Anna Creti

Marianne Verdier
email: marianne.verdier@u-paris10.fr.

Fraud, Investments and Liability Regimes in Payment Platforms

Keywords: G21, L31, L42 Payment card systems, interchange fees, two-sided markets, fraud, liability

In this paper, we discuss how fraud liability regimes impact the price structure that is chosen by a monopolistic payment platform, in a setting where merchants can invest in fraud detection technologies. We show that liability allocation rules distort the price structure charged by platforms or banks to consumers and merchants with respect to a case where such a responsibility regime is not implemented. We determine the allocation of fraud losses between the payment platform and the merchants that maximises the platform's pro…t and we compare it to the allocation that maximises social welfare.

Introduction:

The development of electronic data exchange in the banking industry has generated an increase in fraud and cybercrime. For instance, in the United-States, according to the Consumer Sentinel Network (CSN), 1.2 million complaints of consumer fraud have been recorded in 2008. [START_REF] Arango | Merchants Acceptance, Costs and Perception of Retail Payments: A Canadian Survey[END_REF] As a consequence, banks can make substantial losses because of fraudulent use of payment cards, which di¤er across countries and payment systems (See table 1).

Table 1: Loss rate per $100 payment card transaction value in several countries [START_REF] Bedre-Defolie | Pricing Payment Cards[END_REF]

Country

Spain Australia France UK US which are public laws (e.g. TILA and regulation Z in the United-States) [START_REF] Brown | Toward an Economic Theory of Liability[END_REF] , and by the 'zero liability rule', which has been privately adopted by several payment networks. It follows that, in most payment systems, the burden of fraud losses is shared between banks or platforms and merchants. [START_REF] Cooter | A Theory of Loss Allocation for Consumer Payments[END_REF] The allocation of liability between banks and merchants generally depends on private rules that are chosen by payment platforms. Some networks may even use liability rules to provide merchants with incentives to adopt new technologies. For instance, MasterCard and Visa used liability shift measures to induce merchants to adopt fraud prevention technologies on the internet (MasterCard SecureCode TM and Visa 3-D Secure TM respectively). [START_REF] Chakravorti | Externalities in Payment Card Networks: Theory and Evidence[END_REF] Interestingly, if the merchant implements the 3-D Secure TM technology, the issuer becomes liable for fraud losses for all eCommerce transactions that went through the 3-D Secure TM process. Understanding the impact of fraud losses on payment systems has become a major challenge of the banking industry.

To adress this issue, we consider a monopolistic proprietary payment platform that provides an electronic payment instrument to risk neutral consumers and merchants. Consumers and merchants decide whether or not to adopt the electronic payment instrument based on the price of the payment instrument and on the expected loss that they incur in case of fraudulent transaction. In our setting, we use a broad de…nition of fraud, which is the use of an electronic payment instrument (or its information) by a person other than its owner, to obtain goods and services without authority for such use. [START_REF] Parisi | The Economics of Tort Law: A précis[END_REF] Fraudulent transactions are detected with some probability that is positively related to merchants'investments in fraud prevention technologies.

If a fraud is detected, then the participants do not make losses.

