Fraud, Investments and Liability Regimes in Payment. Platforms

Fraud, Invstmnts and Liability Rgims in Paymnt Platforms Anna Crti and Mariann Vrdir y ptmbr 25, 2011 Abstract In this papr, w discuss how fraud liability rgims impact th pric structur that is chosn by a monopolistic paymnt platform, in a stting whr mrchants can invst in fraud dtction tchnologis. W show that liability allocation ruls distort th pric structur chargd by platforms or banks to consumrs and mrchants with rspct to a cas whr such a rsponsibility rgim is not implmntd. W dtrmin th allocation of fraud losss btwn th paymnt platform and th mrchants that maximiss th platform s pro t and w compar it to th allocation that maximiss social wlfar. JEL Cods G21, L31, L42. Kywords Paymnt card systms, intrchang fs, two-sidd markts, fraud, liability. Univrsité Paris Oust Nantrr and Ecol Polytchniqu. Economix, Bâtimnt G, burau 604, 200 avnu d la Républiqu, 92001 Nantrr Cdx, Franc; E-mail acrtibttoni@u-paris10.fr. y Univrsité Paris Oust Nantrr, Economix, Bâtimnt T, burau 234, 200 avnu d la Républiqu, 92001 Nantrr Cdx, Franc; E-mail mariann.vrdir@u-paris10.fr. 1

1 Introduction Th dvlopmnt of lctronic data xchang in th banking industry has gnratd an incras in fraud and cybrcrim. For instanc, in th Unitd-tats, according to th Consumr ntinl Ntwork (CN), 1.2 million complaints of consumr fraud hav bn rcordd in 2008. 1 As a consqunc, banks can mak substantial losss bcaus of fraudulnt us of paymnt cards, which di r across countris and paymnt systms ( tabl 1). Tabl 1 Loss rat pr $100 paymnt card transaction valu in svral countris 2 Country pain Australia Franc UK U Losss rat 2.24c/ 2.39c/ 5c/ 9.12c/ 9.2c/ Minimizing th occurrnc of fraud in lctronic paymnt systms rquirs costly orts from all th participants to a transaction platforms, banks, consumrs and mrchants. 3 For instanc, consumrs hav to protct thir prsonal data and to rport th fraud rapidly onc it occurs, whras platforms, banks and mrchants may invst substantial amounts in fraud dtction tchnologis. 4 Ths orts in fraud prvntion dpnd on th xpctd amount of losss and thir allocation, which rsponds to svral liability ruls, dtrmind ithr by public laws or by privat ntwork ruls. Currntly, in most paymnt card systms, consumrs hardly bar maningful liability for fraudulnt us of thir paymnt card, bcaus thy ar protctd both by nancial rgulations, which ar public laws (.g. TILA and rgulation Z in th Unitd- tats) 5, and by th zro liability rul, which has bn privatly adoptd by svral paymnt ntworks. It follows that, in most paymnt systms, th burdn of fraud losss is shard btwn banks or platforms and mrchants. 6 Th allocation of liability btwn banks and mrchants 1 ourc Consumr ntinl Ntwork Data Book for January-Dcmbr 2008, Fdral Trad Commission, Fbruary 2009. This rport highlights that crdit card fraud is th most common form of rportd idntity thft amounting at 20% of th rportd fraudulnt transactions. 2 ourc Richard ullivan (2010), Fdral Rsrv Bank of Kansas City, Th Changing Natur of Paymnt Card Fraud Issus for Industry and Public Policy. 3 According to th Fdral Rsrv Board, in th Unitd-tats, "On avrag, by transaction typ, issurs incurrd 2.2c/ pr signatur-dbit transaction for fraud-prvntion and data-scurity activitis and 1.2c/ pr PINdbit transaction. imilarly, ntworks incurrd 0.7c/ pr signatur-dbit transaction for fraud-prvntion and data-scurity activitis and 0.6c/ pr PIN-dbit transaction. Finally, acquirrs incurrd 0.4c/ pr signatur-dbit transaction for fraud-prvntion and data-scurity activitis and 0.3c/ pr PIN-dbit transaction.". ourc Fdral Rgistr / Vol. 75, No. 248 / Tusday, Dcmbr 28, 2010 / Proposd Ruls. 4 According to a survy conductd by th Fdral Rsrv Board in th Unitd-tats, issurs ngag in various fraud-prvntion activitis such as "transaction monitoring and fraud risk scoring systms that may triggr an alrt or call to th cardholdr in ordr to con rm th lgitimacy of a transaction". "Mrchants also hav fraud-prvntion data-scurity costs, including costs rlatd to complianc with paymnt card industry datascurity standards (PCI-D) and othr tools to prvnt fraud, such as addrss vri cation srvics or intrnally dvloppd fraud scrning modls, particularly for card-not-prsnt transactions". 5 For a comparison of consumr protction laws across various countris, s Appndix A. 6 For instanc, in Franc, according to th "Obsrvatoir d la sécurité ds carts d paimnt", fraud losss hav bn shard in 2009 btwn banks (41.1%) and mrchants (53.5%). Mrchants hav bn hld liabl mainly for fraud on intrnt transactions. Consumrs wr hld liabl for only 2.3% of th fraud losss. According to 2

gnrally dpnds on privat ruls that ar chosn by paymnt platforms. om ntworks may vn us liability ruls to provid mrchants with incntivs to adopt nw tchnologis. For instanc, MastrCard and Visa usd liability shift masurs to induc mrchants to adopt fraud prvntion tchnologis on th intrnt (MastrCard curcod TM and Visa 3-D cur TM rspctivly). 7 Intrstingly, if th mrchant implmnts th 3-D cur TM tchnology, th issur bcoms liabl for fraud losss for all Commrc transactions that wnt through th 3-D cur TM procss. This papr adrsss two major issus rlatd to fraud in paymnt systms What is th incidnc of fraud liability rgims on th pric structur that is chargd by paymnt platforms? How do privat liability rgims di r from th socially optimal rgim that would b implmntd by a social plannr? In particular, w analys whthr privat ntwork ruls provid mrchants with su cint incntivs to invst in fraud dtction tchnologis and whthr ths ruls gnrat th socially optimal allocation of fraud losss. In our framwork, w us a broad d nition of fraud, which is th us of an lctronic paymnt instrumnt (or its information) by a prson othr than its ownr, to obtain goods and srvics without authority for such us. 8 W considr a monopolistic propritary paymnt platform that provids an lctronic paymnt instrumnt to consumrs and mrchants. Consumrs and mrchants dcid whthr or not to adopt th lctronic paymnt instrumnt basd on th pric of th paymnt instrumnt and on th xpctd loss that thy incur in cas of fraudulnt transaction. Fraudulnt transactions ar dtctd with som probability that is positivly rlatd to mrchants invstmnts in fraud prvntion tchnologis. If a fraud is dtctd, thn th participants do not mak losss. Our rsults highlight th following trad-o for th paymnt platform. Whn th lvl of liability for mrchants incrass, th numbr of mrchants who accpt th lctronic paymnt instrumnt falls, but mrchants tnd to invst mor in fraud dtction tchnologis, which incrass consumrs willingnss to us th lctronic paymnt mthod. Th paymnt platform trads o btwn incrasing th lvl of liability to minimiz th xpctd loss on fraudulnt transaction and maximizing th transaction volum by ncouraging mrchants and consumrs Furltti (2005), in th Unitd-tats, "consumrs of crdit cards ar shildd from narly $3 billion in fraud losss ach yar". According to a mor rcnt survy conductd by th Fdral Rsrv Board in th U, in 2009, across all typs of dbit card transactions, 57% of fraud losss wr born by issurs and 43% wr born by mrchants. ourc Fdral Rsrv Rgistr, vol. 75 n 248, 2010. 7 Ths srvics provid Intrnt mrchants with th ability to vrify thir consumrs tru idntitis through a scur, lctronic, non fac-to-fac authntication procss. 8 Our modl dos not nabl us to distinguish which typ of fraud is implmntd by th fraudstr. W considr any typ of fraud that can b mpdd by mrchant invstmnt. For instanc, data brachs and phishing do not dpnd on mrchants invstmnts (rathr on platform s invstmnt). On th contrary, idntity thft can b avoidd by th mrchant s ort to vrify th consumr s idntity. 3

to accpt th lctronic paymnt instrumnt. In th short trm, th xistnc of a fraud liability rgim a cts th pricing structur of th paymnts systm. With rspct to th standard pric structur in two-sidd markts (Rocht-Tirol, 2003), th pric structur that w obtain taks into account th platform s trad-o btwn maximising its pro t and minimizing th xpctd loss on fraudulnt transactions. If th zro liability rul for consumrs applis, th allocation of fraud losss that is chosn by th paymnt platform dos not plac nough liability on mrchants to maximis social wlfar. Thrfor, liability rgims can b usd by monopolistic paymnt platforms to xtract rnts from mrchants, as it nabls thm to charg highr prics. W also nd that this rsult dos not hold if invstmnts ar shard btwn th platform and th mrchants. W also dtrmin th incidnc of th liability rgim on th choic of th intrchang f. W nd that, if th issurs ar imprfctly comptitiv, whras th acquirrs ar prfctly comptitiv, th pro t maximising intrchang f dcrass with th lvl of liability that is born by mrchants. Th rst of th papr is organizd as follows. In ction 2, w summariz th litratur rlatd to our study. In ction 3, w dvlop a thortical modl to analyz th optimal allocation of fraud losss btwn th paymnt platform and th mrchants. In sction 4, w dtrmin th pro t maximising allocation of fraud losss. In sction 5, w study th wlfar maximising allocation of fraud losss. In sction 6, w analyz th rol of intrchang fs. In sction 7, w xtnd th modl by studying th optimal allocation of invstmnts btwn th paymnt platform and th mrchants. Finally, w conclud. 2 Rlatd Litratur To our knowldg, this papr is th rst attmpt to modl fraud dtction tchnologis and liability rgims in th litratur on paymnt systms. Our approach thus rlis on thr di rnt strands of litratur th litratur on paymnt platforms, on invstmnt in two-sidd markts, and nally th litratur on liability issus in law and conomics. Most paprs on paymnt systms focus on xplaining th divrgnc btwn th pro t maximising pric structur that is chargd by paymnt platforms and th pric structur that maximiss social wlfar (s Chakravorti (2010) for a rviw). In particular, svral paprs aim at dtrmining whthr paymnt platforms charg xcssiv intrchang fs whn thy maximis banks joint pro t (as survyd by Vrdir, 2011). Our papr contributs to this litratur by xtnding Rocht-Tirol (2003) to study how th allocation of th xpctd fraud loss btwn th platform and th mrchants changs th pro t-maximising pric structur. 4

