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1 On A Way to Improve Cyber-Insurer Profts When a Securty Vendor Becomes the Cyber-Insurer Ranan Pal Unversty of Southern Calforna Emal: rpal@uscedu Leana Golubchk Konstantnos Psouns Unversty of Southern Calforna Emal: {leana, kpsouns}@uscedu Pan Hu T-Labs, Germany Emal: panhu@telekomde Abstract Current research on cyber-nsurance has manly been about studyng the market success of an nsurance-drven securty ecosystem Such an ecosystem comprses of a set of market elements (eg, cyber-nsurers, network users, securty vendors (SVs), regulatory agences, etc,) that coexst together as a system wth the success goal of mutually satsfyng each other s nterests However, exstng works have not explctly consdered SVs as market elements, have analytcally proved the moderate/no success 1 of cyber-nsurance markets due to nsurers not satsfyng ther nterests n makng zero expected profts at tmes In ths paper, we model a securty vendor (eg, Symantec, Mcrosoft) as a cyber-nsurer, thereby makng the former as an explct market element We then propose a novel consumer prcng mechansm for SVs based on ther clent/consumer logcal network consumer securty nvestment amounts, wth the goal to mprove SV profts Our smulaton results show that SVs could mprove up to 5% of ther current proft margns by accountng for clent locaton nvestment nformaton A fracton of the extra profts could then be used up by the SVs to recover costs related to provdng nsurance coverage to ther clents, also to always make strctly postve profts as nsurers Fnally, we propose a dscrete-tme sequental dynamc prcng strategy for an SV, show that t leads to mproved SV profts when compared to a statc prcng strategy Our work demonstrates one partcular way for a cyber-nsurance market to be successful on a more than moderate scale Keywords: nsurance, securty vendor, prcng, centralty I INTRODUCTION The nfrastructure, the users, the servces offered on computer networks today are all subect to a wde varety of rsks These rsks nclude dstrbuted denal of servce attacks, ntrusons of varous knds, eavesdroppng, hackng, phshng, worms, vruses, spams, etc In order to counter the threats posed by the rsks, network users have tradtonally resorted to antvrus ant-spam softwares, frewalls, ntruson-detecton systems (IDSs), other add-ons to reduce the lkelhood of beng affected by threats In practce, a large ndustry (companes lke Symantec, McAfee, etc) as well as consderable research efforts are currently centered around developng deployng tools technques to detect threats anomales n order to protect the cyber nfrastructure ts users from the negatve mpact of the anomales Inspte of mprovements n rsk protecton technques over the last decade due to hardware, software cryptographc methodologes, t s mpossble to acheve a perfect/near-perfect cybersecurty protecton [][7] The mpossblty arses due to a number 1 Currently, the US cyber-nsurance market s worth $800 mllon wth only certan ndustres organzatons buyng nsurance to cover for losses due to cyber-threats A more than moderate success would mply a cyber-nsurance market where common people, apart from ndustres organzatons, would also buy nsurance The term users may refer to both, ndvduals organzatons of reasons: () scarce exstence of sound techncal solutons, () dffculty n desgnng solutons catered to vared ntentons behnd network attacks, () msalgned ncentves between network users, securty product vendors, regulatory authortes regardng each takng approprate labltes to protect the network, (v) network users takng advantage of the postve securty effects generated by other user nvestments n securty, n turn themselves not nvestng n securty resultng n the free-rdng problem, (v) customer lock-n frst mover effects of vulnerable securty products, (v) dffculty to measure rsks resultng n challenges to desgnng pertnent rsk removal solutons, (v) the problem of a lemons market [1], whereby securty vendors have no ncentve to release robust products n the market, (v) lablty shell games played by product vendors In vew of the above mentoned nevtable barrers to 100% rsk mtgaton, the need arses for alternatve methods of rsk management n cyberspace In ths regard, some securty researchers n the recent past have dentfed cyber-nsurance as a potental tool for effectve rsk management Cyber-nsurance s a rsk management technque va whch network user rsks are transferred to an nsurance company (eg, ISP, cloud provder), n return for a fee, e, the nsurance premum Proponents of cyber-nsurance beleve that cyber-nsurance would lead to the desgn of nsurance contracts that would shft approprate amounts of self-defense lablty on the clents, thereby makng the cyberspace more robust Here the term self-defense mples the efforts by a network user to secure ther system through techncal solutons such as ant-vrus ant-spam softwares, frewalls, usng secure operatng systems, etc Cyber-nsurance has also the potental to be a market soluton that can algn wth economc ncentves of cyber-nsurers, users (ndvduals/organzatons), polcy makers, securty software vendors, e, the cyber-nsurers wll earn proft from approprately prcng premums, network users wll seek to hedge potental losses by ontly buyng nsurance nvestng n selfdefense mechansms, the polcy makers would ensure the ncrease n overall network securty, the securty software vendors could go ahead wth ther frst-mover lock-n strateges as well as experence an ncrease n ther product sales va formng allances wth cyber-nsurers A Research Motvaton Recent research works on cyber-nsurance [6][7][10] have mathematcally shown the exstence of neffcent nsurance markets Intutvely, an effcent market (see Secton I-C) s one where all stakeholders (market elements) mutually satsfy ther nterests These works state that cyber-nsurance makes every stakeholder (see Secton I-C) satsfed apart from the regulatory agency (eg, government) sometmes the cyber-nsurer tself The regulatory agency s unsatsfed as overall network robustness s sub-optmal due to network users not optmally nvestng n self-defense mechansms, whereas a cyber-nsurer s unsatsfed due to t potentally makng zero expected profts at tmes Lelarge etal n [7] recommended the use of fnes rebates on cyber-nsurance contracts to make 1