Our results highlight the following trade-o¤ for the payment platform. When the level of liability for merchants increases, the number of merchants who accept the electronic payment instrument falls, but merchants tend to invest more in fraud detection technologies, which increases consumers'willingness to use the electronic payment method. The payment platform trades o¤ between increasing the level of liability to minimize the expected loss on fraudulent [START_REF] Brown | Toward an Economic Theory of Liability[END_REF] For a comparison of consumer protection laws across various countries, see Appendix A. [START_REF] Cooter | A Theory of Loss Allocation for Consumer Payments[END_REF] For instance, in France, according to the "Observatoire de la sécurité des cartes de paiement", fraud losses have been shared in 2009 between banks (41.1%) and merchants (53.5%). Merchants have been held liable mainly for fraud on internet transactions. Consumers were held liable for only 2.3% of the fraud losses. According to Furletti (2005), in the United-States, "consumers of credit cards are shielded from nearly $3 billion in fraud losses each year". According to a more recent survey conducted by the Federal Reserve Board in the US, in 2009, across all types of debit card transactions, 57% of fraud losses were borne by issuers and 43% were borne by merchants. Source: Federal Reserve Register, vol. 75 n 248, 2010. [START_REF] Chakravorti | Externalities in Payment Card Networks: Theory and Evidence[END_REF] These services provide Internet merchants with the ability to verify their consumers'true identities through a secure, electronic, non 'face-to-face'authentication process. [START_REF] Parisi | The Economics of Tort Law: A précis[END_REF] Our model does not enable us to distinguish which type of fraud is implemented by the fraudster. We consider any type of fraud that can be empeded by merchant investment. For instance, data breaches and phishing do not depend on merchants'investments (rather on platform's investment). On the contrary, identity theft can be avoided by the merchant's e¤ort to verify the consumer's identity. transaction and maximizing the transaction volume by encouraging merchants and consumers to accept the electronic payment instrument. In the short term, the existence of a fraud liability regime a¤ects the pricing structure of the payments system. With respect to the standard price structure in two-sided markets (Rochet-Tirole, 2003), the price structure that we obtain takes into account the platform's trade-o¤ between maximising its pro…t and minimizing the expected loss on fraudulent transactions. If the zero liability rule for consumers applies, the allocation of fraud losses that is chosen by the payment platform maximises social welfare if the detection probability is strictly increasing with the fraud prevention e¤ort. However, in other cases, liability regimes can be used by monopolistic payment platforms to extract rents from merchants, as it enables them to charge higher prices. We also …nd that our welfare result does not hold if investments are shared between the platform and the merchants. In this case, the payment platform trades o¤ between providing merchants with incentives to invest in fraud detection technologies and choosing to make itself the fraud prevention e¤ort. The payment platform may choose a level of liability for merchants that exceeds the social optimum, so as to extract the rents that the merchants obtain when the platform invests.

We also relax the assumption that merchants are risk neutral. This assumption is critical to obtain that merchants invest more in self prevention when their share of fraud losses increases.

We show that our welfare result under the zero liability rule does not hold if merchants are risk averse.

Finally, we determine the incidence of the liability regime on the choice of the interchange fee. We …nd that, if the issuers are imperfectly competitive, whereas the acquirers are perfectly competitive, the pro…t maximising interchange fee decreases with the level of liability that is borne by merchants.

The rest of the paper is organized as follows. In Section 2, we summarize the literature related to our study. In Section 3, we develop a theoretical model to analyze the optimal allocation of fraud losses between the payment platform and the merchants. In section 4, we determine the pro…t maximising allocation of fraud losses. In section 5, we study the welfare maximising allocation of fraud losses. In section 6, we extend the model by studying the optimal allocation of investments between the payment platform and the merchants. In section 7, we relax the assumption that merchants are risk neutral. In section 8, we analyze the role of interchange fees. Finally, we conclude.

Related Literature

To our knowledge, this paper is the …rst attempt to model fraud detection technologies and liability regimes in the literature on payment systems. Our approach thus relies on three di¤erent strands of literature: the literature on payment platforms, on investment in two-sided markets, and …nally the literature on liability issues in law and economics.

Most papers on payment systems focus on explaining the divergence between the pro…t maximising price structure that is charged by payment platforms and the price structure that maximises social welfare (see Chakravorti (2010) for a review). In particular, several papers aim at determining whether payment platforms charge excessive interchange fees when they maximise banks' joint pro…t (as surveyed by Verdier, 2011). Our paper contributes to this literature by extending Rochet-Tirole (2003) to study how the allocation of the expected fraud loss between the platform and the merchants changes the pro…t-maximising price structure.

The literature on investment in two-sided markets is scarce. For instance, Verdier (2010) determines the optimal price structure of a payment card platform in which monopolistic banks can invest to improve the quality of the payment service. Her model studies how investments should be allocated between monopolistic banks in four-party payment platforms. In particular, she …nds that a reduction of interchange fees can improve the allocation of investments by encouraging acquirers to invest, when investments increase consumers' demand. Our model departs from that paper, as we consider a monopolistic proprietary payment platform, and we focus on the optimal allocation of fraud losses between the platform and the merchants. The four-party model is used in section 8 of our paper, where we show that the pro…t maximizing interchange fee decreases with the level of liability borne by merchants. The only paper that considers merchants'investments in two-sided platforms is the paper by Peitz and Belle ‡amme (2010), who study the e¤ect of the intermediation mode (for-pro…t competing platforms versus free access) on sellers' investment, in a model where sellers' investment increase the buyers' utility of belonging to the platform. They show that for-pro…t intermediation may lead to overinvestment when innovations increase buyers' surplus, because competing intermediaries react by lowering the access fees on the seller side. Our focus is di¤erent from theirs, as we take the intermediation mode as given, and focus on the impact of liability rules on sellers' investments incentives.