Th litratur on invstmnt in two-sidd markts is scarc. For instanc, Vrdir (2010) dtrmins th optimal pric structur of a paymnt card platform in which monopolistic banks can invst to improv th quality of th paymnt srvic. Hr modl studis how invstmnts should b allocatd btwn monopolistic banks in four-party paymnt platforms. In particular, sh nds that a rduction of intrchang fs can improv th allocation of invstmnts by ncouraging acquirrs to invst, whn invstmnts incras consumrs dmand. Our modl dparts from that papr, as w considr a monopolistic propritary paymnt platform, and w focus on th optimal allocation of fraud losss btwn th platform and th mrchants. Th four-party modl is usd in sction 5 of our papr, whr w show that th pro t maximizing intrchang f dcrass with th lvl of liability born by mrchants. Th only papr that considrs mrchants invstmnts in two-sidd platforms is th papr by Pitz and Bll amm (2010), who study th ct of th intrmdiation mod (for-pro t compting platforms vrsus fr accss) on sllrs invstmnt, in a modl whr sllrs invstmnt incras th buyrs utility of blonging to th platform. Thy show that for-pro t intrmdiation may lad to ovrinvstmnt whn innovations incras buyrs surplus, bcaus compting intrmdiaris ract by lowring th accss fs on th sllr sid. Our focus is di rnt from thirs, as w tak th intrmdiation mod as givn, and focus on th impact of liability ruls on sllrs invstmnts incntivs. Our modl is also rlatd to th vast litratur on tort law whos main goal is to nhanc socially optimal dcisions on th lvl of prcaution (Brown, 1973). Mor prcisly, our framwork shars th sam background of x post liability rgims, whil nglcting th problm of non complianc and nforcmnt of x ant rgulation. 9 In this contxt, strict liability allocats th losss to th injurr by ntitling th victim to compnsation, whras no liability allocats th losss to th victim, by dnying th right to compnsation (Lands and Posnr, 1987). Indd, liability provids incntivs for prcaution. 10 W xtnd this argumnt to th cas of a thr party systm intrrlatd through ntwork cts, which is uncommon in law and conomics modls. In fact, in our framwork, th pric of a transaction implis not only a choic for a consumr, which gnrats a loss risk (as also pointd out by law scholars lik Cootr and Robin, 1987), but also a pricing stratgy by th platform and an incntiv for th mrchant to invst in fraud dtction. 9 Ex ant rgulation is mant to prvnt accidnts from occurring through th nforcmnt of minimum safty standards or complianc rstrictions. Ex post liability, xrcisd aftr an accidnt has occurrd, is a lgal dvic that nabls victims to su for damags, forcing injurrs to intrnaliz part of th harm thy caus. 10 Whn both partis hav to tak prcaution in ordr to avoid an accidnt, strict liability crats no incntivs for victim prcaution, whil no liability would shift th ntir rsidual liability on th victim, inducing optimal victim car. It follows that strict liability and no liability can giv incntivs to tak cint prcaution only to on party, rspctivly ithr th injurr or th victim (Dari-Mattiacci, Parisi, 2006). 5

3 Th modl W build a modl in which a monopolistic paymnt platform o rs an Elctronic Paymnt Instrumnt (hraftr th EPI) to consumrs and mrchants. W xtnd Rocht and Tirol (2003) along svral dimnsions. W considr that thr is an xognous probability that th EPI is fraudulntly usd, in a stting whr mrchants can invst in fraud dtction tchnologis. W d n fraud as th us of an lctronic paymnt instrumnt (or its information) by a prson othr than its ownr, to obtain goods and srvics without authority for such us. Th fraud ntails a lump sum loss which dos not dpnd on th transaction valu. Our framwork nabls us to dtrmin how fraud liability should b allocatd btwn th participants to maximis th platform s pro t. It also nabls us to compar th privat optimal allocation to th on that maximiss social wlfar. Paymnt systm and allocation of fraud A monopolistic paymnt platform provids an lctronic paymnt instrumnt (.g. th paymnt card) to consumrs and mrchants. Th marginal cost of procssing a transaction is dnotd by c. Consumrs and mrchants pay transaction fs to th platform, which ar dnotd by f and m rspctivly. Whn consumrs us th EPI, thr is an xognous probability x 2 (0; 1) that th paymnt instrumnt is intrcptd by fraudstrs. 11 Thr is also a probability q 2 [0; 1] that th fraud is dtctd, which dpnds on mrchants invstmnts. If th fraud is not dtctd, all th participants to th transaction mak an xognous loss that w dnot by L > 0. Th loss is allocatd btwn th consumr, th mrchant and th paymnt platform as follows th consumr (or buyr B) and th mrchant (or sllr ) bar rspctivly a shar B and of th loss, whr + B 2 [0; 1]. Th rst of th loss, P = 1 ( + B ), is born by th paymnt platform. W assum that th paramtr B is dtrmind by public laws and w considr it as xognous to th modl. In particular, if B = 0, th zro liability rul applis for consumrs. Th paramtr is privatly chosn by th paymnt platform. 12 W choos to normaliz th fraud on cash paymnts to zro. 13 11 Th assumption that x is xognous is mad for simplicity. Indd, ndognizing x would introduc anothr trad-o for th mrchant. Highr invstmnts in fraud dtction tchnologis hav two cts on hackrs incntivs to fraud. On th on hand, highr invstmnts in fraud incras th volum of transactions, which incrass th hackrs incntivs to commit fraud. On th othr hand, highr invstmnts incras th probability that a fraud is dtctd, which may discourag hackrs to commit fraud. 12 In our modl, w do not study how th losss ar allocatd btwn banks and th paymnt platform. In practic, paymnt platforms dsign ruls to allocat th losss btwn issuing and acquiring banks and also to allocat th losss btwn banks and th platform itslf. This issu would dsrv a sparat study. W rintroduc banks in sction 4 and choos to focus on th rol of intrchang fs in fraud prvntion issus. 13 Introducing th probability that a fraudulnt paymnt is mad by cash would not chang th trad-o s that w highlight in our modl. W would only hav to modify assumption (A2) to tak into account th losss that ar du to fraud on cash paymnts. 6

Mrchants W considr local monopolist mrchants that supply th sam good to consumrs. Th marginal cost of producing th good is dnotd by d and th pric of th good is dnotd by p. W assum that th non-discrimination rul holds, such that a mrchant cannot not charg a pric that dpnds on th paymnt mthod. Each mrchant can dcid whthr or not to accpt th lctronic paymnt instrumnt. If h dcids to accpt th EPI, th mrchant may invst an amount in fraud dtction tchnologis. Invstmnt to improv fraud dtction costs C ( ) to th mrchant, whr C ( ) is paid pr transaction, C 0 ( ) 0, C 00 ( ) 0 and C 000 ( ) 0. Mrchant s invstmnts incras th probability q that a fraudulnt transaction is dtctd, that is, w assum that dq=d > 0 for all > 0. W also assum that d 2 q=d 2 0 for all 0 and that d 3 q=d 3 0. 14 Th amount invstd in fraud dtction tchnologis is common knowldg, such that banks and consumrs ar awar of th scurity masurs implmntd by th mrchants. 15 By accpting th EPI, ach mrchant obtains a transaction bn t that w dnot by b, whr b > 0. As in Rocht and Tirol (2003), w assum that mrchants ar htrognous ovr thir transaction bn t b which is distributd ovr b ; b according to th probability dnsity h and th cumulativ H. W assum that h 0 0 to nsur dmand (quasi) concavity. W normaliz th bn t of accpting cash to zro. Th mrchant pays a f m to th paymnt platform ach tim a consumr pays with th EPI and bars th cost of invsting in fraud dtction tchnologis. W also assum that mrchants ar risk nutral. Consumrs Consumrs obtain a surplus v > 0 if thy buy th good that is supplid by th mrchants. Thy ar assumd to hold two paymnt instrumnts cash and th Elctronic Paymnt Instrumnt. Each consumr is randomly matchd to on mrchant and chooss btwn paying cash or paying with th EPI, if th mrchant accpts th EPI. W assum that consumrs ar risk nutral and that thy can obsrv mrchants invstmnt in fraud dtction tchnologis bfor dciding whthr or not to us th EPI. If h pays with th EPI, th consumr obtains a transaction bn t b B which is distributd ovr b B ; b B according to th probability dnsity hb and th cumulativ H B. W assum that h 0 B 0 for concavity to hold. Th consumr pays a f f to th paymnt platform, and anticipats that, with som probability x(1 q), h bars a shar B of th loss L, bcaus th EPI is fraudulntly usd without bing dtctd. 16 Th bn t of paying cash is normalizd 14 Th assumption that d 2 q=d 2 0 nsurs that th scond-ordr condition is vri d whn th mrchant chooss its lvl of invstmnt. Th assumption that d 3 q=d 3 0 nsurs that th scond-ordr condition is vri d whn th platform maximiss its pro t. 15 Mrchants can inform consumrs about thir orts to ght fraud. For instanc, onlin sllrs can communicat on th us of a softwar or a spci c tchnology that improvs consumr authntication. 16 W rul out th possibility that consumrs do not anticipat prfctly th dtction probability. Howvr, 7

to zro. It follows that, if a consumr can choos btwn cash and th EPI, undr th nondiscrimination rul, a consumr wishs to us th EPI if and only if b B f B x(1 q)l 0, (1) that is, if his transaction bn t is highr than th cost of th transaction f and th xpctd fraud loss. Additional assumptions h B (x) (A1) Th hazard rat is incrasing. 1 H B (x) 1 (A2) In quilibrium, min v d; h B(f + B (1 q)xl) xl B 1 H B (f + B (1 q)xl). Assumptions (A1) is similar to Wright (2002) and standard in th litratur on paymnt cards. Assumption (A2) nsurs that (i) consumrs obtain a much highr surplus from buying th good that from making a transaction with th Elctronic Paymnt Instrumnt. 17 (ii) th amount of th xpctd shar of th fraud loss for consumrs is not too high, such that it dos not xcd th surplus that consumrs obtain from making a transaction. 18 Timing Th timing of th gam is as follows 1. Th platform chooss th liability lvl and th transaction fs f and m. 2. Th mrchants dcid whthr or not to accpt th EPI and how much to invst in fraud dtction tchnologis. Thy also choos th pric of th good p. 3. Each consumr is matchd randomly to on mrchant. Consumrs dcid on whthr or not to buy th good and how to pay for th good (ithr by cash or with th EPI). In th following sction, w look for th subgam prfct quilibrium and solv th gam by backward induction. in practic, consumrs may ovrract to th risk of fraud. This can b studid in our framwork by doing som comparativ statics about B. 17 Part (i) of Assumption (A2) is standard in th litratur (s Wright (2002)). Formally, this corrsponds to th Assumption that v d h B(f + B(1 q)xl)=(1 H B(f + B(1 q)xl). 18 Part (ii) of Assumption (A2) is nw, as our papr is th rst to modl th incidnc of fraud losss on consumrs and mrchants paymnt choics and platform prics. Formally, this corrsponds to th Assumption that (1=xL B) h B(f + B(1 q)xl)=(1 H B(f + B(1 q)xl). 8

4 Th quilibrium 4.1 tag 3 consumr paymnt dcisions W start by dtrmining th probability that a consumr wishs to us th Elctronic Paymnt Instrumnt. Considr a consumr whos transaction bn t is b B 2 b B ; b B. This consumr is randomly matchd to on mrchant, who may or may not accpt th EPI. If th mrchant accpts th EPI, th consumr chooss his paymnt mthod by comparing his xpctd utility if h pays cash and if h pays with th EPI. Lt us start by th cas in which th mrchant dos not accpt th EPI. If th mrchant sts p v, th consumr wishs to buy th good by paying cash, as his surplus v p is positiv. Othrwis, h dos not buy th good. Now considr th cas in which th mrchant accpts th EPI. If th mrchant sts p v, th consumr wishs to buy th good, as h obtains at last a positiv surplus if h pays cash. H dcids to us th EPI if his xpctd utility is highr than if h pays cash. It follows that, if p v, a consumr wishs to us th EPI if and only if v p + b B f B (1 q)xl v p, that is, if and only if b B f B (1 q)xl 0 If th mrchant sts p > v, th consumr nvr uss cash. Th consumr buys th good and pays with th EPI if and only if v p + b B f B (1 q)xl 0. W dnot by D B th probability that a consumr wishs to us th EPI. Considring consumrs htrognity, it follows from th prvious analysis that 8 < 1 H B (f + B (1 q)xl) if p v D B = 1 H B (f + B (1 q)xl + p v) if p > v. Not that th probability that th consumr wishs to us th EPI dcrass with th transaction f, th consumr s liability, th xpctd amount of fraud loss, but incrass with th probability that th fraud is dtctd. 9