2 each user nvest optmally n self-defense nvestments make the network optmally robust However, there s stll no work that guarantees the strct postveness of nsurer profts at all tmes The noton of makng zero expected profts at tmes s enough for cybernsurers to opt out of the market n future, leadng to an nsurance market falure The mportant queston that thus arses s s there a way for a cyber-nsurer to make strctly postve profts at all tmes, at the same tme ensure optmal network robustness? A postve answer to ths queston would mply cyber-nsurance market success on a larger than moderate scale In addton, n a correlated rsk envronment such as the Internet, an nsurers cannot afford to be rsk-neutral as there are chances t mght go bankrupt due to expected aggregate losses n a perod beng more than what t could afford to compensate As a result t mght hold a safety captal for a certan cost n order to prevent tself from gong bankrupt [3] The queston that arses here s how can the cyber-nsurers recover costs of buyng safety captal? The above two questons motvate us to nvestgate a way n whch cyber-nsurers can always make profts recover ther costs to provde nsurance coverage to clents, at the same tme ensure optmal network robustness B Research Contrbutons We make the followng research contrbutons n ths paper We model rsk-averse securty vendors as cyber-nsurers propose a one-perod statc product prcng scheme for ther consumers based on the consumers logcal network ther securty nvestment amounts Our proposed approach () potentally ncreases the current proft margns of SVs upto 5% allows an SV to make strctly postve profts at all tmes, solely as an nsurer, () ensures the state of optmal network robustness, () allows rsk-averse SV nsurers to recover costs such as ones related to buyng safety captal (See Secton III) We extend the one-perod statc prcng scheme above to a sequental mult-stage dynamc prcng scheme show that t results n an SV cyber-nsurer makng more profts compared to that n the statc one-perod prcng scheme (See Secton IV) C Basc Economc Concepts In ths secton we brefly descrbe some basc economc concepts relevant to the paper Addtonal detals can be found n a stard economcs text such as [8] externalty: An externalty s an effect (postve or negatve) of a purchase of self-defense nvestments by a set of users (ndvduals or organzatons) on other users whose nterests were not taken nto account whle makng the nvestments In ths work, the effects are mprovements n ndvdual securty of network users who are connected to the users nvestng n self-defense rsk probablty: It s the probablty of a network user beng successfully attacked by a cyber-threat ncurrng a loss of a partcular amount market: It s a regulated platform where cyber-nsurance products are traded wth nsurance clents, e, the network users A market may be perfectly compettve, olgopolstc, or monopolstc In a perfectly compettve market there exsts a large number of buyers (those nsured) sellers (nsurers) that are small relatve to the sze of the overall market The exact number of buyers sellers requred for a compettve market s not specfed, but a compettve market has enough buyers sellers that no one buyer or seller can exert any sgnfcant nfluence on premum prcng n the market On the contrary, n a monopolstc market, the sngle nsurer has the power to set clent premums to ts lkng An olgopolstc nsurance market s a specal type of a compettve market where multple nsurance frms exst n a manner so that each nsurer can set clent premums to ts lkng Symbol u ( ) N h x G x B( ) c p x k p k x 1:k 1 Meanng utlty of user (consumer) n consumer-seller model number of consumers of an SV externalty effect of user on user amount of self-defense good consumed by user matrx representng externalty values between user pars vector of self-defense amounts of users apart from Bonacch centralty vector of users n a logcal network constant margnal manufacturng cost to SV prce per unt of self-defense good consumed by self-defense good consumed by user n round k prce per unt of self-defense to user n round k vector of user s self-defense amounts up to round k 1 p 1:k 1 vector of user s prces up to round k 1 φ k sequental order n whch SV vst ts consumers n round k TABLE I LIST OF IMPORTANT SYMBOLS stakeholders: The stakeholders n a cyber-nsurance market refer to enttes whose nterests are affected by the dynamcs of market operaton In our work we assume that the enttes are the network users, a regulatory agency such as the government, securty vendors (also the cyber-nsurers) such as Symantec Mcrosoft When all stakeholders n a market are satsfed, t results n a market success market effcency: A cyber-nsurance market s called effcent f the socal welfare of all nsured network users s maxmzed at the market equlbrum The market s neffcent f t fals to acheve ths condton Here socal welfare refers to the sum of the net utltes of nsured network users after nvestng n self-defense /or cybernsurance At the maxmum socal welfare state, the moral hazard problem 3 n cyber-nsurance s allevated, e, network users adopt safe Internet browsng habts even after after gettng nsured, knowng that they would be covered by ther nsurers II SYSTEM SETUP AND MODEL In ths secton we propose our system model Frst, we qualtatvely descrbe the nsurance envronment under whch our proposed SV prcng mechansms could operate We then follow t up wth a descrpton of the SV prcng envronment Fnally, we defne our system model A lst of mportant symbols relevant to the paper s shown n Table I A Insurance Settng (Envronment) We consder a system of securty vendors exstng n a market offerng cyber-nsurance solutons to ther clents Each clent s locked 4 wth hs correspondng SV wth respect to usng securty products manufactured by the SV, e, he does not use products manufactured by any other frm for self-defense purposes A consumer (network user) of securty products may or may not buy cybernsurance, e, buyng cyber-nsurance s not made matory In general, a rsk-averse consumer would want to buy nsurance, whereas rsk-lovng or rsk-neutral users would not care that much about buyng nsurance A consumer buyng cyber-nsurance s provded a full coverage by hs SV on facng a loss s charged a premum whch s not necessarly far, e, the expected value of the loss Each SV also premum dscrmnates ts clents n the form of chargng 3 Moral hazard s a well known problem n nsurance lterature where an nsured could behave recklessly after gettng nsured, knowng that he could get coverage from hs nsurance company for the losses faced 4 A clent may lock-n wth a securty vendor due to varous aspects such as reputaton, servce qualty etc

