Preferences U i (A i, A i ) = ( C l i (1 R l i ) + C l i R l i C l i ) d ( C l i (1 R l i ) + C l i R l i ) R l i α + ( C l i R l i ) β + C p l i p n d where A i nd A i re the ctions of plyer i nd plyer other thn i respectively. C l i is the coopertion level of plyer i nd R l i is the reputtion clcultion level of plyer i respectively. For C (coopertion) strtegy : C l = 1 nd R l = 0. Becuse these users re lwys cooperting nd not clculting reputtion. Similrly for R (reputtion clcultion) strtegy : C l = 1 nd R l = 1 nd for D (defection) strtegy : C l = 0 nd R l = 0 In the preference function the first term represents the benefit of shring, the benefit of shring the resources cn only be obtined by first user when the second user is either coopertor (C) or when the first plyer is either coopertor or reputtion clcultor user (C nd R) nd second plyer is reputtion clcultor (R) user. The second term represents the cost of shring, the cost of shring will only be imposed when the plyer is either coopertor or he is reputtion clcultor nd second plyer is coopertor. Third term represents the cost of reputtion clcultion cost which is lwys incurred when the first user is reputtion clcultor (R) user. Fourth term is the benefit of reputtion increment. Fifth term is the benefit due
6 TABLE III: Initil Pyment With Distribution To Coopertive Users R (A) C (A) D (A) R(B) C (B) D(B) d α+, +n c d + β+ +n c d α + β+, +n c d α + β+ +n c d + β+, +n c d α+ +n c n p, d p α +n c +n c α, p d +, n +n c + d p, d + n +n c d p d p +n c d p, + p, p +n c to initil pyment distribution. Pyoff mtrix of the gme is shown in tble III. Anlysis In this gme we clim tht if p (1 n d ) (10) then coopertion strtegy profile i.e., (C,C) will be the sh equilibrium nd if p > (1 n d ) (11) then coopertion strtegy profile will be Evolutionrily stble strtegy. If the condition in eqution 10 is not fulfilled nd if p < α (1 n d ) (12) then the defection strtegy profile i.e., (D,D) will be the pure strtegy sh equilibrium profile. The rgument for strtegy profile (C,C) s sh equilibrium, given condition (10), is s follows. We cn observe tht if condition (10) is true, then U 1 (C, C) U 1 (D, C) nd U 1 (C, C) U 1 (R, C) tht mens the pyoff of coopertion strtegy when plyed with itself is lwys greter thn or equl to other two strtegy while plyed with coopertion. The rgument for strtegy profile (C,C) s Evolutionry Stble, given condition (11), is s follows. We cn observe tht if condition (11) is true, then U 1 (C, C) > U 1 (D, C) nd U 1 (C, C) > U 1 (R, C) tht mens the pyoff of coopertion strtegy when plyed with itself is lwys strictly greter thn D nd R strtegy while plyed with C strtegy. The rgument for strtegy profile (D,D) s sh equilibrium s well s Evolutionry stble, given condition (12), is s follows We cn observe tht if condition (12) is true, then U 1 (D, D) > U 1 (R, D) nd U 1 (D, D) > U 1 (C, D) tht mens the pyoff of Defection strtegy when plyed with itself is lwys greter thn R nd C strtegy when plyed with D strtegy. If condition 10 nd 12 is not stisfied, then there is no pure strtegic sh equilibrium in this gme. Therefore, now we will compute the mixed strtegy sh equilibrium profile. For this the expected pyoff of ech strtegy cn be written s P R = d (x R + x C ) (x R + x C ) + x R β α + n d (13) P C = d (x R + x C ) + x R β + (13b) n d P D = x C d p (13c) First we will consider the mixed strtegy in the combintion of two strtegies. If we tke the combintion of only R nd C strtegies, then in this combintion C strtegy lwys domintes R, so no mixed strtegy equilibrium exist. ow we consider the combintion of R nd D strtegies, then if condition (12) is not fulfilled, this results in the domintion of R strtegy over D strtegy nd hence gin mixed strtegy equilibrium does not exist. But if condition (12) is fulfilled, then there exist mixed strtegy sh equilibrium which cn be obtined by equting the expected pyoffs of R nd D strtegy users such tht, x R (d α + β + ( n d α) = p x R (d + β) = α x R = α p ( (d + β) x C = 0 n d ) + (1 x R ) p n d (14) n d ) n d ) x D = 1 α p ( (d + β) (15) (15b) (15c) This equilibrium leds to negtive pyoff so this is not useful from system designer perspective. ow we consider the combintion of C nd D strtegy. If condition 11 is fulfilled, then C strtegy domintes D strtegy nd so no mixed strtegy equilibrium presents. But if p = (1 n d ), then U 1(C, C) = U 1 (D, C) nd U 1 (C, D) = U 1 (D, D). So t this condition lthough coopertion is pure strtegic wek sh equilibrium, but s D strtegy users re lso getting the sme pyoff so there exist mixed strtegy sh equilibrium which cn be obtined by equting the pyoff of D nd C strtegy users such tht, x C (d + ) + (1 x C ) ( + ) = n d n d x C (d p) + (1 x C ) ( p) (16) this drift would be there till the pyoff of reputtion strtegy is lesser thn these two strtegies i.e., x C (d + ) + (1 x C ) ( + ) > n d n d x C (d α + ) + (1 x C ) ( α + ) n d n d (17)