4.2 tag 2 EPI accptanc and invstmnts in fraud dtction 4.2.1 Prics and card accptanc condition W now dtrmin th pric that is chosn by ach mrchant, along with th dcision to accpt th EPI and invst in fraud dtction tchnologis. W start by showing that, bcaus of assumptions (A1) and (A2), th pro t of a mrchant who accpts th EPI is maximisd whn h sts a pric such that cash-usrs ar not xcludd from th markt. It follows that mrchants who accpt th EPI and mrchants who do not accpt th EPI choos th sam pric. This nabls us to driv th EPI accptanc condition. Lmma 1 Each monopolistic mrchant maximiss its pro t by stting p = v. Proof. Appndix B. W ar now abl to driv th condition undr which a mrchant accpts th lctronic paymnt instrumnt. A mrchant accpts th EPI if h maks mor pro t by doing so, that is if v d + D B (f + B (1 q)xl)(b x(1 q)l m C ( )) v d. inc D B (f + B (1 q)xl) 0, this condition is quivalnt to b x(1 q)l m C ( ) 0. (2) Not that a mrchant dos not accpt th EPI if th mrchant f is high or if th amount of th xpctd fraud loss is high. 4.2.2 Invstmnt in fraud dtction tchnologis A mrchant that accpts th EPI can invst in fraud dtction tchnologis. Th amount of invstmnt in fraud dtction tchnologis, which w dnot by, maximiss th mrchant s pro t undr th constraint that th mrchant accpts th EPI. Lmma 2 If th mrchant f is not too high, all mrchants such that b b ( ; B ; x; L; m; f) accpt th lctronic paymnt instrumnt, whr b ( ; B ; x; L; m; f) 2 b ; b. Th pro t maximising invstmnt for a mrchant who accpts th EPI solvs xl dq d C( 0 ) = [b x(1 q)l m C ( )] Bj ; (3) 10

whr B = dd B=d D B = ort. dnots th lasticity of th consumr s dmand to th invstmnt Proof. Appndix C. Th mrchant chooss its fraud prvntion ort so as to qualiz th marginal bn ts of invstmnts in fraud dtction tchnologis and th marginal cost of invstmnts. Th marginal bn ts of invstmnts ar qual to th marginal gains from lowr xpctd fraud losss (trm xl(dq=d ) in (3)), and to th marginal bn ts that ar du to an incras in th volum of lctronic transactions (trm [b x(1 q)l m C ( )] ( Bj = ) in (3)). Lt us dtail ach of th two cts that will b rfrrd to as th xpctd loss ct and th transaction volum ct. First, if th mrchant invsts in fraud dtction tchnologis, this incrass th probability that a fraudulnt transaction is dtctd, and thrfor, this rducs th amount of th xpctd loss that h has to bar whn h accpts th EPI. Th xpctd loss ct has a positiv impact on mrchant s invstmnts. cond, if consumrs bar a positiv shar of fraud losss, th probability that a consumr wishs to us th lctronic paymnt instrumnt is impactd positivly by th mrchant s invstmnts, as th xpctd loss dcrass. Th transaction volum ct has also a positiv impact on mrchant s invstmnts. Rmark that, bcaus of th transaction volum ct (if B 6= 0), th mrchants invst in fraud prvntion tchnologis vn if thy bar no liability for fraud, that is if = 0. In two-sidd markts, th liability rgim is not th only incntiv that can b usd to ncourag mrchant invstmnt, as mrchants car about th transaction volum, which is rlatd to consumr dmand. This ct is not prsnt in th litratur on law and conomics that w mntiond in sction 2. Not also that mrchants xrt a positiv xtrnality on th paymnt platform and on consumrs if B 6= 0, bcaus thir invstmnt in fraud dtction tchnologis rducs th amount of thir xpctd fraud loss. If B = 0, th zro liability rul applis for consumrs. In this cas, all mrchants who accpt th EPI invst th sam amount in fraud dtction tchnologis, which is implicitly d nd by xl dq d = C( 0 ) (4) As invstmnts do not impact consumr dmand, th transaction volum ct is null undr th zro liability rul. A mrchant who obtains a highr transaction bn t dos not hav highr invstmnt incntivs, as th marginal bn ts obtaind through a highr transaction volum ar qual to zro. 11

4.2.3 Comparativ statics In Lmma 3, w giv som comparativ statics to xplain how a mrchant s invstmnt in fraud dtction tchnologis vary with th transaction fs, th liability lvls and th bn t that a mrchant obtains of bing paid with th lctronic paymnt instrumnt. Lmma 3 If B > 0, th mrchant s invstmnts in fraud dtction tchnologis incras with th consumr liability, th consumr transaction f, th mrchant s transactional bn t, and th mrchant s liability, but thy dcras with th mrchant f. Proof. Appndix D. W provd in Lmma 2 that a mrchant s invstmnts in fraud dtction tchnologis ar chosn such that th marginal bn ts ar qual to th marginal costs of invstmnts. If th mrchant f incrass (rsp. if th mrchant s transactional bn t incrass), all othr things bing qual, th marginal bn ts from invstmnt dcras, bcaus of a rduction of th transaction volum ct. Th mrchant racts by rducing its invstmnts in fraud dtction tchnologis. If th mrchant s liability incrass, this incrass th xpctd loss ct, bcaus th mrchant has mor to sav whn a fraud is dtctd, whras this dcrass th transaction volum ct, as th mrchant s margin pr transaction is rducd. Undr Assumption (A2), th rst ct dominats and th mrchant racts by incrasing its invstmnts in fraud dtction tchnologis. Morovr, if th consumr liability incrass or if th consumr f incrass, this incrass th transaction volum ct, bcaus th impact of mrchant s invstmnts on consumr dmand incras. Thrfor, th mrchant s invstmnts incras. If th zro liability rul applis, from (4), th mrchant s invstmnts in fraud dtction tchnologis do not dpnd on th transaction fs that ar chosn by th paymnt platform. Thy only dpnd on th mrchant s liability and th xpctd loss. As whn B > 0, thy dcras with th mrchant s liability and it can b shown that thy dcras with th xpctd fraud loss. In Lmma 4, w dtrmin how th transaction fs and th liability lvls impact th probability that a mrchant accpts th lctronic paymnt instrumnt. Lmma 4 Th probability that a mrchant accpts th EPI dcrass with th mrchant f, with th consumr f and with th lvl of liability that is born by mrchants or by consumrs. Proof. Appndix E. 12

A highr mrchant f lowrs th transaction margin that th mrchant obtains if h accpts th EPI, whras it rducs th mrchant s incntivs to accpt th EPI, which is a standard ct in th litratur on paymnt cards. Morovr, in our modl, th probability that a mrchant accpts th EPI also dpnds on th consumr f, bcaus mrchants xrt a positiv xtrnality on consumrs whn thy choos to invst in fraud dtction tchnologis. Indd, this intraction, which is novl in th litratur on paymnt platforms, ariss whn B 6= 0 and this is spci c to our modl stting. Finally, a highr consumr f dcrass th probability that a consumr wishs to us th EPI, which rducs th marginal bn ts of invsting in fraud dtction tchnologis and th bn ts of accpting th EPI for th mrchant. Thrfor, th probability that a mrchant accpts th EPI dcrass with th consumr f. Most importantly, our modl is th rst to highlight th impact of liability rgims on mrchants accptanc of paymnt mdia. W show in Appndix E that th lvl of liability has an ambiguous impact on mrchants choic to accpt th lctronic paymnt instrumnt. On th on hand, a highr liability lvl incrass th loss in cas of a fraudulnt us of th EPI, which discourags mrchants to accpt th EPI. On th othr hand, it incrass th lvl of ort mad by mrchants, which rducs th probability that th EPI is fraudulntly usd - and thus incrass th probability that a consumr wishs to us th EPI. From assumption (A2), th rst ct dominats in our framwork, and thrfor, th probability that a mrchant accpts th EPI dcrass with his liability lvl. 4.3 tag 1 Prics and liability lvls At th rst stag, th paymnt platform choss th prics that maximis its pro t, P = (f + m c)v P EL P, whr V P dnots th transaction volum, as follows V P = Z bs EL P dnots th avrag xpctd loss, or cb h(b )(1 H B (f + B xl(1 q ))db ; (5) and Z bs EL P = P xl (1 cb q )h(b )(1 H B (f + B xl(1 q )))db ; (6) q = q( ). 13

If B = 0, as q dos not dpnd on b, w hav EL P = P xl(1 q )V P (7) Not that, for all B 2 [0; 1], th transaction volum dcrass with th consumr transaction f and with th mrchant f. Whil this ct is standard in th litratur, anothr qustion ariss in our framwork, that is th impact of th transaction prics and th mrchants liability on th xpctd fraud loss that is born by th paymnt platform. 4.3.1 Variations of th xpctd loss with th prics W start by dtrmining how th xpctd fraud loss is impactd by th choic of transaction fs and by th lvl of liability that is born by mrchants. Proposition 1 Th xpctd loss incurrd by th paymnt platform on fraudulnt transactions (EL P ) dcrass with th consumr transaction f and with th lvl of liability that is born by mrchants. EL P dcrass with th mrchant f only if th lasticity of th mrchant s ort to th mrchant f is small or if th lasticity of th mrchant s dmand to th mrchant f is high. Proof. Appndix F. An incras in th consumr f dcrass th numbr of mrchants who accpt th EPI, whras it incrass mrchants invstmnts in fraud dtction tchnologis. It follows that a highr consumr f dcrass th xpctd loss that is incurrd by th paymnt platform. Morovr, a highr lvl of liability for mrchants dcrass th xpctd loss that is born by th paymnt platform, as it dcrass mrchants accptanc of th EPI, whras it incrass mrchants invstmnt in fraud dtction tchnologis. An incras in th mrchant f has two cts on th xpctd loss that is incurrd by th paymnt platform. Th highr th mrchant f, th lowr th numbr of mrchants who accpt th EPI, and th lowr th transaction volum. This ct rducs th xpctd loss that is incurrd by th paymnt platform. At th sam tim, a highr mrchant f dcrass th mrchants invstmnt in fraud dtction tchnologis, which incrass th xpctd loss that is born by th paymnt platform. Th impact of an incras in th mrchant f on th xpctd loss dpnds on how both cts compnsat ach othr. 4.3.2 Th pro t maximising pric structur Proposition 2 givs th pro t maximising pric structur for a givn lvl of mrchants liability. 14

Proposition 2 Th pro t maximising pric structur r cts th platform s trad-o btwn balancing pro ts btwn both sids of th markt and minimizing th xpctd loss on fraudulnt transactions. Th total pric is implicitly d nd by f + m f c = 1 " V B (f) + @EL P =@f f@v P =@f ; and th pric structur vri s f m = 1 " V (m) + @EL P =@m m@v P =@M 1 " V B (f) + @EL P =@f f@v P =@f whr " V B (f) = (@V P =@f)(f=v P ) and " V (m) = (@V P =@m)(m=v P ) dnot th lasticity of th transaction volum to th consumr f and th mrchant f rspctivly. ; Proof. W dnot by M P = f + m c th paymnt platform s gross margin. Assum that thr is an intrior solution. olving for th rst-ordr conditions of pro t maximisation yilds @ P @f = M P @V P @f + V P @EL P @f = 0; and @ P @m = M @V P P @m + V P @EL P @m = 0 Ths quations can b rwrittn as f + m f c = V P f@v P =@f + @EL P =@f f@v=@f ; (8) and f + m m c = V P m@v P =@m + @EL P =@m m@v P =@m (9) Introducing th lasticitis " V B (f) = (@V P =@f)(f=v P ) and " V (m) = (@V P =@m)(m=v P ) and dividing th rst quation by th scond quation yilds th rsult of Proposition 2. In Appndix G-A, w show that th scond-ordr conditions of pro t maximisation ar vri d if B = 0. It is intrsting to compar th prics that w nd in an intrior solution with th prics obtaind in th standard two-sidd markt monopoly pricing formula obtaind by Rocht and Tirol (2003). Equations (8) and (9) show that with rspct to th standard pric structur in two-sidd markts, th pric structur that w obtain ncompasss an additional trm that taks into account th platform s trad-o btwn maximising its pro t and minimizing th 15

xpctd loss on fraudulnt transactions. Notic that if th zro liability rul applis for consumrs (that is if B = 0), from (7), th xpctd loss only dpnds on th transaction prics through th transaction volum. It follows that, in this cas, th pric structur is th sam as th on obtaind by Rocht and Tirol (2003), that is f m = "V B (f) " V (m); and th total pric is implicitly d nd by f + m c (1 )xl(1 q ) f = 1 " V B (f) For instanc, if B = 0, with uniforms distribution on [0; 1] for b B and b, with a cost function C ( ) = k( ) 2 =2, with q( ) = and c = 0, w prov in Appndix H that th pro t maximising transaction fs ar m = 1 + xl(1 3 ) + (xl)2 (2 2 ) k ; (10) 3 and f = 1 + xl + (2 2 )(xl) 2 2k 3 Not that th consumr f is highr than th mrchant f if 6= 0, as w hav f m = xl 1 q xl 2 + 0 (11) 2k If th dmands ar uniform and symmtric, in th standard cas of th litratur on paymnt cards, th pro t maximising transaction fs ar such that f = m. Equation (11) shows that, if > 0, th paymnt platform tnds to lowr th mrchant f to provid mrchants with incntivs to invst in fraud dtction tchnologis. Th pric structur changs in favor of mrchants. This is not ncssarily th cas if dmands ar not symmtric, or if B 6= 0. If B 6= 0, th paymnt platform can us th transaction prics on both sids of th markt to ncourag mrchants to invst in fraud dtction tchnologis, bcaus of th transaction volum ct that w highlightd in Lmma 2. 16