3 fnes/rebates atop premums Here we assume that the SVs are located n a compettve market settng where each SV can afford to premum dscrmnate ts clents n a certan manner wthout the fear of losng consumer dem to other SVs Based on works n [6][7], premum dscrmnatng cyber-nsurance clents s one way to allevate the moral hazard problem, enables the nsured s to optmally nvest n self-defense nvestments, maxmzes the socal welfare of the nsured s n the network We assume that each cyber-nsurer n the market wants to maxmze socal welfare of ts clents as a regulatory constrant mposed upon t by a regulator such as the government, therefore premum dscrmnates ts clents Remark: Snce our focus n the paper s on SV prcng, the prcng step s a pre-cursor to the clent premum chargng step, we do not mathematcally model the nsurance aspects n the paper By the term pre-cursor we mean that the net premums (ncludng fnes rebates) charged to clents wll depend on the self-defense nvestment amounts of users whch n turn depend on the prces set by the SV on ts securty products However, t s mportant to state down the type of nsurance envronment/s where our proposed prcng mechansm would ft n B SV Prcng Envronment We consder SVs adoptng a product prcng mechansm that s based on the logcal network of ts consumers ther correspondng securty nvestments The purpose of an SV to prce products n ths way s to make addtonal profts to cover up for the costs of provdng nsurance to clents as well as make strctly postve profts at all tmes as an nsurer It has been stated n [6][7] that cyber-nsurance wth premum dscrmnaton my not lead to the nsurer makng strctly postve profts at all tmes An mportant queston that arses here s: why would an SV need to prce dfferently based on consumer logcal network hs securty nvestment amount when t could transfer a part of hs current profts to the nsurance busness always make strctly postve profts? The answer to ths queston has three parts: () a ratonal frm would not mnd fndng a way to make more profts than ther current scenaro, from prncples of basc economcs, prcng based on ncreased clent nformaton results n more profts [8], () the amount of securty nvestments of users results n unpad postve externaltes for other users n the logcal network these externaltes need to be accounted for n some manner to ensure a certan prce farness amongst consumers 5, () specfcally when an SV s the cyber-nsurer, our proposed prcng phlosophy would allow approprate contracts (wth fnes/rebates) to be hed out to consumers buyng nsurance For example, a consumer who generates a hgh amount of externalty would be prced less for SV products than a consumer generatng a low amount of externalty, as a result the latter mght end up payng a fne, whereas the fomer consumer mght get a rebate on hs premum C Defnng The Model We assume that each SV (seller) has a set of clents N (consumers) connected va a logcal network usng self-defense products manufactured by the SV Each consumer ɛ N has an utlty functon, u ( ), whch s gven as u (x, x, p ) = α x β x + x X h x p x, (1) where x s the amount 6 of non-negatve self-defense goods (manufactured by the SV) consumed by user, s the vector of x 5 A SV can get nvestment related nformaton from ts nsurance clents ( hence estmate externaltes) through dsclosure agreements sgned between the clent the SV as part of some mate mposed by the government [3] 6 We assume a contnuous amount varable for purposes of analyss In realty SV products are bundled n a package whch s prced as a sngle tem nvestments of users other than, p s the prce charged by the SV to user per unt of good consumed Here p s the equlbrum market prce set by the SV after competng wth other SVs n the securty product busness α, β, h are constants α β can be thought of as ndrect representatve parameters of advertsed cybernsurance contracts, whereas h s the amount of externalty user exerts on user through hs per unt nvestments Here h 0 h = 0, x s assumed to be contnuous for analyss tractablty reasons The frst second term n the utlty functon denotes the utlty to a user solely dependent on hs own nvestments, the thrd term s the postve externalty effects of nvestments made by other users n the network on user, the fourth term s the prce user pays for consumng x unts of self-defense goods manufactured by the SV We assume here that x s bounded The quadratc nature of the utlty functon allows for a tractable analyss a nce secondorder approxmaton of concave payoffs The SV accounts for the strategc nvestment behavor (after t would have set ts prces) of users t provdes servce to, decdes on an optmal prcng scheme that arses from the soluton to the followng optmzaton problem X max p p x cx, where p are the vectors of prces charged by the SV to ts consumers, x s the amount of self-defense good consumed by consumer after the SV sets ts prces c s the constant margnal cost to the SV to manufacture a unt of any of ts products III STATIC PRICING STRATEGY In ths secton we frst descrbe our two-stage prcng game between an SV ts consumers We then state the results of the prcng game A Two-Stage Prcng Game Defnton The game has the followng two steps 1) The SV chooses a prce vector p so as to maxmze ts profts va the followng optmzaton problem X max p p x cx, where p s prce vector charged by the SV to ts consumers, per