Our model is also related to the vast literature on tort law whose main goal is to enhance socially optimal decisions on the level of precaution (Brown, 1973). More precisely, our framework shares the same background of ex post liability regimes, while neglecting the problem of non compliance and enforcement of ex ante regulation. [START_REF] Dionne | Self-Insurance, Self-Protection and Increased Risk-Aversion[END_REF] In this context, strict liability allocates the losses to the injurer by entitling the victim to compensation, whereas no liability allocates the losses to the victim, by denying the right to compensation (Landes and Posner, 1987). Indeed, liability provides incentives for precaution. [START_REF] Douglas | An examination of the fraud liability shift in consumer cardbased payment systems[END_REF] We extend this argument to the case of a three party system interrelated through network e¤ects, which is uncommon in law and economics models. In fact, in our framework, the price of a transaction implies not only a choice for a consumer, which generates a loss risk (as also pointed out by law scholars like Cooter and Robin, 1987), but also a pricing strategy by the platform and an incentive for the merchant to invest in fraud detection.

The model

We build a model in which a monopolistic payment platform o¤ers an Electronic Payment Instrument (hereafter the EPI) to consumers and merchants. We extend Rochet and Tirole (2003) along several dimensions. We consider that there is an exogenous probability that the EPI is fraudulently used, in a setting where merchants can invest in fraud detection technologies.

We de…ne fraud as the use of an electronic payment instrument (or its information) by a person other than its owner, to obtain goods and services without authority for such use. The fraud entails a lump sum loss which does not depend on the transaction value. Our framework enables us to determine how fraud liability should be allocated between the participants to maximise the platform's pro…t. It also enables us to compare the private optimal allocation to the one that maximises social welfare.

Payment system and allocation of fraud: A monopolistic payment platform provides an electronic payment instrument (e.g. the payment card) to consumers and merchants. The marginal cost of processing a transaction is denoted by c. Consumers and merchants pay transaction fees to the platform, which are denoted by f and m respectively.

When consumers use the EPI, there is an exogenous probability x 2 (0; 1) that the payment instrument is intercepted by fraudsters. [START_REF] Ehlrich | Market Insurance, Self-Insurance & Self-Protection[END_REF] There is also a probability q 2 [0; 1] that the fraud [START_REF] Dionne | Self-Insurance, Self-Protection and Increased Risk-Aversion[END_REF] Ex ante regulation is meant to prevent accidents from occurring through the enforcement of minimum safety standards or compliance restrictions. Ex post liability, exercised after an accident has occurred, is a legal device that enables victims to sue for damages, forcing injurers to internalize part of the harm they cause.

1 0 When both parties have to take precaution in order to avoid an accident, strict liability creates no incentives for victim precaution, while no liability would shift the entire residual liability on the victim, inducing optimal victim care. It follows that strict liability and no liability can give incentives to take e¢ cient precaution only to one party, respectively either the injurer or the victim (Dari-Mattiacci, Parisi, 2006). 1 1 The assumption that x is exogenous is made for simplicity. Indeed, endogenizing x would introduce another trade-o¤ for the merchant. Higher investments in fraud detection technologies have two e¤ects on hackers' incentives to fraud. On the one hand, higher investments in fraud increase the volume of transactions, which is detected, which depends on merchants' investments. If the fraud is not detected, all the participants to the transaction make an exogenous loss that we denote by L > 0. The loss is allocated between the consumer, the merchant and the payment platform as follows: the consumer (or buyer B) and the merchant (or seller S) bear respectively a share B and S of the loss, where S + B 2 [0; 1]. The rest of the loss, P = 1 ( S + B ), is borne by the payment platform. We assume that the parameter B is determined by public laws and we consider it as exogenous to the model. In particular, if B = 0, the zero liability rule applies for consumers. The parameter S is privately chosen by the payment platform. [START_REF] Federal | Debit Card Interchange Fees and Routing: Proposed Rule[END_REF] We choose to normalize the fraud on cash payments to zero. [START_REF] Furletti | The Laws, Regulation and Industry Practices that Protect Consumers who use Electronic Payment Systems: Policy Considerations[END_REF] Merchants: We consider local monopolist merchants that supply the same good to consumers.