4.3.3 Th pro t maximising lvl of liability W hav assumd that th paymnt platform has th opportunity to choos th mrchant s lvl of liability at th sam stag as th transaction prics. Thus, w start by dtrmining how th mrchant s lvl of liability impacts th plaform s pro t. W know from Proposition 1 that th xpctd loss that is born by th paymnt platform dcrass with th lvl of liability born by mrchants. It rmains to study how th lvl of liability born by mrchants impacts th transaction volum. W hav @V P @ = @ b h ( @ b b )(1 H B (f + B xl(1 q ))) {z } Trm I Z bs + h (b ) @D B(f + B xl(1 q )) db cb @ {z } Trm II Th rst trm of (12) is ngativ. It r cts th fact that fwr mrchants accpt th EPI whn th lvl of liability that is born by mrchants incrass. Th scond trm of (12) is positiv. It shows that mor consumrs wish to pay with th EPI whn mrchants invst in fraud dtction tchnologis. It follows that a highr lvl of liability for mrchants has an ambiguous impact on th transaction volum. Not that if th lasticity of th mrchants dmand to thir liability lvl is small (that is, if trm I is small), th transaction volum may incras with th mrchants lvl of liability. Morovr, if th zro liability rul applis for consumrs, th scond trm of (12) is null, and th transaction volum dcrass with th mrchant s lvl of liability. Proposition 3 givs th pro t maximising lvl of liability for mrchants. (12) Proposition 3 A monopolistic paymnt platform chooss a lvl of liability for mrchants that r cts a trad-o btwn minimizing th xpctd loss on fraudulnt transactions and maximising th transaction volum. Th intrior solution for th pro t maximising lvl of liability for mrchants solvs (f + m c) @V P @ = @EL P @ If th transaction volum incrass with th liability lvl that is born by mrchants, thr is a cornr solution such that th paymnt platform lts th mrchants bar all th losss. Proof. Th paymnt platform chooss th lvl of liability that maximiss its pro t. olving for th rst-ordr condition of pro t maximisation yilds @ P @ = (f + m c) @V P @ @EL P @ 17

In an intrior solution, w hav (f + m c) @V P @ = @EL P @ From Proposition 1, w know that th xpctd loss dcrass with th lvl of liability that is born by th mrchants. It follows that, if th transaction volum incrass with th lvl of liability born by mrchants, th pro t maximising liability lvl is a cornr solution, with th mrchants baring th maximum shar of th loss. In Appndix G-B, w show that th scond-ordr conditions of pro t maximisation ar vri d if B = 0. Proposition 3 shows that th paymnt platform has an incntiv to shar th losss on fraudulnt transactions with th mrchants, as this ncourags mrchants to accpt th lctronic paymnt instrumnt, unlss mrchants dmand is inlastic to th lvl of liability. Howvr, th choic of a liability rgim is also a mans for th paymnt platform to xtract rnts from mrchants if th lasticity of th mrchants dmand to th liability lvl is small. In Appndix H, w prov that, if B = 0, with uniforms distribution on [0; 1] for b B and b, with a cost function C ( ) = k( ) 2 =2, and a dtction probability q( ) =, th pro t maximising lvl of liability for mrchants is qual to 1 This rsult is not gnral undr th zro liability rul. In othr cass, th liability for fraud is shard btwn th platform and th mrchants. 19 5 Wlfar maximising liability lvls To study wlfar maximizing liability lvls, w assum that th mrchant s lvl of liability is dcidd by a social plannr at th rst stag, who maximiss th sum of th platform s pro t, th consumr surplus and th mrchant surplus. Thn, th paymnt platform chooss th transaction fs at th following stag. Our aim is to compar th pro t maximising lvl of liability for mrchants, which is chosn by th paymnt platform, to th wlfar maximising lvl of liability for mrchants. W start by analyzing th simpl cas in which consumrs bar zro liability on fraudulnt transactions. 20 For this purpos, w nd to dtrmin how th liability lvl that is born by mrchants impacts th transaction fs that ar chosn by th paymnt platform. 19 A gnral rsult undr th zro liability rul is that th paymnt platform chooss th lvl of liability for mrchants that maximiss th probability of fraud dtction ( Appndix G-B). 20 In th futur, our analysis will b xtndd to th cas in which consumrs bar som rsponsibility for fraud. 18

Lmma 5 If th zro liability rul applis for consumrs, th transaction fs chosn by th paymnt platform dcras with th lvl of liability that is born by mrchants. Proof. Appndix I-A and I-B. Whn th lvl of liability that is born by mrchants incrass, this has two cts on th paymnt platform s pro t. First, this rducs th shar of th xpctd loss that is born by th paymnt platform, which amounts to a rduction of its marginal cost. Th paymnt platform may dcid to pass through this marginal cost rduction to th usrs by rducing th transaction fs. cond, mrchants invst mor in fraud prvntion tchnologis, which rducs th amount of th xpctd loss that is born by th paymnt platform for ach transaction. This ct can b rinforcd if th paymnt platform dcids to rduc th transaction fs paid by th usrs, as this incrass th transaction volum. Thrfor, if th lvl of liability that is born by mrchants incrass, th paymnt platform has an incntiv to lowr th transaction fs on both sids of th markt. Th paymnt platform loss som rnts from th transaction fs, but this loss is compnsatd by highr rnt xtraction through th liability rgim, which ncourags mrchant invstmnt. W ar now abl to compar th pro t maximising lvl of liability and th wlfar maximising lvl of liability for mrchants if consumrs do not bar any liability for fraudulnt transactions. W assum that social wlfar is a concav function of th transaction fs. 21 Proposition 4 Undr th zro liability rul for consumrs, if social wlfar is a concav function of, th pro t maximising lvl of liability for mrchants is lowr than (or qual to) th wlfar maximising lvl of liability. Proof. Appndix J-B. W showd in Proposition 5 that th transaction fs paid by th usrs dcras with th lvl of liability that is born by mrchants. A dirct consqunc of Proposition 5 is that consumr and mrchant surplus incras whn mrchants liability incras. It follows that, from th point of viw of total usr surplus maximisation, it is socially optimal to lt th mrchants bar th maximum liability on fraudulnt transactions. Howvr, if th rgulator taks into account th paymnt platform s pro t, th wlfar maximising lvl of liability for mrchants is not ncssarily qual to on. 21 W is concav in for instanc if b and b B ar uniformly distributd on [0; 1] undr som assumptions about th cost of fraud prvntion and th snsitivity of th dtction probability which ar prcisd in Appndix J. In gnral, it is possibl to prov that P is concav in, howvr, th total usr surplus is not ncssarily concav in. 19

Th paymnt platform dos not plac nough liability on mrchants to maximis social wlfar, xcpt in th cas whr it is maximiss its pro t by ltting th mrchants bar th maximum liability on fraudulnt transactions. This is bcaus th paymnt platform intrnalizs imprfctly th impact of th liability rgims on consumr and mrchant surplus. Not that this rsult is drivn by th assumption that th probability to dtct a fraudulnt transaction only dpnds on mrchants invstmnt. Th rsult could chang if th invstmnts wr shard by th paymnt platform and by th mrchants. 6 Th rol of intrchang fs In this sction, w xamin an important rgulatory challng, which is th impact of mrchant liability on th lvl of intrchang fs. 22 This issu has bn xamind in th Unitd-tats aftr th vot of th Dodd-Frank act in July 2010, which givs to th Fdral Rsrv Board th powr to rgulat intrchang fs on dbit card transactions. Among th rgulatory ruls, th "fraud adjustmnt rulmaking" provids th Board with th opportunity to assss how card ntworks authorization choics and fraud procdurs may burdn th mrchant community and potntially incras th volum of dbit card fraud. Th rulmaking also givs th Board th opportunity to promot th us of th fraud adjustmnt mchanism as a mans of crating incntivs for banks and mrchants to migrat to mor ctiv fraud dtction tchnologis. To study this issu, w modify our modl stting, by making th standard assumption that th paymnt platform is now composd of imprfctly comptitiv issurs and prfctly comptitiv acquirrs. 23 W also assum for simplicity of th modl that consumrs bar no liability on fraudulnt transactions ( B = 0). Th issurs charg a f f (c I a) to th consumrs, whras th acquirrs charg mrchants with thir prcivd marginal cost, that is m = a+c A. As in th litratur, w mak th standard assumption that f is dcrasing with a, and that th pass-through rat is lowr than on, that is @f =@a 1. At th rst stag of th gam, th paymnt platform chooss th lvl of intrchang f that maximiss banks joint pro t. Thn banks choos th transaction prics, mrchants invst in fraud dtction tchnologis and consumrs mak thir paymnts dcisions. W dnot th pro t maximising intrchang f by a P, and study how th pro t maximising intrchang f is impactd by th lvl of liability that is born by mrchants. Proposition 5 If th issurs ar imprfctly comptitiv and if th acquirrs ar prfctly comptitiv, th pro t maximising intrchang f dcrass with th lvl of liability that is born 22 Intrchang fs ar paid by th acquiring bank to th issuing bank ach tim a consumr maks a transaction. 23 For instanc, this assumption is also mad in Rocht and Tirol (2002). 20

by mrchants. Proof. Appndix K. Proposition 4 has important implications for rgulatory dcisions about intrchang fs. It mans that, if mrchants bar a highr shar of th loss on fraudulnt transactions, th pro t maximising intrchang f bcoms lowr. Th rsult of Proposition 4 may chang if consumrs ar hld liabl for fraudulnt transactions. In this cas, mrchants invstmnts ar impactd by th transaction fs and by th intrchang f that is chosn by th paymnt platform. Th paymnt platform may dcid ithr to lowr or to incras th intrchang f to provid mrchants with incntivs to incras thir invstmnt in fraud dtction tchnologis, dpnding on th rlativ importanc of th xpctd loss ct and th transaction volum ct that w highlightd in Lmma 2. Anothr intrsting aspct of th problm is that rgulators may wish to x a maximum lvl for th intrchang f, but th paymnt platform can ract by adjusting th lvl of liability that is born by mrchants for fraudulnt transactions. In Appndix K, w show in a simpl xampl that, if th rgulator chooss a low lvl for th intrchang f, th paymnt platform racts by choosing a high lvl of liability for mrchants, which may not b dsirabl from th point of viw of social wlfar. 7 Platform s invstmnts W analyz if our wlfar rsult undr th zro liability rul holds in an xtnsion of th modl that allows th paymnt platform to invst. W assum that th paymnt platform invsts an amount P in fraud dtction tchnologis, which costs C P ( P ) pr transaction, whr C P is a convx cost function. Th probability to dtct a fraudulnt transaction, which w dnot by q( ; P ), incrass with th platform s invstmnts, that is @q=@ P 0. Th platform chooss its lvl of invstmnt at th sam stag as th prics, and mrchants ar abl to obsrv this dcision bfor dciding on whthr or not to accpt th lctronic paymnt instrumnt. In a supplmntary not, which is availabl upon authors rqust, w show that th wlfar rsult obtaind undr th zro liability rul dos not hold whn th platform s invstmnts ar takn into account. 24 This is bcaus th prics chosn by th paymnt platform do not ncssarily dcras with th lvl of liability born by mrchants. Th intuition of this rsult is th following. Th platform now trads o btwn ncouraging th mrchants to invst in fraud dtction tchnologis and choosing to mak itslf th fraud 24 Excpt in th cas whr th platform s cost function is linar and if th dtction probability is linar in th platform s invstmnt ort. 21