unt of nvestment, x s the amount of self-defense good consumed (nvested) by user after the SV sets ts prces We consder three types of consumer prcng scenaros n the paper: () Scenaro 1 - here, the SV does not prce dscrmnate amongst ts consumers all elements of p are dentcal, e, p = p,, () Scenaro (bnary prcng) - here, the SV charges two types of prces per unt of user nvestment: a regular prce denoted as p reg for each user n a partcular category, a dscounted prce, denoted as p dsc for other users, () Scenaro 3 - here, the SV charges dfferent prces to dfferent consumers the elements of p are non-dentcal c s the constant margnal cost to the SV to manufacture a unt of any of ts products ) Consumer chooses to consume x unts of self-defense products, so as to maxmze hs utlty u (x,, x, p ) gven the prces chosen by the SV Snce the game conssts of two stages, we wll analyze the subgame perfect Nash equlbra of ths game, nstead of ust focussng on smple Nash equlbra 3

4 B Results In ths secton we state the results related to our statc prcng strategy We frst comment on the equlbrum of the second stage of the two-stage prcng game, gven a vector of prces p Gven p, the second stage of our prcng game s a subgame we denote t as G sub We the have the followng theorem The proof of the theorem s n the Appendx Theorem 1 G sub has a unque Nash equlbrum s represented n closed form as x = BR( α p x ) = + 1 X h x, () β β where BR( x ) s the best response of user when other users n the network consume x In the case when SV does not prce dscrmnate ts consumers, the Nash equlbrum vector of user nvestments s gven by ɛn x = (Q G) 1 ( α p 1 ), (3) where p s the optmal per unt nvestment prce charged by the SV to all ts consumers, matrx Q takes values β at locaton (, ) f = zero otherwse Theorem Intuton Implcatons: The ntuton behnd a unque Nash equlbrum s the fact that ncreasng one s consumpton ncurs a postve externalty on hs peers, whch further mples that the game nvolves strategc complementartes 7 [8], therefore the equlbra are ordered Ths monotonc orderng results n a unque NE The unque equlbrum mples that the SV ust needs to be concerned about the sngle equlbrum vector of user consumpton amounts base ts optmal strategy on that equlbrum vector If there would be multple equlbra to G sub, t would be cumbersome for the SV to decde on ts optmal strategy Why? because t would be dffcult for non-cooperatve users to decde n the frst place whch equlbrum s the best then ontly play the best equlbra As a result the SV mght not be sure that the best equlbrum would be played by the users However, f the SV s able to compute the best equlbra, t could base ts prcng strategy on that one rrespectve of the equlbra played by the users We now dscuss the optmal prcng strategy for the SV gven that the users self-protect accordng to the Nash equlbrum of G sub Before gong nto the detals we frst defne the concept of a Bonacch centralty n a network of heterogenous users The Bonacch centralty measure [4] s a socologcal graph-theoretc measure of network nfluence It assgns relatve nfluence scores to all nodes n the network based on the concept that connectons to hgh-scorng nodes contrbute more to the score of the node n queston than equal connectons to low-scorng nodes In our work, the Bonacch measure of a user reflects hs nfluence on other users of the network va the externaltes generated by hm through hs self-defense nvestments Formally, let G be a matrx defnng the logcal network of N users (consumers), havng n ts entres the h values Let D be a dagonal matrx, w be a weght vector The weghted Bonacch centralty vector s gven by B(G, D, w ) = (I GD) 1 w, (4) where (I GD) 1 s well-defned non-negatve We now have our frst result regardng the optmal prces charged by the SV to ts consumers The result addresses prcng scenaros 1 3 together as Scenaro 1 s a specal case of Scenaro 3 The proof of theorem s n the Appendx 7 In economcs game theory, the decsons of two or more players are called strategc complements f they mutually renforce one another Theorem The optmal prce vector p charged by the SV s gven by p = α + c 1 +GQ 1 B(G, Q 1, w ) G T Q 1 B(G, Q 1, w ), (5) w = α c 1 where G = G+GT In the case when the SV does not prce dscrmnate ts consumers, the optmal prce (same for every consumer) charged per consumer s gven by p = 1 1 T (Q G) 1 ( α + c 1 ) 1 T (Q G) 1 1 Theorem Intuton Implcatons: The optmal prce vector n the no prce dscrmnaton case s ndependent of ndvdual node centraltes, whereas n the prce dscrmnaton case the optmal prce vector depends on the Bonacch centralty of ndvdual users The ntuton behnd the result s the fact that users tend to nvest n securty mechansms proportonal to ther Bonacch centralty ( n turn generate proportonal amount of network externaltes) n the Nash Equlbrum [11][1] Therefore t makes sense for the SV to charge users based on ther Bonacch centraltes when prce dscrmnaton s possble Proft Rato Rato Upper Bound Rato Obtaned Rato Lower Bound Influence values Fg 1 Proft Rato Its Bounds We now state the followng result regardng proft amounts made by a SV from ts cyber-nsurance busness for prcng scenaros 1 3 The proof of the theorem s n the Appendx Theorem 3 The profts made by an SV from ts cyber-nsurance busness when the latter does not (does) account for user nvestment externaltes are gven by 8 <! T! 9 = P 0 = (Q G) 1 : ; (7) 8 <! T! 9 = P 1 = (Q G ) 1 : ;, (8) We plot the rato of P 0 P 1 for preferental attachment (PA) graphs n Fgure 1 We choose preferental attachment graphs as they represent socal/logcal network nteractons For networks of sze 100, we generate 50 PA graphs for each dfferent value of µ rangng from 0 to 1, where µ reflects the nfluencng nature of a user n a PA graph wrt the postve externalty effects of hs securty nvestments made A µ value of 1 ndcates that the user s nfluenced by all hs neghbors but does not nfluence any one of them, whereas a µ value of zero ndcates the opposte Each pont n the plot s the average of the 50 P 0 P 1 values obtaned per value of µ Our plot results show (6) 4