The marginal cost of producing the good is denoted by d and the price of the good is denoted by p. The non-discrimination rule holds, such that a merchant cannot not charge a price that depends on the payment method. Merchants are risk neutral [START_REF] Gates | Payments Fraud: Perception Versus Reality -A conference Summary[END_REF] S (e S ) 0. 15 Merchant's investments increase the probability q that a fraudulent transaction is detected, that is, we assume that dq=de S > 0 for all e S > 0. We also assume that d 2 q=d 2 e S 0 for all e S 0 and that d 3 q=d 3 e S 0. [START_REF]Observatoire de la sécurité des cartes de paiement[END_REF] The amount invested in fraud detection technologies is common knowledge, such that banks and consumers are aware of the security measures implemented by the merchants. [START_REF] Rochet | The Theory of Interchange Fees: A synthesis of recent contributions[END_REF] By accepting the EPI, each merchant obtains a transaction bene…t b S > 0. As in Rochet and increases the hackers'incentives to commit fraud. On the other hand, higher investments increase the probability that a fraud is detected, which may discourage hackers to commit fraud. 1 2 In our model, we do not study how the losses are allocated between banks and the payment platform. In practice, payment platforms design rules to allocate the losses between issuing and acquiring banks and also to allocate the losses between banks and the platform itself. This issue would deserve a separate study. We reintroduce banks in section 4 and choose to focus on the role of interchange fees in fraud prevention issues. 1 3 Introducing the probability that a fraudulent payment is made by cash would not change the trade-o¤s that we highlight in our model. We would only have to modify assumption (A2) to take into account the losses that are due to fraud on cash payments. 1 4 The assumption that merchants are risk neutral is discussed in section 7, where we assume that merchants are risk averse. 1 5 The literature on insurance markets makes a distinction between "self insurance" and "self protection". In our setting, merchants invest in "self protecting" measures but cannot reduce the amount of the expected loss. Under the assumption that merchants are risk neutral, both investments are equivalent as L and (1 q) play the same role in the merchant's pro…t. This is not the case if merchants are risk averse. 1 6 The assumption that d 2 q=d 2 eS 0 ensures that the second-order condition is veri…ed when the merchant chooses its level of investment. The assumption that d 3 q=d 3 eS 0 ensures that the second-order condition is veri…ed when the platform maximises its pro…t. 1 7 Merchants can inform consumers about their e¤orts to …ght fraud. For instance, online sellers can communicate on the use of a software or a speci…c technology that improves consumer authentication. Tirole (2003), merchants are heterogenous over their transaction bene…t b S which is distributed over b S ; b S according to the probability density h S and the cumulative H S . We assume that h 0 S 0 to ensure demand (quasi) concavity. We normalize the bene…t of accepting cash to zero.

The merchant pays a fee m to the payment platform each time a consumer pays with the EPI and bears the cost of investing in fraud detection technologies.

Consumers: Consumers obtain a surplus v > 0 if they buy the good that is supplied by the merchants. Each consumer is randomly matched to one merchant and may choose between paying cash or paying with the EPI, if the merchant accepts the EPI. We assume that consumers are risk neutral and that they can observe merchants'investment in fraud detection technologies before deciding whether or not to use the EPI. [START_REF] Rochet | Cooperation Among Competitors: The Economics of Payment Card Associations[END_REF] If he pays with the EPI, the consumer obtains a transaction bene…t b B which is distributed over b B ; b B according to the probability density h B and the cumulative H B . We assume that h 0 B 0 for concavity to hold. The consumer pays a fee f to the payment platform, and anticipates that, with some probability x(1 q), he bears a share B of the loss L, because the EPI is fraudulently used without being detected. The bene…t of paying cash is normalized to zero. It follows that, if a consumer can choose between cash and the EPI, under the nondiscrimination rule, a consumer wishes to use the EPI if and only if

b B f B x(1 q)L 0, (1)
that is, if his transaction bene…t is higher than the cost of the transaction fee and the expected fraud loss.