prvntion ort. Th rsult of this trad-o is impactd by th rlativ cost of invstmnt for th platform and th mrchants, and by th fact that thir tchnological choics may b ithr indpndnt or may in unc ach othr. 7.1 Indpndnt invstmnts W start by analyzing th cas in which th mrchant s invstmnts and th platform s invstmnt ar b indpndnt. This cas can b illustratd by assuming for instanc that q( ; P ) is linar and sparabl in and P, that is q( ; P ) = d + h P ; whr d 0 and h 0. To undrstand bttr th impact of th platform s invstmnts on our wlfar rsult undr th zro liability rul for consumrs, w spcify quadratic invstmnt cost functions for th mrchants and th platform, such that C ( ) = k ( ) 2 =2 and C P ( P ) = k P ( P ) 2 =2, whr k 0 and k P 0. W also assum uniform distributions on [0; 1] for b and b B. Undr ths assumptions, at th quilibrium of stag 2, ach mrchant invsts an amount = dlpx =k in fraud dtction tchnologis. At stag 1, th prics chosn by th platform ar f = 1 3 (1 + c + Lx) k P 6 ( P ) 2 k 6 ( ) 2 2 ; m = 1 3 (1 + c + Lx(1 3 )) k P 6 ( P ) 2 (1 6 ) k 6 ( ) 2 2 4 ; whr ( P ) = hlp=k P dnots th optimal invstmnt of th platform. Not that this illustration shows that th consumr f is not ncssarily highr than th mrchant f, unlik our prvious xampl with uniform distributions undr th zro liability rul. W hav @f @ = d2 L 2 x 2 (1 ) 3k ; (13) and @m @ = xl 3k P k 2k (h 2 xl k P ) + d 2 k P xl (14) From (13), th consumr f dcrass with th lvl of liability born by mrchants, whras from (14), th mrchant f incrass with th lvl of liability born by mrchants if th invstmnt cost of th platform is high and if th mrchants contribution to incras th dtction probability is small (through th paramtr d). This rsult can b xplaind as follows. A highr lvl of liability for mrchants has two cts on th platform s incntivs to 22

invst in fraud dtction tchnologis. First, it dcrass mrchants accptanc, which rducs th marginal bn ts of invsting in fraud dtction tchnologis for th paymnt platform. cond, it rducs fraud losss, which amounts to a rduction of th platform s marginal cost. This ct impacts th platform s invstmnts in two opposit dirctions. On th on hand, it dcrass th platform s incntivs to invst, as th platform bars a lowr shar of fraud losss. On th othr hand, it incrass th platform s margin pr transaction, which can rsult in highr invstmnt incntivs. From th point of viw of mrchants, an incras in thir liability raiss th valu of th platform s invstmnts, as this improvs th quality of srvic providd by th platform. Th paymnt platform trads o btwn xtracting this additional surplus from th mrchants through th mrchant f and incrasing th transaction volum through lowr fs. Th variation of th mrchant f with th mrchants shar of fraud losss r cts this trad-o, which is not prsnt on th consumr sid. 7.2 Rlatd invstmnts W now analyz th cas in which th platform s dcision to invst in fraud dtction tchnologis impacts positivly th mrchant s invstmnt ort. This cas can b illustratd by assuming for instanc that q( ; P ) is a product of th mrchant s invstmnt ort and th platform s invstmnt ort, that is q( ; P ) = d P + h P ; whr d 0 and h 0. At th quilibrium of stag 2, th mrchant s invstmnts in fraud dtction tchnologis ar positivly rlatd to th platform s prvntion ort, and w hav = d P xl =k. Thrfor, th platform taks into account this ct in its trad-o btwn ncouraging mrchants invstmnts and choosing to bar itslf th fraud prvntion ort. At stag 1, th platform chooss th transaction fs f = 1 3 (1 + c + Lx) hlx P 6 ; m = 1 3 (1 + c + Lx(1 3 )) + k 2 ( ) 2 hlx 3 P (1 3 ) + hlx P ; 6 whr P = hk Lx=(k P k d 2 L 2 x 2 (2 )). Not that th platform s lvl of invstmnt incrass with th shar of liability born by mrchants. This rsult can b xplaind as follows. An incras in th lvl of liability born by mrchants amounts to a rduction of th platform s marginal cost, which rsults in highr invstmnt incntivs for th platform. A highr lvl of liability also raiss th impact of th platform s ort on mrchants invstmnt incntivs, 23

which provids th platform with an additional incntiv to incras its lvl of prvntion ort. 8 Conclusion and discussion Our rsults highlight th fact that liability rgims can b usd by monopolistic paymnt platforms to xtract rnts from mrchants. From th point of viw of a social plannr, paymnt platforms do not plac nough liability on mrchants for invstmnts that only dpnd on th mrchants sid undr th zro liability rul. This rsult changs if th platform shars th cost of invstmnts with mrchants. Anothr issu that dsrvs furthr rsarch is th problm of complianc in paymnt systms. This papr has considrd only prics and liability rgims as an incntiv to ncourag mrchant invstmnt. Howvr, w think that it would b intrsting to compar th impact of di rnt masurs on invstmnts and fraud losss such as complianc ruls, pric incntivs and liability shifts. 9 Rfrncs Rfrncs [1] ARANGO, C. & TAYLOR, V. (2008) "Mrchants Accptanc, Costs and Prcption of Rtail Paymnts A Canadian urvy," Bank of Canada, Discussion Papr 2008-12. [2] BEDRE-DEFOLIE O. & CALVANO E. (2009) "Pricing Paymnt Cards", ECB Working Papr No 119. [3] BELLEFLAMME, P. & PEITZ, M. (2010) "Platform Comptition and llr Invstmnt Incntivs", Forthcoming in Europan Economic Rviw. [4] BROWN, J. P. (1973) "Toward an Economic Thory of Liability," Th Journal of Lgal tudis, 2,2 323-349 [5] COOTER R. & ROBIN E. (1987), A Thory of Loss Allocation for Consumr Paymnts, Txas Law Rviw, 63-129 [6] CHAKRAVORTI (2010) Extrnalitis in Paymnt Card Ntworks Thory and Evidnc, Rviw of Ntwork Economics, vol.9, Issu 2, Articl 3. [7] DARI-MATTIACCI G. & PARII F. (2006), Th Economics of Tort Law A précis. Th Elgar Companion to Law and Economics (2nd d.), Edward Elgar Publishing 24

[8] DOUGLA D.B. (2009) An xamination of th fraud liability shift in consumr cardbasd paymnt systms, Fdral Rsrv Bank of Chicago, Economic Prspctivs, First Quartr 2009. [9] FEDERAL REERVE REGITER "Dbit Card Intrchang Fs and Routing Proposd Rul", Vol. 75, No. 248, Tusday, Dcmbr 28, 2010. [10] FURLETTI M. (2005) Th Laws, Rgulation and Industry Practics that Protct Consumrs who us Elctronic Paymnt ystms Policy Considrations, Discussion Papr, Paymnt Cards Cntr, Fdral Rsrv Bank of Philadlphia. [11] GATE, T. & JACOB K. (2009) Paymnts Fraud Prcption Vrsus Rality A confrnc ummary, Ti any Gats and Katy Jacob, 1st Quartr 2009, Economic Prspctivs, Fdral Rsrv Bank of Chicago. [12] LANDE, W.M. and PONER, R.A. (1987), Th Economic tructur of Tort Law, Cambridg (MA) Harvard Univrsity Prss. [13] Obsrvatoir d la sécurité ds carts d paimnt, Rapport Annul 2009. [14] ROCHET J-C. (2003) "Th Thory of Intrchang Fs A synthsis of rcnt contributions," Rviw of Ntwork Economics, vol. 2, no. 2, Jun, pp. 97-124. [15] ROCHET J-C. & TIROLE J. (2002) "Coopration Among Comptitors Th Economics of Paymnt Card Associations," Th RAND Journal of Economics, vol. 33, no. 4, wintr, pp. 549-570. [16] ROCHET J-C & TIROLE (2003) "Platform Comptition in Two-idd Markts," Journal of th Europan Economic Association, vol. 1, no. 4, pp. 990-1029. [17] ROCHET J-C. & TIROLE J. (2006) "Two-sidd markts a progrss rport," RAND Journal of Economics, vol. 37, no. 3, March, pp. 645-667. [18] ULLIVAN, R. J. (2010) "Th Changing Natur of U. Card Paymnt Fraud Industry and Public Policy Options, Economic Rviw, cond Quartr 2010, Fdral Rsrv Bank of Kansas City. [19] VERDIER M. (2010) "Intrchang Fs and Incntivs to Invst in Quality of a Paymnt Card ystm" (2010), Intrnational Journal of Industrial Organization, vol.28, pp. 539-554. [20] VERDIER M. (2011) "Intrchang Fs in Paymnt Card ystms a urvy of th litratur", Journal of Economic urvys, Vol.25, Issu 2, pp.273-297. 25

[21] WRIGHT J. (2002) "Optimal Paymnt Card ystms," Europan Economic Rviw, vol. 47, no. 4, August, pp. 587-612. [22] WRIGHT J. (2004) "Dtrminants of Optimal Intrchang Fs in Paymnt ystms," Journal of Industrial Economics, vol. 52, no. 1, March, pp. 1-26. 10 Appndix Appndix A Consumr Protction Laws in Various Countris. Th following tabl provids som xampls of consumr protction laws in various countris. Th common fatur of consumr protction laws is that consumr bar hardly maningful rsponsibility for fraudulnt us of cards in all countris. Country Nam of th Law Consumr Protction UA TILA/Rg Z for crdit cards Cappd at $50 for all unauthorizd transactions. Dbit Cards If th cardholdr fails to notify th card issur within 2 days, th cardholdr s maximum liability is $500, of which only $50 can b attributd to fraud occurring during th rst 2 days aftr th cardholdr larnt th loss or thft. Europ Paymnt rvic Dirctiv Th cardholdr has 13 months to contst an unauthorizd transaction. Th cardholdr s liability is cappd at 150 uros if h has faild to kp th prsonnalizd scurity masurs saf. If th cardholdr was a victim from an idntity thft, h cannot b hld liabl. No liability in all cass aftr th fraud is rportd. Right for paymnt srvic usrs to njoy immdiat rfund of unauthorizd transactions following th stablishmnt of th proof. Appndix B Proof of Lmma 1. W prov in Lmma 1 than th mrchants who accpt th EPI and th mrchants who do not accpt th EPI st th sam pric p = v. Thr ar two cass ithr a mrchant rfuss th EPI or h accpts it. Lt us start by th rst cas. If 26

a mrchant rfuss th EPI, all consumrs pay cash, and h maks pro t 25 = p d. In this cas, th mrchant s pro t is maximisd whn h sts p = v, and w hav that = v d. In th scond cas, th mrchant accpts th EPI. If h sts p v, h attracts both EPI and cash usrs. In this cas, h maks pro t = p d + D B (f + B (1 q)xl)(b x(1 q)l m C ( )) This pro t is maximisd at p = v. If h sts p > v, th mrchant attracts only EPI usrs. In this cas, h maks pro t = (p d + b x(1 q)l m C ( ))D B (f + p v + B (1 q)xl) (15) W now show that th mrchant always maks mor pro t by stting p = v. For this purpos, w prov that and that for any p > v, w hav lim p!v d dp < 0, d dp < 0 From (15), w hav d dp = D B(f+p v+ B (1 q)xl) h B (f+p v+ B (1 q)xl)(p d+b x(1 q)l m C ( )) W also hav lim p!v d dp = D B(f + B (1 q)xl) h B (f + B (1 q)xl)(v d+b x(1 q)l m C ( )) 25 Our rsults would not chang if w addd a paramtr q 0 to modl th probability that th mrchant is paid with countrfit nots and coins. [FOR U In this cas, w would only hav to modify assumption (A2) hb(f + B(1 q)xl) v d q 0v 1 H B(f + B(1 q)xl) ] 27