5 that the provable proft rato bounds are qute tght are less than 1 8, mplyng the fact that an SV does not do that badly n terms of proft when t s not nformed about consumer externalty values ther network locaton propertes, compared to the case when t does have full nformaton Theorem Intuton Implcatons: As observed from the plot n Fgure 1, the profts to the SV are greater when t accounts for externaltes than when t does not, an SV could make up to 5% extra profts wth complete nformaton Ths s ntutve n the sense that the SV has more user nformaton when knowng about the externaltes can prce optmally to ncrease ts profts However, n realty t s dffcult to measure/observe the externaltes Thus, n spte of gettng topologcal nformaton from the nsurer, an SV mght have to prce ts products wthout takng externaltes nto account The profts for the non prce dscrmnaton scenaro s encapsulated as a specal case of P 1 when G has all entres equal except the zero dagonal entres We also emphasze here that even n the absence of complete nformaton, partal nformaton wll boost SV profts n a proportonal manner General Remarks: In practce, t s really dffcult to compute the h values One can at best approxmate or stochastcally estmate t The Bonacch centralty measure can however be exactly computed gven the socal network structure of consumers As mentoned before, even partal nformaton on consumer nvestments related externaltes wll mprove proft margns for SVs In the case a logcal network conssts of dsont large subnetworks or sparsely overlappng networks, nsurance sellers (SVs) would segment the market at equlbrum exercse monopoly prcng power n ther respectve localtes When networks are consderably overlappng t would be dffcult for any SV to exercse monopoly prcng of ts products However, n realty there s some heterogenety between product types of sellers whch leads to one beng more popular than others, as a result sellers could have a slght prcng power over ther consumers The case for two prces (bnary prcng): In realty, chargng multple dfferent prces to varous consumers may not be very practcal to mplement To make thngs smpler, a SV can opt to charge two types of prces for two dfferent classes of consumers: () a dscounted prce, p dsc, for consumers who have sgnfcant postve nfluence on the securty of a network based on ther network locaton the amount of nvestments made, () a regular prce, p reg for the other consumers Thus, the frst goal of an SV s to determne the subset of consumers that should be offered the dscounted prce so as to maxmze ts own profts Gven that p reg p dsc are exogenously specfed, the proft optmzaton problem for an SV s gven by Maxmze ( p c 1 ) T (Q G) 1 ( α p ) st p ɛ {p reg, p dsc }, ɛ N Note here that the expresson, (Q G) 1 ( α p ), n the obectve functon s the NE nvestment amount of users n self-defense mechansms Thus, we have a combnatoral optmzaton problem for maxmzng the profts of an SV In order to nvestgate the tractablty of the problem, we formulate t n the followng manner: OP T : Maxmze (δ y + c 1 ) T (Q G) 1 ( α δ y ) st y ɛ { 1, 1}, ɛ N Here δ = p reg p T, where p T = preg+p dsc, a = a p T, c = p T c δ Note that usng these varables, the feasble prce allocaton can be expressed as p = δ y + p T Our next result comments on the ntractablty of solvng OPT The proof of the 8 Here 1 s the trval upper bound result whch s n the Appendx, s based on the reducton of OPT from the well-known MAX-CUT problem [5] Theorem 4 Gven that p reg p dsc are exogenously specfed n the bnary prcng case, an SV s proft optmzaton problem, OPT, s NP-Hard We plan to desgn approxmaton schemes to the optmal soluton as part of future work IV DYNAMIC PRICING STRATEGY In the prevous secton, we analyzed the case when there s ust one round of prcng game played between the consumers the SV In ths secton we propose a sequental dynamc prcng strategy for a securty vendor, where the consumers the SV nteract n multple rounds The motvaton for consderng multple rounds s that securty products often have dfferent versons comng out at perodc tme ntervals, gven our framework, the consumers would want take nto account past actons of themselves others n the network, e, nvestment amounts, modfy ther actons n the present Lkewse the SV would want to optmally prce ts product accountng for ts consumers present behavors Our goal s to study whether playng multple rounds of the two-stage prcng game results n more profts for an SV A Prcng Model We assume that the SV adopts a dscrete-tme sequental prcng strategy where the prce vector s changed once every round The SV approaches each consumer n a partcular order per round Formally, let φ k := {φ k 1,, φ k n= N } be the adopted order of approachng consumers n round k for the SV Here φ k ɛ N, φ k φ k for any Let p k be prce charged by the SV to consumer n round k, x k be the securty good consumpton by user (consumer) n round k Regardng a consumer s nformaton on other consumer nvestments pror to round k, we smplstcally assume he has accurate knowledge of the quanttes consumed by other users n the network pror to hs own purchase n round k Let I k be the nformaton avalable to user (consumer) n round k Then we could mathematcally represent t as I k = x (1:k 1),, X, x k ɛ N k p (1:k 1) where N k N s the set of consumers that purchase self-defense mechansms before consumer n round k, x (1:k 1) s the k 1 dmensonal vector of nvestments of consumer from rounds 1 to k 1, p (1:k 1) s the k 1 dmensonal vector of per unt of nvestment prces charged to consumer from rounds 1 to k 1, X s a - dmensonal matrx storng the nvestments of each of N users from rounds 1 to k 1, x k s the nvestment made by consumer n round k Let y k = P k =1 xk be the cumulatve self-defense quantty consumed by user up to round k The utlty functon of consumer n round k s then gven by where «, u (x k, I k, p k ) = A B + C + D E, (9) C = D = A = α y k 1 + x k, B = β y k 1 + x k, y k 1 + x k X ɛn X y k 1 + x k ɛn k h y k 1, h x k 1, 5