7 By solving bove inequlity we get x D < α (18) This shows tht till the frction of defectors remins lesser thn the rtio of reputtion cost nd cost of shring, reputtion users will not be there in the system. This is becuse when the defectors re less in the society, then pying the reputtion cost seems less useful. But s defectors increse, the pyoff to reputtion strtegy increses nd users muttes to the reputtion strtegy. ow we will find out the mixed strtegy with ll the three strtegies. For this equilibrium, the expected pyoff to ll three strtegies should be equl. (d α + β + n d ) x R + (d α + ) x C + ( α)(1 x R x C ) n d n d = (d + β + ) x R + (d + ) x C n d n d +( + )(1 x R x C ) n d = ( p) x R + (d p) x C + (1 x R x C ) ( p) (19) By solving bove equlity x D = α x R = ( p ( d + β x C = ( α) n d )) n d ) p ( (d + β) (20) (20b) (20c) Putting this frction of D strtegy users in the frction of reputtion nd coopertion strtegy we got x R = ( p ( α )) d + β x C = ( α) p ( α ) (d + β) (21) (21b) In this mixed strtegy equilibrium described by (20), following things cn be observed:- With the increment in cost of the reputtion clcultion α, D strtegy users increses, C nd R strtegy users decreses iesulting mixed strtegy equilibrium With the increment in cost of shring, R strtegy users increses wheres D strtegy users decreses iesulting mixed strtegy equilibrium The frction of R strtegy is inversely proportionl to initil pyment p wheres the frction of C strtegy users is directly proportionl to p. Moreover, the mixed strtegy equilibrium is not defined for p greter thn (1 x D ) The frction of R strtegy users decreses, frction of C strtegy users increses nd frction of D strtegy users remin sme with the increment in benefit of reputtion increment β The resoning of first observtion is, s the cost of the reputtion clcultion α increses, the expected pyoff to R () Fig. 2: prmeter vs Frction of popultion in mixed sh equilibrium for the network with () vs cost of shring with other prmeters {d = 8, α = 2, β = 4, p = 0.5} vs cost of reputtion clcultion lph with other prmeters {d = 8, = 3, β = 4, p = 0.5} vs benefit of reputtion increment with other prmeters { d = 8, α = 2, = 3, p = 0.5} vs round bsed initil pyment p with other prmeters {β = 4, α = 2, = 3, d = 8} strtegy users decreses so they switches to coopertion nd defection users. As the coopertion increses the pyoff to D strtegy users increses nd s R strtegy users decreses the pyoff to C strtegy users lso decreses so C strtegy users lso switches to D strtegy. As D strtegy users increses the pyoff to reputtion nd coopertion users slightly increses becuse now they re getting benefit of the pyment p. As defection increses nd coopertion decreses the pyoff to defection lso decreses nd so they switches to R strtegy users. As R strtegy users increses the pyoff to C strtegy users increses so some D strtegy users now switches to coopertion. This whole process shifts the equilibrium where x R nd x C decreses nd x D increses. Unlike previous gme in this gme frction of R strtegy users decreses s α increses becuse s defection increses the pyoff to coopertion lso increses due to pyment so some R strtegy users switches to coopertion in equilibrium s shown in figure 2b The resoning behind second observtion is, s cost of shring increses, the expected pyoff to R nd C strtegy users decreses, but the pyoff to C strtegy users decreses more so C strtegy users switches to D strtegy users which currently hs highest pyoff. As the frction of C strtegy users decreses the expected pyoff to D strtegy users lso decreses nd it comes to lowest. So now D strtegy users switches to C strtegy users nd R strtegy users. Like previous gme in this gme the next equilibrium comes t point where frction of R strtegy users increses nd frction