Additional assumptions:

(A1) The hazard rate

h B (x) 1 H B (x) is increasing. (A2) In equilibrium, min v d; 1 xL B h B (f + B (1 q)xL) 1 H B (f + B (1 q)xL) .
Assumptions (A1) is similar to Wright (2002) and standard in the literature on payment cards. Assumption (A2) ensures that:

1 8 Our parameter B could be also used to understand the impact of these assumptions. First, assume that consumers are risk averse, and that their utility function takes the form of a constant risk aversion function (CARA). An increase in consumers'aversion for risk implies a reduction of the elasticity of the consumer demand to the fraud prevention e¤ort. An approximation of this e¤ect in our model would be to reduce B . Second, if consumers do not observe the merchants' investments in fraud detection technologies, their demand is inelastic to their investment e¤ort. This situation is captured in our model by setting B = 0. Therefore, more generally, B could be interpreted as a parameter that impacts the consumer's demand sensitivity to the fraud prevention e¤ort.

(i) consumers obtain a much higher surplus from buying the good that from making a transaction with the Electronic Payment Instrument. [START_REF] Tirole | Platform Competition in Two-Sided Markets[END_REF] (ii) the amount of the expected share of the fraud loss for consumers is not too high, such that it does not exceed the surplus that consumers obtain from making a transaction. [START_REF] Rochet | Two-sided markets: a progress report[END_REF]

Timing:

The timing of the game is as follows:

1. The platform chooses the liability level S and the transaction fees f and m.

2. The merchants decide whether or not to accept the EPI and how much to invest in fraud detection technologies. They also choose the price of the good p.

3. Each consumer is matched randomly to one merchant. Consumers decide on whether or not to buy the good and how to pay for the good (either by cash or with the EPI).

In the following section, we look for the subgame perfect equilibrium and solve the game by backward induction. 4 The equilibrium: Let us start by the case in which the merchant does not accept the EPI. If the merchant sets p v, the consumer wishes to buy the good by paying cash, as his surplus v p is positive.

Otherwise, he does not buy the good. Now consider the case in which the merchant accepts the EPI. If the merchant sets p v, the consumer wishes to buy the good, as he obtains at least a positive surplus if he pays cash.

He decides to use the EPI if his expected utility is higher than if he pays cash. It follows that, 1 9 Part (i) of Assumption (A2) is standard in the literature (see Wright (2002)). Formally, this corresponds to the Assumption that v d hB(f

+ B (1 q)xL)=(1 HB(f + B (1 q)xL).
2 0 Part (ii) of Assumption (A2) is new, as our paper is the …rst to model the incidence of fraud losses on consumers and merchants' payment choices and platform prices. Formally, this corresponds to the Assumption that:

(1=xL B ) hB(f + B (1 q)xL)=(1 HB(f + B (1 q)xL).
if p v, a consumer wishes to use the 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 the merchant sets p > v, the consumer never uses cash. The consumer buys the good and pays with the EPI if and only if

v p + b B f B (1 q)xL 0.
We denote by D B the probability that a consumer wishes to use the EPI. Considering consumers' heterogeneity, it follows from the previous analysis that

D B = 8 <
:

1 H B (f + B (1 q)xL) if p v 1 H B (f + B (1 q)xL + p v) if p > v.
Note that the probability that the consumer wishes to use the EPI decreases with the transaction fee, the consumer's liability, the expected amount of fraud loss, but increases with the probability that the fraud is detected.

Stage 2: EPI acceptance and investments in fraud detection

Prices and card acceptance condition

We now determine the price that is chosen by each merchant, along with the decision to accept the EPI and invest in fraud detection technologies. We start by showing that, because of assumptions (A1) and (A2), the pro…t of a merchant who accepts the EPI is maximised when he sets a price such that cash-users are not excluded from the market. It follows that merchants who accept the EPI and merchants who do not accept the EPI choose the same price. This enables us to derive the EPI acceptance condition.

Lemma 1 Each monopolistic merchant maximises its pro…t by setting p = v.

Proof. See Appendix B.

We are now able to derive the condition under which a merchant accepts the electronic payment instrument. A merchant accepts the EPI if he makes more pro…t by doing so, that is if

v d + D B (f + B (1 q)xL)(b S S x(1 q)L m C S (e S )) v d. Since D B (f + B (1 q)xL) 0, this condition is equivalent to b S S x(1 q)L m C S (e S ) 0. (2)
Note that a merchant does not accept the EPI if the merchant fee is high or if the amount of the expected fraud loss is high.