6 E = kx p x, =1 The terms A, B, C, D, E have the same nterpretaton as the correspondng terms of Equaton 1, except that now we are consderng cumulatve nvestments up to round k, nstead of ust nvestments n one round, as that n Equaton 1 For each round k, consumer wll look to maxmze u (x k, I k, p k ), whereas the SV wll choose {p k } to maxmze P ɛ N pk x k cx k We thus have the followng optmzaton problem, the soluton to whch gves the optmal orderng that maxmzes the SV profts n round k argmax k ɛ P erm(φ k )P k ( k ) where P erm(φ k ) s the set of all permutatons of the elements of φ k, e, the set of all possble orderngs of consumers n round k, k := {1 k,, n} k be the adopted optmal order of approachng consumers n round k that maxmzes the obectve functon Note that P k ( ) sts for the optmal soluton to the SV profts n round k gven any partcular order It s the soluton to another optmzaton problem n tself, where we are requred to fnd the optmal user nvestment amounts that maxmze SV profts gven any partcular order, φ k We formulate the optmzaton problem as follows: where Ĩk = st x k φ k P k (φ k ) = max x (1:k 1) nx =1 p k φ k x k φ k cx k = argmax z 0 u φ k z, Ĩk φ k, p k φ k «, p (1:k 1), X x, k ɛ N k,, s the nformaton avalable to consumer φ k n round k x k p k are the optmal consumed amount of securty good prce of consumer n round k respectvely B Results Note that n order to solve the proft maxmzaton problem for the SV, n each round we need to consder n! possble orderngs to fnd the optmal order, whch makes the state space sze ntractable for analyss For tractablty of analyss, we assume that the h values are symmetrc c = 0 We now state two results (theorems) on the outcome of the prcng game played for multple rounds The proofs of these theorems are n the Appendx Theorem 5 The optmal soluton to x k (φ k ) the correspondng P k (φ k ) value are unque, ndependent of the orderng n φ k The closed form of the optmal soluton for x k (φ k ) n each round k s gven by x k (φ k ) = [Q G] 1 [I (Q G)(Q G) 1 ] k 1 α, (10) where matrx Q takes values β at locaton (, ) f = zero otherwse The optmal per unt consumpton prces charged by the SV to each consumer n round k s gven by p k = α β ỹ k 1 + X ɛ N h ỹ k 1 β x k + X < h x k (11) Theorem Intuton Implcatons: The ndependence of the optmal consumer nvestments wth respect to consumer orderng follows from the fact that h values are symmetrc We also observe that the optmal consumpton amounts the prces charged by the SV do not depend on the centralty measures Ths s agan due to the fact the h values are symmetrc amongst users (consumers) Externalty effects rely on the topologcal structure - thus t s evdent that wth symmetrc externalty effects, topology does not affect the optmal consumpton prce structure However, n realty there wll be asymmetrc externalty effects, we conecture that the optmal consumer consumpton SV prces wll depend on the locaton of network users We now have the followng theorem related to the comparson of SV profts between the statc prcng strategy the dynamc prcng strategy cases Theorem 6 The proft obtaned by an SV va our proposed dynamc prcng strategy strctly domnates that obtaned due to the statc prcng strategy (even f the strategy prces unformly for all consumers) Theorem Intuton Implcatons: The theorem follows from the fact that an SV has more nformaton from consumer nvestment hstory prces ts consumers accordngly to ncrease profts In ths work we have assumed symmetrc externalty effects n the dynamc prcng case, whch s equvalent to havng partal nformaton about consumer nvestments We observe that n the absence of complete nformaton on consumer nvestment hstory, an SV can make mproved profts wth partal nformaton when compared to the statc prcng case Wth more nformaton on consumer nvestments, SVs can only ncrease ther profts, accordngly allocate some profts to ts cyber-nsurance busness C Consumer Farness Through Dynamc Prcng We have already shown n the dynamc prcng settng above that the order n whch a consumer s approached by a SV does not have an mpact on hs optmal prcng strategy, under symmetrc externalty effects In ths case we can explore ths nvarance of orderng property to ensure farness amongst the network users By the term farness we mean dstrbutng the net utlty of all consumers n a network n a far manner over tme Farness s an mportant parameter from a consumer perspectve An SV could ncrease ts reputaton consumer popularty ( n turn sales) by ensurng farness amongst ts network users In ths paper we consder the max-mn farness allocaton concept (not relevant for unform consumer prcng) The goal of the SV s to choose the consumer vst order n each round so as to acheve the max-mn farness goal over tme We emphasze agan that rrespectve of the vst order, the SV keeps ts optmal profts fxed Consder the dynamc prcng strategy where the SV approaches ts consumers n a same fxed order every round Assume the order to be {1,,, n = N } The prce n round k, charged to a consumer, he beng the mth person vsted by the SV, s gven by where a s gven by p k = α(β (n m)h) ak 1, ɛ N, (1) 4β (n 1)h a = β < 1 (13) 4β (n 1)h Note that n case of a symmetrc graph α = α, β = β It s evdent from the equaton that prces charged to consumers ncrease lnearly n the order of vst Ths generates a wde dsparty n the ndvdual user utlty wth the nth vsted consumer havng the least utlty n all rounds Thus we do not get a far net utlty allocaton amongst the consumers over tme To allevate the farness ssue, we make the followng smple change n the dynamc prcng protocol: the SV chooses any arbtrary orderng (assume {1,,, n}) n the frst round, but uses a maxmn farness crtera to fnd the order n subsequent rounds For a symmetrc graph, for each k > 1, we choose the orderng φ k = {φ k 1,, φ k n} such that the followng holds φ k = argmn ɛ N {φ k 1,,φ k 1 }u(xk, I k, p k ) 6