8 of D strtegy users decreses but s compre to previous gme the rte of chnge is low becuse now the R strtegy users lso getting benefit of the pyment from D strtegy users. The resoning behind third observtion is, s the initil pyment p increses, the expected pyoff of C nd R strtegy users increses wheres the expected pyoff to D strtegy users(d x C p) decreses. As the pyoff to R nd C strtegy users increses the D strtegy users switches to R nd C strtegy users. As D strtegy users decreses nd R strtegy users increses the pyoff to C strtegy users increses more thn R strtegy users due to this R strtegy users switches to C strtegy users. As R strtegy users decreses nd C strtegy users increses the pyoff to D strtegy users increses so some R nd C strtegy users switches to D strtegy. Due to this process frction of R strtegy users decreses, C strtegy users increses nd D strtegy users remins sme in new sh equilibrium s shown in Figure 2d. If the initil pyment stisfies the condition 11 i.e., C strtegy is evolutionry stble, then mixed strtegy equilibrium is not defined nd hence, it cn be observed in figure 2d tht the R strtegy users frction becomes negtive for initil pyment greter thn (1 x D ) i.e., 0.99. The resoning behind fourth observtion is, s the benefit of reputtion β increses the expected pyoff of C nd R strtegy users increse eqully wheres the expected pyoff of D strtegy users remin sme. Due to this D strtegy users switches to R nd C strtegy users. As the frction of R strtegy users increses nd D strtegy users decreses, the pyoff of C strtegy users increses more thn R strtegy users so R strtegy users lso switches to C strtegy. As C strtegy users increses the pyoff of D strtegy users gin increses nd C nd R strtegy users lso switches to D strtegy. Due to this whole process frction of R strtegy users decreses, frction of C strtegy users increses nd frction of D strtegy users remin sme in the new sh equilibrium s shown in Figure 2c. C. Reputtion system with round bsed initil pyment distributed to reputtion clcultor R users In the lst subsection, we hve nlyzed the reputtion system in which initil pyment is distributed mong the coopertion nd reputtion with coopertion peers (C nd R strtegy). ow, in this subsection, we will nlyze the reputtion system with round bsed initil pyment imposed to the peers, which is distributed mong only reputtion (R) strtegy users. The gme cn be defined s follows. Plyers:- User1, User2 Strtegies:- Reputtion with coopertion (R), Coopertion (C), Defection (D) Preferences u i (A i, A i ) = ( C l i (1 R l i ) + C l i R l i C l i ) d ( C l i (1 R l i ) + C l i R l i ) R l i α + ( C l i R l i ) β + R p l i C l i p (22) n d TABLE IV: Initil Pyment with Distribution To Reputtion Users R (A) C (A) D (A) R(B) C (B) D(B) d α + β+ ( ) p, d α + β+ ( ) p d + β p, d α+ ( ) p d α+ ( ) p, d + β p ( ) p α, p d p p,d p d p, p, ( nr) p α d p, p p, p where A 1 nd A 2 re the ctions of plyer 1 nd plyer 2 respectively. C l i is the coopertion level of plyer i nd R l i is the reputtion clcultion level of plyer i respectively. For C (coopertion) strtegy : C l = 1 nd R l = 0. Becuse these users re lwys cooperting nd not clculting reputtion. Similrly For R (reputtion clcultion) strtegy : C l = 1 nd R l = 1 For D (defection) strtegy : C l = 0 nd R l = 0 In the preference function the first term represents the benefit of shring, the benefit of shring the resources cn only be obtined by first user when the second user is either coopertor (C) or when the first plyer is either coopertor or reputtion clcultor user (C nd R) nd second plyer is reputtion clcultor (R) user. The second term represents the cost of shring, the cost of shring will only be imposed when the plyer is either coopertor or he is reputtion clcultor nd second plyer is coopertor. Third term represents the cost of reputtion clcultion cost which is lwys incurred when the first user is reputtion clcultor (R) user. Fourth term is the benefit of reputtion increment. Fifth term is the benefit due to initil pyment distribution. Pyoff mtrix of the gme is shown in tble IV. Anlysis In this gme if p > α (23) then we clim tht reputtion (R) strtegy will be pure strtegic strict sh equilibrium nd hence evolutionrily stble. Where is the number of R strtegy users in the popultion. If (23) is not fulfilled nd if p < α (24) then we clim tht D strtegy is pure strtegy sh equilibrium, nd hence evolutionrily stble. The rgument for strtegy profile (R,R) s strict sh equilibrium, given condition (23), is s follows. We cn observe tht if condition (23) is true, then U 1 (R, R) > U 1 (C, R) nd U 1 (R, R) U 1 (D, R) tht mens the pyoff of R strtegy when plyed with itself is lwys greter thn or equl to other two strtegy while plyed with R. The rgument for strtegy profile (D,D) s strict sh equilibrium, given condition (24), is s follows. We cn observe tht if condition (24) is true, then