This quantity is ngativ if and only if v d + b x(1 q)l m C ( ) 1 H B(f + B (1 q)xl) (16) h B (f + B (1 q)xl) As th mrchant accpts th EPI, w hav that b x(1 q)l m C ( ) 0. It follows that (A2) is a su cint condition for (16) to hold. W can now prov that for any p > v, d < 0 To simplify th notations, w dnot by dp gd B = D B (f + p v + B (1 q)xl) W hav d dp = g D B h B (f + p v + B (1 q)xl)(p d + b x(1 q)l m C ( )) < D g B h B (f + p v + B (1 q)xl)(v d + b x(1 q)l m C ( )) = D g h B (f + p v + B (1 q)xl) B 1 1 H B (f + p v + B (1 q)xl) (v d + b x(1 q)l m C ( )) W hav D g B 0. Thrfor, a su cint condition for d < 0 to hold is that th trm into dp brackt is ngativ. Th trm into brackts is ngativ if and only if v d + b x(1 q)l m C ( ) 1 H B(f + p v + B (1 q)xl) h B (f + p v + B (1 q)xl) As by assumption (A1) th hazard rat is incrasing, w hav that, for any p > v, 1 H B (f + p v + B (1 q)xl) h B (f + p v + B (1 q)xl) 1 H B(f + B (1 q)xl) h B (f + B (1 q)xl) From assumption (A2), w hav that It follows that v d + b x(1 q)l m C ( ) 1 H B(f + B (1 q)xl) h B (f + B (1 q)xl) v d + b x(1 q)l m C ( ) 1 H B(f + p v + B (1 q)xl). h B (f + p v + B (1 q)xl) Thrfor, w hav that, for any p > v, d < 0. It follows that th mrchant maks mor pro t dp by stting p = v, which nabls him to attract cash-usrs and EPI usrs. W can conclud that all mrchants choos a pric such that p = v. Appndix C proof of Lmma 2. W procd in two stps. First, w dtrmin th pro t maximising lvl of invstmnt of a mrchant who accpts th EPI. cond, w prov that, if 28

th mrchant f is not too high, som mrchants accpt th EPI. W start by th rst stp. A mrchant who accpts th EPI chooss th lvl of invstmnt in fraud dtction tchnologis that maximiss its pro t, = p d + D B (f + B (1 q)xl)(b x(1 q)l m C ( )) olving for th rst-ordr condition of pro t-maximisation yilds " xl dq # d C( 0 ) D B + [b x(1 q)l m C ( )] dd B d = 0. (17) W d n B = dd B=d th lasticity of th consumrs dmand to th fraud dtction D B = tchnology. Th mrchant s invstmnt in fraud dtction tchnologis is implicitly d nd by xl dq d C( 0 ) = [b x(1 q)l m C ( )] Bj " Th scond-ordr condition must b vri d at, that is, C ( " ) + xl d2 q d 2 whr M = b x(1 q)l m C ( ). # " D B + 2 xl dq # d C( 0 ) dd B d d 2 D B + M d 2 0; (18) Undr th assumption that C is convx and q is concav, th rst trm of this inquality is ngativ. To dtrmin th sign of th scond trm, w us quation (17). inc B j is ngativ and sinc th mrchant s margin is positiv if h accpts th EPI, w conclud that xl dq C 0 d ( ) 0 at th pro t maximising lvl of invstmnt. W hav and @ 2 D B @ 2 @D B @ = h B ( B xl(1 q ) + f) B xl dq d ; (19) = h 0 B( B xl(1 q ) + f)( B xl dq d ) 2 + h B ( B xl(1 q ) + f) B xl d2 q d 2 (20) From (19), w hav dd B d 0. It follows that th scond trm of (18) is ngativ. Finally, sinc M 0 and sinc d2 D B d 2 0 from (20), th last trm of (18) is ngativ. It follows that th scond-ordr condition is always vri d at. 29

W now show that mrchants accpt th EPI if thir transactional bn t b is such that b b ( ; B ; x; L; m; f), which is th scond stp of our proof. A mrchant accpts th EPI if and only if b m x(1 q)l C ( ) 0. (21) Lt us considr th function M (y) = y x(1 q)l C ( ), whr y = b m. Not that (21) dos not hold if y < 0, which happns if th mrchant f is too high. W hav that M 0 (y) = 1 + d dy xl dq! d C 0 ( ) From (17), w hav that xl dq d C 0 ( ) 0. W can also prov, using (17) and th nvlop thorm that d dy 0. It follows that M is incrasing in y for all y 0. Not that M (0) 0 and that th sign of M (y), whr y = b m, dpnds on m. Thr ar thr cass. Lt us start by th rst cas, in which th mrchant f m is su cintly high, such that M (y) < 0. As M is incrasing in y, for all b 2 b ; b and for all y = b M (y) < 0. It follows that no mrchant accpts th EPI. In th scond cas, th mrchant f m is su cintly low, such that b m, w hav m > 0 and M (y) 0; whr y = b m. As M is incrasing in y, for all b 2 b ; b and for all y = b m, w hav M (y) 0. It follows that all mrchants accpt th EPI. In th third cas, th mrchant f is such that M (y) > 0 and M (y) < 0. As M is incrasing in y, from th bijction thorm, thr xists a thrshold that w dnot by b ( ; B ; x; L; m; f) such that mrchants accpt th EPI for all b b ( ; B ; x; L; m; f). Appndix D proof of Lmma 3. From th nvlop thorm, w hav that, for any z 2 f B ; ; f; m; b g @ @z = @ 2 @ 2! 1 @ 2 @ @z! As from th scond-ordr condition @ 2 =@ 2 0, it follows that @ =@z has th sam sign as @ 2 @ @z. Lt us study th variation of th mrchant s invstmnts with th mrchant f. W hav that @ 2 @ @m = h B(f + B (1 q)xl) B xl dq 0 d From th nvlop thorm, @ =@m has th sam sign as @2 =@ @m. It follows that th 30

mrchant s invstmnt always dcrass with th mrchant f. imilarly, w hav that @ 2 @ @b = h B (f + B (1 q)xl) B xl dq d 0 (22) It follows that th mrchant s invstmnts in fraud dtction tchnologis incras with th mrchant s transactional bn t. W now study th variation of th mrchant s invstmnts with th consumr transaction f. Using th sam rasoning, w know that @ =@f has th sam sign as @2 =@ @f. W hav @ 2 @ @f = xl dq C 0 d ( ddb ) df +[b x(1 q)l m C ( )] B xl dq d h 0 B(f+ B (1 q)xl) " From th rst-ordr condition, w hav that xl dq # d C 0 ( ) 0. W also hav dd B =df 0. It follows that @ 2 =@ @f 0 sinc h 0 B is positiv. W can conclud that th mrchant s invstmnt incrass with th transaction f that is paid by th consumr. W dtrmin th variation of th mrchant s invstmnts with th consumr liability. Using th sam rasoning, @ =@ B has th sam sign as @ 2 =@ @ B. W hav @ 2 = xl dq C 0 @ @ B d ( ddb ) +[b x(1 q)l m C ( d )] B (1 q)x 2 L 2 dq B d h 0 B Exactly lik in th prvious proof, w hav that @ 2 =@ @ B 0 sinc h 0 B is positiv. It follows that th mrchant s invstmnt incrass with th consumr liability. Lt us study th variation of th mrchant s invstmnts with his lvl of liability. From th rasoning abov, @ =@ has th sam sign as @ 2 =@ @. W hav that @ 2 @ @ = xl dq d D B j x 2 L 2 B (1 q( )) dq d h B (f + B (1 q( ))xl) " # dq = xl D B j 1 xl B (1 q( d )) h B(f + B (1 q( ))xl) D B j From assumption (A2), w hav that h B (f + B (1 q( ))xl) 1 D B j xl B 31

As 1 q( ) 2 [0; 1], it follows that h B (f + B (1 q( ))xl) D B j 1 xl B (1 q( )) Thrfor, w hav that 1 xl B (1 q( )) h B(f + B (1 q( ))xl) D B j 0. As dq d 0 and xl D B j 0, w can conclud that @ 2 @ @ 0 It follows that, from assumption (A2), th mrchant s invstmnts in fraud dtction tchnologis incras with his liability lvl. Appndix E Proof of Lmma 4. Impact of th lvl of liability born by mrchants on EPI accptanc (2), th thrshold abov which mrchants accpt th EPI solvs From b m xl(1 q ) C ( ) = 0 Di rntiating this quation with rspct to, w obtain that @ b "! # 1 + xl dq @ d C( 0 d! ) = xl(1 q ) + d C 0 db d ( ) xl dq d (E-1) From (17), w hav that C 0 ( ) xl dq d 0. From Lmma 3, w know that d =d 0. It follows that th right-hand sid of th quality is positiv. that Lt us now dtrmin th sign of th lft-hand sid of th quality. From Lmma 3, w know d db = @ 2 @ 2! 1 @ 2 @ @b! (E-2) 32

Rplacing for d =db in (??), w obtain that! 1+ xl dq d C( 0 d ) = db @ 2 Rplacing for @ 2 (from (18)) and brackts in (E-3) is qual to " C ( " ) + xl d2 q d 2 @2 @ 2 @ 2 @ @b! 1 " @ 2 @ 2 + # " D B + 3 xl dq # d C( 0 )! xl dq d C( 0 @ 2 ) @ @b (E-3) (from (22)), w obtain that th trm into dd B d d 2 D B + M d 2 ; # whr M = b m xl(1 q ) C ( ) As xl dq d C 0 ( ) 0, dd B d 0 from (19) and d2 D B d 2 0 from (20), it follows that th trm into brackts in (??) is ngativ. As @2 @ 2 0, w can conclud that! 1 + xl dq d C( 0 d ) 0 db It follows from (E-2) that @ b @ 0. Thrfor, th probability that a mrchant accpts th EPI dcrass with th lvl of liability that is born by th mrchants. Impact of th transaction bn t rcivd by mrchants on EPI accptanc imilarly, w hav that @ b " 1 + xl dq @f d! C( 0 ) d db # = d df C 0 ( ) xl dq d! As d =df 0, it follows that @ b =@f 0. @ b =@ B 0. It can b also provd in a similar way that 33

Impact of th transaction fs on EPI accptanc @ b " 1 + xl dq @m d! C( 0 ) d db # W also hav that! = 1 + d dm C ( 0 ) xl dq d @ 2 @ 2 + dd B C 0 d ( ) xl dq d = @ 2 0 @ 2 It follows that @ b =@m 0. Appndix F Proof of Proposition 1. W start by dtrmining th variation of th xpctd loss with th consumr f. To that nd, w dnot by (b ; f; m; ; B ) th function d nd by (b ; f; m; ; B ) = (1 q )h(b )(1 H B (f + B xl(1 q )) From (6), Z bs EL P = P xl (b ; f; m; ; B )db cb Hnc, w hav that 26 @EL P @f = P xl " @ b @f Z ( b bs ; f; m; ; B ) + cb @(b ; f; m; ; B ) @f db # ; whr @(b ; f; m; ; B ) @f = dq @ d @f (1 H B(f + B xl(1 q ))h(b ) 1 (1 q )h(b )h B (f + B xl(1 q )) (1 q ) B xlh B + 1 H B From Lmma 3, w hav @ =@f 0. W also hav that dq=d 0. From assumption (A2), w hav 1 (1 q ) B xlh B 1 H B 0 It follows that @(b ; f; m; ; B )=@f 0. inc, from Lmma 4, @ b =@f 0, w conclud that @EL=@f 0. 26 Th conditions to us th Libniz rul apply. 34