7 We then obtan the net utlty of a consumer vsted n the mth poston n any round k as «u m(x k m, Im, k p k m) = a α (β mh)a (β (n m)h) + 4β 1 a (14) Ths equaton mples that any other order n round k > 1 n whch consumer n s not scheduled n the begnnng wll lead to hm havng a lower net utlty compared to the orderngs n whch he s scheduled n the begnnng Thus, we acheve a max-mn farness over tme, n regard to allocatng net utlty of consumers V CONCLUSION AND FUTURE WORK Cyber-nsurance markets, be t monopolstc, compettve, or olgopolstc mght not turn out to be very proftable for cybernsurer/s Ths s more so the case because these markets are applcable to nterdependent correlated rsk envronments In such envronments, nsurers n the worst case could go bankrupt, generally mght make zero or non-negatve expected profts even under clent contract dscrmnaton, n the process of sustanng a certan level of socal welfare requrement from a regulatory agency such as the government To allevate ths problem a securty vendor can be a part of the cyber-nsurance drven securty ecosystem by beng the nsurer tself channelng the extra profts obtaned from ts securty product busness to ts cyber-nsurance busness Strct postve profts at all tmes for a cyber-nsurer wll satsfy ts nterest n an nsurance-drven securty ecosystem result n more than moderate success of cyber-nsurance markets The channelng process can be made possble by SVs by gettng clent nformaton on ther logcal network locaton as well as the amount of externaltes caused due to ther securty nvestments Accordng to basc mcroeconomc theory, prcng technques based on such addtonal clent nformaton (be t perfect or mperfect) generates extra profts for SVs compared to ther tradtonal prcng methods In the process, an SV could also ensure a lock-n effect amongst ts nsurance clents by enforcng the latter to buy securty products only from hs SV, n turn ncrease SV dem for securty products In ths paper we have shown that () prce dscrmnatng consumers n proporton to the Bonacch centralty of ndvdual users results n maxmum profts for an SV, () an SV could make up to 5% addtonal profts wth perfect clent nformaton, () the problem of prce dscrmnatng consumers n order to maxmze SV profts when there are only two prce categores, e, regular dscounted, s NP-Hard We have also showed that a prcng game played between consumers an SV for multple rounds can result n greater maxmum profts for both, the SV an nsurer when compared to the outcome of the prcng game played for ust one round Fnally, for the dynamc prcng game, we showed that for a gven optmal prcng strategy that s nvarant of the order n whch the SV vsts ts consumers, t s possble for the SV to allocate the latters net utlty n a maxmn far manner over tme by changng the order of vsts n each round Mantanng the same order of consumer vsts would result n optmal profts for the SV but would not be far to certan users who would always (n every round) be vsted near the end accrue lower utltes than ther counterparts who were vsted earler As part of future work we plan to desgn an approxmaton algorthm that provdes a nce approxmaton guarantee to the bnary prcng problem Note that n ths paper, we assumed the bnary prces p reg p dsc to be exogenously specfed for the bnary prcng problem In ths regard, we also am to compute the optmal bnary prces that maxmzes SV profts when they are not exogenously specfed We also am to extend our dynamc prcng strategy to account for asymmetrc externalty effects of consumer nvestments VI ACKNOWLEDGEMENTS We would lke to express our sncere thanks to Darren Shou hs research group at Symantec Research for ther nsghtful comments REFERENCES [1] G A Akerlof The market for lemons - qualty uncertanty the market mechansm Quarterly Journal of Economcs, 84(3), 1970 [] R Anderson T Moore Informaton securty economcs beyond In Informaton Securty Summt, 008 [3] R Bohme G Schwartz Modelng cyber-nsurance: Towards a unfyng framework In WEIS, 010 [4] P B Bonacch Power centralty: A famly of measures Amercan Journal of Socology, 9, 1987 [5] M R Garey D S Johnson Computers Intractablty: A Gude to The Theory of NP-Completeness WH Freeman Company, 1979 [6] A Hoffman Internalzng externaltes of loss preventon through nsurance monopoly Geneva Rsk Insurance Revew, 3, 007 [7] M Lelarge J Bolot Economc ncentves to ncrease securty n the nternet: The case for nsurance In IEEE INFOCOM, 009 [8] A Mascollel, M D Wnston, J R Green Mcroeconomc Theory Oxford Unversty Press, 1985 [9] C D Meyer Matrx Analyss Lnear Algebra SIAM Press, 000 [10] NShetty, GSchwarz, MFeleghyaz, JWalr Compettve cybernsurance nternet securty In WEIS, 009 [11] R Pal P Hu Cyber-nsurance for cyber-securty: A topologcal take on modulatng nsurance premums In Performance Evaluaton Revew, 01 [1] R Pal P Hu On dfferentatng cyber-nsurance contracts: A topologcal perspectve In IEEE/IFIP Network Management, 013 [13] D M Topks Supermodularty Complementarty Prnceton Unversty Press, 1998 VII APPENDIX In ths secton, we prove Theorems 1-6 Proof of Theorem 1 The proof reles on the results on the followng lemmas We frst state prove the relevant lemmas requred for the proof of Theorem 1 follow t up wth the proof of the man theorem Lemma 1 The game G sub s supermodular 9 Proof The payoff/utlty functons are contnuous, the strategy sets are compact subsets of real space, for any two consumers u x x, ɛn, 0 Hence G sub s supermodular Lemma The spectral radus of Q 1 G s smaller than 1, the matrx I Q 1 G s nvertble Proof Let v be an egenvector of Q 1 G wth λ beng the correspondng egenvalue, wth v > v for all ɛ N We have the followng equaton due to the fact that (Q 1 G) v = λ v λv = (Q 1 G ) v X (Q 1 G) v 1 v X h < v β ɛ N ɛ N (15) Here (Q 1 G) denotes the th row of (Q 1 G) Snce the equaton holds for any egenvalue-egenvector par, the spectral radus of (Q 1 G) s strctly smaller than 1 Now observe that each egenvalue of I Q 1 G can be wrtten as 1 λ Snce the spectral radus of Q 1 G s strctly smaller than 1, none of the egenvalues of I Q 1 G s zero, thus the matrx s nvertble Now we contnue wth the proof of Theorem 1 Snce G sub s a supermodular game, the equlbrum set has a mnmum a maxmum element [13] Let x denote the maxmum of the equlbrum set let S be such that x > 0 only f ɛ S If S = φ there cannot be another equlbrum pont, snce x = 0 s the maxmum of the equlbrum set Assume for a contradctory purpose that S φ there s another equlbrum x, of the game By the supermodularty property of G sub, x x, ɛ N Allow k to equal argmax ɛ N x x Snce x x are not equal, we have 9 In supermodular games, the margnal utlty of ncreasng a player s strategy ncreases wth the ncreases n the other players strateges 7