9 U 1 (D, D) > U 1 (C, D) nd U 1 (D, D) U 1 (R, D) tht mens the pyoff of D strtegy when plyed with itself is lwys greter thn or equl to other two strtegy while plyed with D. ow we will compute the mixed strtegy sh equilibrium profile. For this the expected pyoff of ech strtegy cn be written s P R = d (x R + x C ) (x R + x C ) + β x R + p α (25) P C = d (x R + x C ) + β x R p (25b) P D = d x C p (25c) In this gme if condition (23) is fulfilled, then no mixed strtegy equilibrium presents s R is strictly dominting strtegy. If this condition is not fulfilled, then we check for multiple equilibrium in the system. Let us first tke the combintion of C nd D strtegies. In this combintion D strtegy lwys domintes the C strtegy, hence no mixed strtegy equilibrium presents. If we tke the combintion of R nd D strtegy, then if p = α, then (D,D) will be pure strtegic wek sh equilibrium nd (R,R) will be pure strtegic strict sh equilibrium. In this condition no mixed strtegy equilibrium presents. If the condition is p < nr α, then the negtive pyoff mixed strtegic sh equilibrium presents but this is of no use to the system designers. ow we tke the combintion of R nd C strtegies. In this if p = nr α, then drift occurs mong these two strtegies nd it will be continued till the expected pyoff of D strtegy will be less thn the expected pyoff to these strtegies, x R (d + β α + p) + (1 x R ) (d α + p) > x R ( p) + (1 x R ) (d p) (26) By using bove eqution we got x R > d + β (27) This mens tht when R strtegy users re more in the system, then plying D strtegy will not be lucrtive. In the combintion of R nd C strtegy if p < nr α, then C strtegy domintes R strtegy nd so no mixed strtegy equilibrium presents in this condition. ow let us tke the combintion of ll three strtegies for the mixed strtegy equilibrium. For this, the equlity is, (d α + β + ( ) p )x R + (d α + ( )p ) x C + ( ( ) p α)(1 x R x C ) = (d + β p)x R + (d p) x C + ( p)(1 x R x C ) = ( p) x R + (d p) x C +(1 x R x C ) ( p) (28) By this equlity we got x R = d + β x C = 1 α p x D = α p d + β (29) (29b) (29c) In this mixed strtegy equilibrium described by (29), following things cn be observed:- With the increment in cost of the reputtion clcultion α, D strtegy users increses, C strtegy users decreses nd R strtegy users remins sme iesulting mixed strtegy equilibrium With the increment in benefit of reputtion increment β, the frction of R nd D strtegy users decreses wheres frction of C strtegy users increses With the increment in the initil pyment p, the frction of D strtegy users decreses, wheres the frction of C strtegy users increses With the increment in cost of shring, C strtegy users decreses, R nd D strtegy users increses iesulting mixed strtegy equilibrium. The resoning behind first observtion is tht s cost of reputtion clcultion α increses, then the pyoff to R strtegy users decreses nd therefore the R strtegy becomes less lucrtive to the users so they switches to C strtegy nd D strtegy users. As R strtegy users decreses nd C strtegy users increses, then the pyoff to C strtegy users lso decreses nd pyoff to D strtegy users increses. Due to this the C strtegy users lso switches to D strtegy users. As D strtegy users increses the pyoff to C strtegy users decreses more nd comes to lower thn R strtegy. Due to this C strtegy users now switches to R strtegy till the pyoff of ll the strtegy equlizes. Due to this in new equilibrium D strtegy users increses, C strtegy users decreses nd R strtegy users remins sme s shown in figure 1b. The resoning behind second observtion is tht s the benefit of reputtion increment β increses the pyoff to R nd C strtegy users increses wheres the pyoff to D strtegy users remins sme due to this D strtegy users switches to R nd C strtegy users. As the frction of D strtegy users decreses nd frction of R strtegy users increses the benefit of pyment to R strtegy users decreses so they lso switches to C strtegy users. This whole process continues till the pyoff to ll three strtegies equlizes. This results in the increment to the C strtegy frction nd decrement in the R nd D strtegy frction of popultion. The resoning behind third observtion is tht s the initil pyment p increses the pyoff to R strtegy users increses wheres the pyoff to C nd D strtegy users decreses. Due to this the C nd D strtegy users switches to R strtegy users. As R strtegy users increses the pyoff to C strtegy increses due to this R strtegy users switches to C strtegy users till the pyoff to ll three strtegies equlizes. This process results in increment in the frction of C strtegy, decrement in the frction of D strtegy nd remin sme in the frction of R strtegy.