W now dtrmin th variations of th xpctd loss with th mrchant f. W hav @EL P @m 2 = P xl 6 4 @ b @m ( b b ; f; m; ; B ) {z } T rma Z bs + @(b ; f; m; ; B ) db 7 cb @m 5 ; {z } T rmb 3 whr @(b ; f; m; ; B ) @m = dq @ d @m (1 H B(f + B xl(1 q ))h(b ) 1 (1 q ) B xlh B 0. 1 H B From Lmma 3, w hav @ =@m 0. From Lmma 4, @ b =@m 0. From Assumption (A2), w hav 1 (1 q ) B xlh B 1 H B 0 It follows that trm A is ngativ, whras trm B is positiv. Thrfor, an incras in th mrchant f has an ambiguous impact on th xpctd loss that is born by th paymnt platform. Lt us now study how th lvl of liability that is born by mrchants impacts th xpctd loss. W hav whr @EL P @ = EL P P 2 @ b + P xl 6 ( b 4 @ ; f; m; ; B ) {z } T rmc Z bs + @(b ; f; m; ; B ) db 7 cb @ 5 ; {z } T rmd 3 @(b ; f; m; ; B ) = dq @ (1 H B (f + B xl(1 q ))h(b ) 1 @ d @ (1 q ) B xlh B 0 1 H B From Assumption (A2), 1 (1 q ) B xlh B 1 H B 0 As dq=d 0, and sinc @ b =@ 0 from Lmma 4, it follows that @EL P @ 0 Appndix G cond-ordr conditions if B = 0. Appndix G-A scond-ordr conditions if th paymnt platform chooss th transaction prics. W provid hr th scond-ordr conditions of pro t maximisation if 35

B = 0. Th rst-ordr conditions of pro t maximisation ar @ P @m = D B(f) [D (b m ) M P h (b m )] = 0; (23) and @ P @f = D (b m ) [D B (f) M P h B (f)] = 0 (24) Th scond drivativs of th platform s pro t with rspct to th prics and th liability lvl ar @ 2 P @m 2 = 2h D B h 0 D B M P ; (25) @ 2 P @f 2 = 2h B D h 0 BD M P ; @ 2 P @m@f = h B D h D B + M P h h B ; @ 2 P = xl(1 q ) @2 P @m@ @m 2 (1 )xl @q h D B ; @ @ 2 P = xl(1 q ) @2 P (1 )xl @q h B D ; @f@ @m@f @ @ 2 P @ 2 = 2xLD B h xl(1 q ) 1 q + (1 ) @q @q + M P D B xlh (xl(1 q )) 2 h 0 @ @ " ( +xlv P 2 @q @ 2 )# q 2 @ + (1 ) @ @ 2 + @q @ 2 @ @ @ 2 W dnot by dt M th dtrminant of th Hssian matrix at th pro t maximising transaction fs. It can b chckd that th scond-ordr conditions of pro t maximisation ar vri d as h 0 0 and h0 B 0. From (23) and (24), w hav that, at th pro t maximising prics, D = M P h and D B = M P h B. Thrfor, w hav dt Mj (f ;m ) = 2h h B D D B + 2h 0 BD B D 2 + 2h 0 D D 2 B + h 0 h 0 BD B D M 2 P + h 2 D 2 B > 0; (26) and @ 2 @m 2 < 0, (f ;m ) which provs that th conditions for a maximum to xist at (f ; m ) hold. Appndix G-B scond-ordr conditions if th paymnt platform chooss th transaction prics and th lvl of liability for mrchants. W provid hr th conditions undr which th scond-ordr conditions ar vri d at x = (f ; m ; ) by computing 36

th co cints of th Hssian matrix. 0 1 a 1 b c Dnoting th Hssian matrix at x = (f ; m ; ) by H = B b a @ 2 d C, th scond-ordr A c d a 3 conditions ar vri d if a 1 0, a 2 0, a 1 a 2 b 2 0, a 1 a 3 c 2 0, a 3 a 2 d 2 0 and dt H 0 ( hraftr). If ths conditions ar vri d, this provs that th Hssian matrix is smi-d nit ngativ at x = (f ; m ; ). Lt us start by th cas in which thr is an intrior solution. From (25), as h 0 and h0 B ar positiv, w hav that a 1 0 and a 2 0. W alrady provd in Appndix G-A that a 1 a 2 b 2 0. W now prov that a 1 a 3 c 2 0 and that a 3 a 2 d 2 0. At x = (f ; m ; ), if th solution is intrior, from (23) and (24), w hav that D = M P h and D B = M P h B Th rst-ordr condition of pro t maximisation with rspct to is @ P @ = (f + m c) @V P @ @EL P @ = 0 From (7), w hav that @EL P @ = xl(1 q )V P + (1 )xlq @V P @. From (5), w hav that with rspct to writs @V=@ = D B (f)h ( b )xl(1 q ). Thrfor, th rst-ordr condition M P h D B xl(1 q ) = xld B D 1 q + (1 ) dq d As at x = (f ; m ; ) w hav D = M P h, in intrior solution, w hav that xl(1 q ) = xl 1 q + (1 ) dq ; d that is dq d = 0. x It follows that, in an intrior solution, th paymnt platform chooss th lvl of liability that maximiss th probability of fraud dtction. W dnot by = ( 2 @q @ 2 ) q 2 @ + (1 ) @ @ 2 + @q @ 2 @ @ @ 2 (GB-1) 37

Lmma 6 W hav @ 2 =@2 0 Proof. From th implicit function thorm, w hav @ @ = @ 2 @ 2! 1 @ 2 @ @! = = xl(@q=@ )D B D B ( xl d2 q C 00 ( )) d 2 xl(@q=@ ) ( xl d2 q C 00 ( )) d 2 It follows that @ 2 @ 2 = xln ; ( xl d2 q C 00 ( ))2 d 2 whr d 2! q N = xl + C 00 ( d 2 q ) " d 2 xl d3 q d 3 d 2 @ @ + C 000 ( ) @ @ # xl dq d inc d3 q 0, d 3 d 2 q 0, and C 000 ( ) 0, w hav N 0. Thrfor, w can conclud d 2 that @ 2 =@ 2 0. inc @ 2 =@ 2 0; @q=@ 0 and @ 2 q=@ 2 0, from (GB-1), w hav 0. As dq d = 0 and from (25), w hav that x @ 2 P @ 2 @ 2 P @ @f x x = (xl(1 q )) 2 2h + M P h 0 DB + xlv P ; = xl(1 q )h D B ; and @ 2 P @ @m = xl(1 q )D B (2h + h 0 M P ) x 38

W now comput a 1 a 3 c 2 and a 3 a 2 d 2 at x = (f ; m ; ). W hav a 1 a 3 c 2 = @ 2 P @f 2 x @ 2 P @ 2 x @ 2 P @ @f x = D 2 B [xl(1 q )] 2 (3h 2 + (M P h 0 ) 2 + 4M P h 0 h ) D 2 BD xl 2h + M P h 0 0 2 W also hav a 3 a 2 d 2 = @ 2 P @m 2 x @ 2 P @ 2 x @ 2 P @ @m = D 2 BD xl 2h + M P h 0 0 x 2 W now show that dt H 0 at x = (f ; m ; ). From th rul of arrus, w hav dt H = a 1 a 2 a 3 + 2bdc c 2 a 2 b 2 a 3 d 2 a 1 = a 1 (a 3 a 2 d 2 ) + 2bdc c 2 a 2 b 2 a 3 At x = (f ; m ; ), sinc a 1 = D (2h B + M P h 0 B ), w hav a 1 (a 3 a 2 d 2 ) = D 2 BD 2 xl 2h + M P h 0 2hB + M P h 0 B W also hav 2bdc = 2(xL) 2 (1 q ) 2 h 2 D 3 B 2h + M P h 0 ; and c 2 a 2 b 2 a 3 = 2(xL) 2 (1 q ) 2 h 2 DB 3 2h + M P h 0 xld 3 BD h 2 Using th fact that, at x = (f ; m ; ), w hav M P h = D and M P h B = D B, w obtain that dt H = D 2 BD xlm P 3hB h 2 + h 0 h 0 BM 2 P h + 2h h B h 0 M P + 2h 2 h 0 BM P (27) inc 0, w can conclud that dt H 0 at x = (f ; m ; ). Thrfor, th Hssian matrix is smi-d nit ngativ at x = (f ; m ; ) and th scond-ordr conditions ar vri d at x = (f ; m ; ). Appndix H An illustration of Proposition 2. W mak th following assumptions C ( ) = k( ) 2 =2, q( ) =, uniform distributions on [0; 1] for b and b B. In this cas, from quation (4), w hav 39

= xl ; k whr xl 2 =k 1. Th mrchant s ort incrass with th liability lvl, th probability that thr is a fraudulnt transaction, th losss born by th participants, and th marginal impact of invstmnts on th probability of fraud dtction. Th probability that a mrchant dtcts a fraudulnt transaction is implicitly d nd by q( ) = ( xl 2 )=k In this cas, th dmands ar D B = 1 f, and D = 1 m xl + ( xl) 2 Using th 2k standard pric structur/ratio formula, w nd that th prics vrify f = m + xl ( xl) 2 ; 2k and f + m c (1 )xl(1 q) f = 1 f 27 f olving for f and m, w obtain that m = 1 + c + xl(1 3 ) + (xl)2 (2 2 ) k ; 3 and f = 1 + c + xl + (2 2 )(xl) 2 2k 3 W can comput th marginal mrchant Th mrchant dmand is b = m + xl ( xl) 2 2k 1 + c + xl + (2 2 )(xl) 2 = 2k 3 D ( b 2(1 + c) xl (2 2 )(xl) 2 ) = 2k 3 40

If is chosn by th paymnt platform (at th sam stag as th prics), w hav that @ P @ = M P " @D ( b # b ) D B (f) @ + @M P @ D B (f)d ( b ) As M P = 1 f = D B = D at th optimal prics, w hav @ P = 2M P D B (f) @D ( b ) 0 @ @ In this cas, w nd that th platform s pro t is maximisd by choosing = 1. Appndix I Impact of th mrchants liability on transaction prics. Appndix I-A impact of th mrchants liability on transaction prics (gnral cas if B = 0). In this Appndix, w xamin how th lvl of liability born by mrchants impacts th transaction fs that ar chosn by th paymnt platform, if th zro liability rul applis for consumrs. conditions with rspct to, w obtain that By di rntiating quations (23) and (24) that d n th rst-ordr @ 2 P @m @m 2 + @2 P @f + @2 P = 0; (28) @ @m@f @ @m@ and @ 2 P @f 2 @f + @2 P @m + @2 P = 0 (29) @ @m@f @ @f@ olving for @m =@ and @f =@ in (28) and (29), w obtain that @m = 1 xl(1 q )( dt M) (1 )xl @q R ; @ dt M @ and @f = 1 @ dt M (1 )xl @q T ; @ whr and R = h B D @ 2 P @m@f T = h D B @ 2 P @m@f h D B @ 2 @f 2 ; h B D @ 2 P @m 2 41