8 x k x k > 0 Snce at NE no consumer has an ncentve to ncrease or decrease hs consumpton, we have x k x k 1 G k ( x x ) = 1 X h k (x x ), (16) β k β k where G k s the k th row of G But we have 1 X h k (x x ) x k x k X h k < x k x k (17) β k β k Thus, we reach a contradcton G sub has a unque Nash equlbrum Proof of Theorem We have from Lemma that Q G s non-sngular as a result the followng equaton holds p = α (Q G) Q G GT G (18) Equaton 18 can we rewrtten as p = α I GT G (Q G) 1 By the matrx nverson lemma [9], we have I GT G (Q G) 1 = I+ GT G Thus, from Equaton 19 t follows that p = α + c 1 GT G Q GT + G Q GT + G α c 1 (19) (0) (1) Applyng Equaton 1 usng the defnton of weghted Bonacch centralty, we get α + c 1 p = +GQ 1 B(G, Q 1, w ) G T Q 1 B(G, Q 1, w ) thus prove Theorem Proof of Theorem 3 The optmal prce vector of the SV wthout wth the consderaton of externalty effects are gven by the followng equatons α + c 1 p0 = () p1 = α (Q G) Q G + GT (3) The correspondng consumpton vectors are gven by x 0 = (Q G) 1 α c 1 (4) x 1 = (Q G ) 1 α c 1 (5) It then follows that P 0 = ( p 0 c 1 ) T x 0 = (Q G) 1 α c 1 (6) P 1 = ( p 1 c 1 ) T x 1, (7) P 1 can be re-wrtten as P 1 = X Y, where X = Y =! T «R + R T 1!,! T «R + R T 1! Thus, we have 8 <! T P 1 = (Q G ) 1 : we prove our theorem! 9 = ;, Proof of Theorem 4 The well known MAX-CUT problem [5] s as follows: max X W (1 x x ) (,) ɛ E st x ɛ { 1, +1}, ɛ V, where W denotes a matrx of bnary weghts consstng of 0s 1s The soluton to ths problem corresponds to a cut as follows: let S be the agents who were assgned value 1 n the optmal soluton Then t s straghtforward to see that the value of the obectve functon corresponds to the sze of the cut defned by S V S The problem can also be re-wrten as P 0 : mn x T W x st x ɛ { 1, +1}, ɛ V Now consder the followng related problem: P 1 : mn x T W x st x ɛ { 1, +1}, ɛ V, where W s a symmetrc matrx wth ratonal entres that satsfy 0 < W T = W < 1 In the proof of Theorem 4, we wll frst show that P 1 s NP-Hard by reducng from MAX-CUT We wll then reduce P1 to OPT to clam the correctness of our theorem Lemma 3 P1 s NP-Hard Proof We prove our clam by reducng P1 from P0 Let W be the weght matrx n an nstance of P0 Let W ɛ = 1 (ɛ + W ), where ɛ s a ratonal number between 0 1 V = n Observe that n for any feasble x n P0 or P1, t follows that x T W ɛ( x ) nɛ x T W x x T W ɛ( x ) + nɛ Because the obectve of P0 s always an nteger n ɛ < 1, the cost of P0 for any feasble vector x can be obtaned from the cost of P1 by scalng roundng Therefore, snce P0 s NP-Hard, t follows that P1 s also NP-Hard Havng proved P1 to be NP-Hard, we now prove Theorem 4 by reducng OPT from P1 We consder specal nstances of OPT where G = G T, c = 0, α = [α,, α], α = p reg + p dsc OPT can then be re-casted as OP T : mn x T (Q G) x st x ɛ { 1, +1}, ɛ N Now consder an nstance of P1 wth W > 0 Note that because x = 1, P1 s equvalent to mn x T (W + I) x st x ɛ { 1, +1}, ɛ V, 8

9 where we choose as an nteger such that > 4 max{ρ(w ), X, W, mn,w } ρ( ) s the spectral radus of ts argument The defnton of mples that the spectral radus of W s less than 1 Therefore we have (W + I) 1 = 1 1 W I (W ) W (W W «3 ) (8) Snce all entres of W W ( W ) are postve, the above equalty mples that the off-dagonal entres of (W + I) 1 are negatve Therefore (W + I) 1 = (Q G) for some dagonal matrx Q some G 0 Thus, t follows that ((Q G) 1 1 ) k = I W + W ««1 (9) k Snce W > 0, we have ((Q G) 1 ) k 1 From the defnton of t follows that The above nequalty mples that ((Q G) 1 ) k W 1 ) W! 1 X l=0 «!! l 1 4 W 1 ( ( P = 1 k 1! W ) ) 1 «> 0 (30) 3 Thus, P1 can be reduced to an nstance of OPT by defnng Q G accordng to (W + I) 1 = (Q G) Therefore t follows that OPT, hence OPT, are NP-Hard Proof of Theorem 5 Consder the frst round We fx order φ 1 = {φ 1 1,, φ 1 n} The proft of the SV s gven as P 1 (φ 1, x 1 ) = X p φ 1x φ 1 = X (α φ 1 β φ 1x φ 1+ X h φ 1,φ 1x φ 1)x φ 1 P Note that the sum 1 k=1 α T Q 1 [Q(Q G) 1 ] k 1 α converges < due to the fact that the spectral radus of Q(Q G) 1 ] k 1 α s (31) Let x less than 1 Thus, the optmal proft obtaned by the SV s gven as 1 be the optmal consumpton of user n round 1 Then for symmetrc h values, we have P D = 1 α T (Q G) 1 (I (Q G)(Q G) 1 ) 1 α (43) where α k = α β φk φ kỹk 1 + X φ k h φ k,φ kỹk 1 φ k Thus, the optmal consumpton amount of user n round k s gven by α k 4β xk + X < h φ k,φ k xk φ k + X h φ k,φ k xk = 0, (36) > whch s ndependent of order n φ Thus, we have x = (Q G) 1 [I (Q G)(Q G) 1 ] k 1 α (37) The total consumpton as the number of rounds goes to nfnty s gven as lm ỹk X = (Q G) 1 [I (Q G)(Q G) 1 ] k 1 α (38) k k=1 Smplfyng, we get Thus, we have proved Theorem 5 lm ỹk = (Q G) 1 α (39) k Proof of Theorem 6 The asymptotc proft, P D, obtaned by the SV va ts dynamc prcng strategy s expressed as the followng sequence of equaton X P D = P k (φ k, x k ); (40) P D = 1 k=1 X α T [(Q G) 1 Q] k 1 (Q G) 1 [Q(Q G) 1 ] k 1 α ; k=1 P D = 1 (41) X α T Q 1 [Q(Q G) 1 ] k 1 α (4) k=1 α 4β x 1 + X h x 1 = 0, (3) whch s ndependent of φ Thus x 1 = (Q G) 1 α The optmal proft for the SV n round 1 s gven by P 1 (φ 1, x 1 ) = 1 α T (Q G) 1 α (33) The results for rounds to k can be proved nductvely We have the followng equatons for a partcular round k P k (φ k, x k ) = X Φx φ k (34) The optmal proft, P S, obtaned by the SV n the statc prcng case s gven as P S = 1 4 α T (Q G) 1 α (44) Thus, the dfference n profts of the SV between the dynamc prcng the statc prcng cases s P D P S, s gven by P D P S = 1 α T 3 4 Q G α (45) Snce the entres n ( 3 Q G) are non-negatve, we have PD PS > 0 Thus we have proved Theorem 6 where Φ = (α φ k β φ kỹ k 1 + X φ k h φ k,φ kỹk 1 β φ k φ kx k φ + X h φ k,φ kx φ k) < Now snce the spectral radus of matrx [I (Q G)(Q G) 1 ] k 1 α ] s less than 1, we have α k = [I (Q G)(Q G) 1 ] k 1 α, (35) 9