10 () Fig. 3: Frction of popultion in mixed sh equilibrium for the network () vs benefit of reputtion increment β with other prmeters {p = 0.5, α = 2, = 3, d = 8} vs cost of shring with other prmeters {p = 0.5, α = 2, d = 8, β = 4} vs round bsed initil pyment p with other prmeters {α = 2, = 3, d = 8, β = 4} vs cost of reputtion clcultion α with other prmeters {p = 0.5, = 3, d = 8, β = 4} The resoning behind fourth observtion is tht s the cost of shring increses the pyoff to C strtegy users nd R strtegy users decreses nd pyoff to D strtegy users remins constnt. This results in switching of C nd R strtegy users to D strtegy. As the frction of C strtegy users decreses this results in the decrement the pyoff to D strtegy users nd increment the pyoff to R strtegy users s they re getting benefit from initil pyment. So now the users switch to R strtegy users till the pyoff to ll three strtegies equlizes nd in new equilibrium frction of the D nd R strtegy users increses wheres the frction of C strtegy users decreses. IV. UMERICAL AALYSIS OF DIFFERET MODELS OF REPUTATIO GAME All the bove three explined system model is nlyzed by simultion s well. By simultion, we hve shown the finl evolution of the system. The simultion experiments hve been conducted for 10000 nodes. We ssume the fully connected topology of the network in which ny two peer in the network cn interct with ech other t rndom in lrge nd well-mixed popultion. On the prt of ech user, system constitutes three strtegy viz. R (Reputtion clcultion with coopertion), C (Coopertion) nd D (Defection). To show the reltionship between the finl evolved frction nd the initil frction of different strtegy users, we hve tken different initil frctions of popultion for different strtegy users nd plotted their evolution seprtely. The evolution process contins the repetition of three phses viz. selection phse, trnsction phse nd reproduction phse. Initilly ech node selects ny of the other node for pirwise interction with equl probbility so the probbility tht user will interct with ny other strtegy user is the frction of tht strtegy users in the popultion. This phse is clled selection phse. After this phse ech node simultneously clcultes its pyoff using the utility function bsed on gme. This phse is clled trnsction phse. After ech trnsction, there is reproduction phse in which ll users imitte ny other strtegy with the probbility proportionl to the difference between the strtegy s expected pyoff nd the popultion expected pyoff. In our system model we ssume tht ech node hs the knowledge of ll his neighbor s pyoff nd strtegy. So the nodes dopts new strtegy ccording to the nturl selection. For the simultion we lso chooses the prmeter vlues viz. d (benefit of shring), (cost of shring), α (cost of reputtion clcultion), β (benefit of reputtion increment) nd p (initil pyment). In the selection of the prmeter vlues we follow constrints tht is necessry nd sufficient for modeling this gme viz. the cost of shring should lwys be less thn or equl to the benefit of the shred resources d nd greter thn cost of reputtion clcultion α. We hve exmined these prmeters for different vlues in ordinl fshion nd observed tht finl evolution is still sme. A. First Reputtion Gme 1) Reputtion cost α negligible: In first scenrio we hve tken α negligible. We run our simultion for different initil frctions of R, C nd D strtegies nd fixed prmeter vlues (d = 8, = 3, β = 4, α = 0). Figure 4 nd 4b shows tht the finl frction of R nd C strtegy depends on the initil frction of the strtegies, which substntite the theoreticl nlysis of the first gme. We observe tht s α is zero so the pyoff to R strtegy users is greter thn the pyoff to C strtegy until D strtegy user s frctioe greter thn zero nd becomes equl to the pyoff of C strtegy users when ll D strtegy users dies out. At first we run the simultion with 0.1, 0.2, 0.7 frction of R, C nd D strtegies users respectively (figure 4). As mentioned erlier, in the selection phse user selects other user with equl probbility so the probbility tht it will meet with R strtegy users is 0.1, with C strtegy users is 0.2 nd with D strtegy users is 0.7. Then in trnsction phse ech node simultneously clcultes the pyoff. ode