W provd in Appndix G-A that dt M 0. W now prov that R 0 and T 0. W hav R = h 2 BD 2 + h B h D B D + M P h 2 Bh D + h h 0 BD D B M P ; and T = h B h D B D h 2 D 2 B + M P h 2 h B D B + h B h 0 D B D M P Using th rst-ordr condition, w hav that, at th pro t maximising prics, M P h = D and M P h B = D B. It follows that, at th pro t maximising prics, R = h B h D B D + h h 0 BD D B M P ; and T = h B h D B D + h B h 0 D B D M P inc h 0 and h0 B ar positiv, w hav R 0 and T 0. As @q =@ 0 and DtM > 0, it follows that @m =@ 0 and @f =@ 0 Not that, sinc b = m + xl(1 as @ b =@ = xl(1 q ). q ) + C ( ), w hav db = @ b + @m (30) d @ @ d b d = 1 dt M (1 )xl @q @ R 0; Appndix I-B Application to th cas of uniform distributions for b B and b. If b B and b ar uniformly distributd on [0; 1], from th rst ordr conditions, at th pro t maximising prics, w hav D B = D = M P. In this cas, w hav R = T = D 2 B. Thrfor, from (26), w hav dt M = 3DB 2. It follows that, in this xampl, w hav It follows from (31) that d b d = df = (1 ) d 3 xl @q @ (31) d 2 b b d 2 = d2 f d 2 = xl 3 @q @ (1 ) @2 q @ 2 0 (32) From Lmma 6, w hav @ 2 q =@ 2 0. Thrfor, w can conclud that d 2 b b =d 2 0 in th cas of uniform distributions on [0; 1] for th transactional bn ts. 42

Appndix J ocial wlfar analysis. Appndix J-AVariation of th consumr and th mrchant surplus with th lvl of liability born by mrchants. W start by computing th consumr surplus. Consumrs who pay cash do not obtain any surplus from making a transaction, as a monopolistic mrchant sts a pric p = v. A consumr of transactional bn t b B who pays with th EPI obtains a surplus b B f B xl(1 q ) Agrgating this xprssion ovr all b B 2 f + B xl(1 q ); b B and ovr all b 2 obtain th agrgat consumr surplus, that is h b b ; b i, w B = Z bs cb h(b )E(b B f B xl(1 q )=b B f + B xl(1 q ))db ; whr E(b B f B xl(1 q )=b B f + B xl(1 q )) dnots th mathmatical xpctancy conditional on b B f + B xl(1 q ). W hav @ B @ = X + Y; whr X = Y = db h( b d )E(b B f B xl(1 q )=b B f + B xl(1 q )); Z bs cb h(b ) @ @ E(b B f B xl(1 q )=b B f + B xl(1 q ))db ; whr from th Libniz rul, @ E(b B f B xl(1 q )=b B b @ c B ) = @ @ = Z bb cb B Z bb cb B (b B f B xl(1 q ))h B (b B )db B h B (b B )db B 0 @f @ + B xl dq d @ @ First cas B = 0. In this cas, from proposition 5, th transaction fs dcras with th lvl of liability that is born by mrchants. Thrfor, Y is positiv. Trm X is also positiv, sinc d b =d 0 from (30). It follows that th consumr surplus incrass with th lvl of liability that is born by th mrchants. cond cas B 6= 0 [TO DO] imilarly, w comput th agrgat mrchant surplus by agrgating th mrchants pro t 43

for all b 2 b ; b. W hav Z bs = (v d)(b b )+ cb h(b )(b m xl(1 q ) C ( ))(1 H B (f + B xl(1 q )))db From th Libniz rul, w hav @ @ = = Z bb h(b ) @ f(b m xl(1 q ) C ( @ ))(1 H B (f + B xl(1 q )))g db @m h(b ) + xl(1 q ) D B ( b cb @ c B ) + (b m xl(1 q ) C ( ))h 0B( b c B ) dq d cb Z b db 2 First cas B = 0. In this cas, from proposition 5, th transaction fs dcras with th lvl of liability that is born by mrchants. From (30), w hav @m =@ + xl(1 q ) 0. It follows that th mrchant surplus incrass with th lvl of liability that is born by mrchants. cond cas B 6= 0 [TO DO] Appndix J-B Th social wlfar maximising lvl of liability if B = 0. W start by proving that th paymnt platform pro t is concav in at th pro t maximising prics (f ; m ), which ar chosn at stag 2 (aftr a bnvolnt social planr chooss th liability lvl for mrchants). From (25), w hav @ 2 P @ 2 = 2xLD B h xl(1 q ) 1 q + (1 ) @q @ +xlv P " ( 2 @q @ 2 q @ + (1 ) @ @ 2 @ + M P D B xlh @q 2 + @q @ @ 2 @ 2 )# @ (xl(1 q )) 2 h 0 From Appndix J-A, at th pro t maximising prics, w hav that M P h B = D B and M P h = D. It follows that @ 2 P @ 2 (f ;m ) = 2(xL) 2 D B h (1 q ) 1 q + (1 ) @q @ +xlv P " ( @q @ 2 q @ + (1 ) @ @ 2 @ 2 + @q @ @ 2 @ 2 M P D B (xl(1 )# q )) 2 h 0 inc h 0 0, @2 q=@ 2 0, and @2 =@2 0 from Lmma 6, w conclud that @ 2 P @ 2 0. (f ;m ) W now study th concavity of th total usr surplus. For this purpos w nd to dtrmin 44

th sign of @ 2 f =@ 2 and @2 b b =@ 2. With uniform distributions for b B and b on [0; 1], this sign is positiv, and, from Appndix J-B, w hav @ 2 f @ 2 = @2 b b @ 2 = xl 3 @q @ (1 ) @2 q @ 2 0. In gnral, th total usr surplus is not ncssarily a concav function of. W hav @ 2 B @ 2 = 2 4 @2 b b @ 2 h ( b ) @ b! 3 2 h 0 @ ( b ) 5 E(b B f B xl(1 q )=b B b c B ) +2 @ b @f h ( b @ @ )D B (f) @ 2 f @ 2 D B (f)d ( b ); and @ 2 @ 2 = @2 b b @ 2 D B (f)d ( b ) + @ b! 2 h ( b @ )D B (f) With uniform distributions on [0; 1] for b B and b, sinc D B (f) = D ( b ), w hav that @ 2 B @ 2 = 3 @ 2 b b D 2 2 B(f) + 2 @ 2 @ b! 2 D B (f), @ and @ 2 @ 2 = @2 b b DB(f) 2 + @ 2 @ b! 2 D B (f) @ W now prov that a su cint condition for total usr surplus to b concav in is that C( 00 ) (xl)2 dq 2 + xl d2 q 3 d d 2 Th total usr surplus is a concav function of if and only if @ 2 @ 2 + @2 B @ 2 0 As at th pro t maximising prics, with uniform distributions, D B = D, w hav that, at th pro t maximising prics, @ 2 @ 2 + @2 B @ 2 = 5 @ 2 b b D 2 2 B(f) + 3 @ 2 @ b! 2 D B (f) @ 45

It follows that th total usr surplus is a concav function of if and only if @ 2 b b @ 2 D B (f) 6 5 @ b! 2 @ As D B blongs to [0; 1], a su cint condition for th total surplus to b concav in is that @ 2 b b @ 2 6 5 @ b! 2 @ From (31) and (32), this condition is quivalnt to @q @ (1 ) @2 q @ 2 (1 ) 2 3 @q 2 xl ; @ that is @q 1 @ (1 ) 2 3 @q xl @ (1 ) @2 q @ 2 0 As @ 2 q =@ 2 0 from Lmma 6, a su cint condition for this inquality to hold is that 1 (1 ) 2 3 @q xl 0 (33) @ If B = 0, w hav It follows that (33) holds if @q @ = @q 2 xld B @ xl @2 q @ 2 + C 00 ( ) C( 00 ) (xl)2 @q 2 + xl @2 q 3 @ @ 2 W dnot by P th lvl of liability that maximiss th platform s pro t and by W lvl of liability that maximiss social wlfar. W hav th @W @ P = @ @ P + @ B @ P 0 = @W @ W If W is concav in, it follows that P W. Appndix K Th rol of intrchang fs. In this sction, w look at th impact of mrchants liability on pro t-maximising and wlfar maximising intrchang fs. As con- 46

sumrs bar no liability on fraudulnt transactions, mrchants invstmnts in fraud dtction tchnologis do not dpnd on th transaction fs that ar paid by th usrs. W hav b = m + xl(1 q ) + C ( ); whr solvs (4). As m = a + c A, and from th nvlop thorm, w hav d b da = 1 As th acquirrs mak zro pro t, banks joint pro t is qual to th issurs pro t, I = (f (c I a) + a c I (1 )xl(1 q ))D B (f)d ( b ) Not that th lvl of invstmnt that is chosn by th mrchants dpnds nithr on th transaction fs nor on th intrchang f. From th nvlop thorm, as @ I @f = 0, w hav f d I da = @ I @a + @ I @m @m @a olving for th rst-ordr condition of pro t maximisation yilds d I df da = da + 1 D B (f)d ( b ) (f (c I a) + a c I (1 )xl(1 q ))D B (f)h ( b ) Th scond-ordr condition is d 2 I da 2 d 2 = D f df B da 2 D 2 da + 1 h ( b ) (f (c I a) + a c I (1 )xl(1 q ))h 0 ( b ) 0 inc h 0 ( b ) 0, a su cint condition for th scond-ordr condition to hold is that d 2 f =da 2 0. For instanc, th scond-ordr condition holds with uniforms distributions on [0; 1] for b B and b and if th issur is a monopolist, as in this cas f = (1 a + c I )=2. In an intrior solution, th pro t maximising intrchang f is implicitly d nd by (f (c I a P ) + a P c I (1 )xl(1 q )) = D ( b ) h ( b ) df da a P + 1 Th pro t maximising intrchang f r cts a trad-o btwn incrasing th transaction volum by ncouraging mrchants to accpt th EPI and maximising th margin pr transaction. For instanc, with uniforms distributions on [0; 1] for b B and b and if th issur is a monopolist, 47

w hav a P = c I c A xl(1 q ) C ( ) 2 + (1 )xl(1 q ) From th implicit function thorm, w hav da P d = d 2 1 I @ 2 I da 2 ; @a@ whr @ 2 I df = D B xl(1 q ) @a@ da D B (f)h ( b ) a P + 1 h ( b ) + (f + a P c I (1 )xl(1 q ))h 0 ( b ) xl(1 q ) + (1 ) dq d d d As h 0 ( b ) 0, @f=@a 0, and @ =@ 0, w hav @ 2 I @a@ 0 It follows that th intrchang f dcrass with th lvl of liability that is born by mrchants. If consumrs could b hld liabl for fraudulnt transactions, th mrchants fraud prvntion ort could dpnd on th intrchang f. In that cas, th paymnt platform could dcid to lowr th intrchang f to ncourag mrchants to invst in fraud prvntion tchnologis. W now show in an xampl that if th intrchang f is chosn by a rgulator at stag 1, th paymnt platform can ract at stag 2 by adjusting th lvl of mrchant liability. For instanc, assum that b B and b ar uniformly distributd on [0; 1] and that th issur is a monopolist. In this cas, w hav that f = (1 lvl of liability for mrchants that maximiss th issur pro t, a + c I )=2. Th paymnt plaform chooss th I = ( 1 + a 2 c I (1 )xl(1 q ))( 1 + a c I )(1 a c A xl(1 q ) C ( 2 )) With a linar probability such that q( ) =, whr 0 < 1, with a cost fonction such that C ( ) = k 2 =2, w hav = xl k olving for th rst-ordr condition of pro t maximisation with rspct to, from th 48

nvlop thorm, w obtain that 1 + a (xlr + (1 )xl) [1 a c A xlr C ( )] = 2 ci (1 )xlr xlr ; whr r = 1 q. Th scond-ordr condition writs (1 + xl)d (b m ) 2(xL) 2 (1 q )(1 q + (1 )) (1 )(xl) 2 (1 q ) 0. From th implicit function thorm, @ =@a has th sam sign as @ 2 I =@a@. W hav @ 2 I @a@ = 2xL(1 q ) (1 )xl 0 W nd that th liability lvl that is chosn by th paymnt platform dcrass with th lvl of intrchang f. This xampl shows that if a rgulator chooss a lvl of intrchang f that is quit low, th platform can ract by incrasing th lvl of liability that is born by mrchants. 49