selects other strtegy with probbility proportionl to the difference between his neighbor s pyoff nd his pyoff. Ech node imittes to the higher pyoff strtegy with positive probbility. In this scenrio initilly s x R < i.e., 0.1 < 0.25 nd x R > (x R+x C )+α i.e., 0.1 > 0.075, results in the expected pyoff order s P R > P D > P C, so C strtegy imittes to D nd R strtegy wheres D strtegy imittes to R strtegy. This results in increment in the R strtegy nd D strtegy frction nd decrement in C strtegy frction initilly. As the frction of R strtegy users increses nd becomes greter thn 0.25, then the expected pyoff of D strtegy users comes to lower thn C strtegy users which results in pyoff order s P R > P C > P D nd so from now D strtegy users imittes to R nd C strtegy, nd C strtegy users imittes to
11 () () Fig. 5: Round vs Frction of popultion for the network with d=8, =3, β = 4, α = 2 (e) Fig. 4: Round vs Frction of popultion for the network with d=8, =3, β = 4, p = 0.5, α = 0 only R strtegy. This results in increment to R nd C strtegy users wheres decrement in D strtegy frction. After this s D strtegy frction becomes zero which is equl to α, then the pyoff order becomes P R = P C > P D which further results in constnt frction of ll three strtegies. We observe tht in this scenrio s discussed in the model 1, in finl evolution system composed with only R nd C strtegy users s in figure 4. After this we run the simultion with 0.1, 0.7, 0.2 initil frction of R, C nd D strtegies users respectively (figure 4b). The sme process s of explined erlier gin hppens but in this scenrio the frction of C strtegy users re lmost greter thn 0.35 when the frction of R strtegy becomes greter thn. So in finl evolution gin there is only R nd C strtegy but in this C strtegy users re higher thn previous. In third initil setting, we took very less R nd C user frction i.e., 0.002 nd 0.001 respectively nd we observe tht C strtegy dies out before the frction of R strtegy becomes greter thn so only R strtegy users remins in the finl evolution. With this simultion scenrio, it cn be observed tht s (x R+x C )+α is zero when ll the popultion imittes to D strtegy, therefore the pyoff of R nd D strtegy is equl when x R, x C = 0 nd x D = 1. As x R slightly increses, the pyoff of R strtegy users becomes greter thn D strtegy users nd D strtegy users imittes (f) to R strtegy s in figure 4c nd 4d tht even with very smll initil frction from muttion, R strtegy is there in the finl evolution. In figure 4c, 4d nd 4e with initil R strtegy frction s 0.002, 0.005 nd 0.0005 respectively, it cn lso be observed tht s the initil frction of R strtegy decreses, the finl evolution time of the system increses. In figure 4f from beginning, the frction of D strtegy users decrese becuse from beginning initil frction of R strtegy remins greter thn 0.25 vlue. 2) Reputtion cost α is not negligible: In the second scenrio α is not negligible. Results of this scenrio lso substntite the theoreticl results of Gme 1. First we hve tken the frction of different strtegies s 0.6, 0.2, 0.2 frction of R, C nd D strtegies users respectively s shown in figure 5. Initilly when the R strtegy users re more thn nd (x R +x C )+α, nd D strtegy users re lesser thn α so the order of expected pyoff becomes P C > P R > P D. Due to this initilly R strtegy users imittes to C strtegy users wheres D strtegy users imittes to R nd C strtegy users. The frction of R strtegy users increses till the expected pyoff of R strtegy remins greter thn the expected pyoff α of popultion. This is the point where x R > α+x D ( ) i.e., 0.61. After this point s the frction of R strtegy decreses nd comes to lower thn i.e., 0.25, then the expected pyoff order becomes P D > P C > P R so now R strtegy users strt to imitte to both C nd D strtegy users s shown in figure 5. At this point s C strtegy users re more thn 70% nd D strtegy users re lesser thn 10% so R strtegy users intercts with more C strtegy users nd so they imittes to more C strtegy users. This continues till the expected pyoff of C strtegy users is greter thn the expected pyoff of the popultion which is the point where x R > ( )+. α x D After this point more R nd C strtegy imittes to D strtegy