Investment in fraud detection technologies

A merchant that accepts the EPI can invest in fraud detection technologies. The amount of investment in fraud detection technologies, which we denote by e S , maximises the merchant's pro…t under the constraint that the merchant accepts the EPI.

S xL dq de S e S C 0 S (e S ) = [b S S x(1 q)L m C S (e S )] B j e S e S ; (3)
where B = dD B =de S D B =e S denotes the elasticity of the consumer's demand to the investment e¤ ort.

Proof. See Appendix C.

The merchant chooses its fraud prevention e¤ort so as to equalize the marginal bene…ts of investments in fraud detection technologies and the marginal cost of investments. The marginal bene…ts of investments are equal to the marginal gains from lower expected fraud losses (term S xL(dq=de S ) in ( 3)), and to the marginal bene…ts that are due to an increase in the volume

of electronic transactions (term [b S S x(1 q)L m C S (e S )] ( B j e S =e S ) in (3)).
Let us detail each of the two e¤ects that will be referred to as the expected loss e¤ ect and the transaction volume e¤ ect. First, if the merchant invests in fraud detection technologies, this increases the probability that a fraudulent transaction is detected, and therefore, this reduces the amount of the expected loss that he has to bear when he accepts the EPI. The expected loss e¤ect has a positive impact on merchant's investments. Second, if consumers bear a positive share of fraud losses, the probability that a consumer wishes to use the EPI is impacted positively by the merchant's investments, as the expected loss decreases. The transaction volume e¤ect has also a positive impact on merchant's investments.

Remark that, because of the transaction volume e¤ect (if B 6 = 0), the merchants invest in fraud prevention technologies even if they bear no liability for fraud, that is if S = 0. In two-sided markets, the liability regime is not the only incentive that can be used to encourage merchant investment, as merchants care about the transaction volume, which is related to consumer demand. This e¤ect is not present in the literature on law and economics that we mentioned in section 2.

Note also that merchants exert a positive externality on the payment platform and on consumers if B 6 = 0, because their investment in fraud detection technologies reduces the amount of their expected fraud loss.

If B = 0, the zero liability rule applies for consumers. In this case, all merchants who accept the EPI invest the same amount in fraud detection technologies, which is implicitely de…ned by

S xL dq de S e S = C 0 S (e S ): (4)
As investments do not impact consumer demand, the transaction volume e¤ect is null under the zero liability rule. A merchant who obtains a higher transaction bene…t does not have higher investment incentives, as the marginal bene…ts obtained through a higher transaction volume are equal to zero.

Comparative statics

In Lemma 3, we give some comparative statics to explain how a merchant's investment in fraud detection technologies vary with the transaction fees, the liability levels and the bene…t that a merchant obtains of being paid with the electronic payment instrument.

Lemma 3 If B > 0, the merchant's investments in fraud detection technologies increase with the consumer liability, the consumer transaction fee, the merchant's transactional bene…t, and the merchant's liability, but they decrease with the merchant fee.

Proof. See Appendix D.

We proved in Lemma 2 that a merchant's investments in fraud detection technologies are chosen such that the marginal bene…ts are equal to the marginal costs of investments. If the merchant fee increases (resp. if the merchant's transactional bene…t increases), all other things being equal, the marginal bene…ts from investment decrease, because of a reduction of the transaction volume e¤ect. The merchant reacts by reducing its investments in fraud detection technologies.

If the merchant's liability increases, this increases the expected loss e¤ect, because the merchant has more to save when a fraud is detected, whereas this decreases the transaction volume e¤ect, as the merchant's margin per transaction is reduced. Under Assumption (A2), the …rst e¤ect dominates and the merchant reacts by increasing its investments in fraud detection technologies.

Moreover, if the consumer liability increases or if the consumer fee increases, this increases the transaction volume e¤ect, because the impact of merchant's investments on consumer demand increase. Therefore, the merchant's investments increase.

If the zero liability rule applies, from (4), the merchant's investments in fraud detection technologies do not depend on the transaction fees that are chosen by the payment platform.

They only depend on the merchant's liability and the expected loss. As when B > 0, they decrease with the merchant's liability and it can be shown that they decrease with the expected fraud loss.

In Lemma 4, we determine how the transaction fees and the liability levels impact the probability that a merchant accepts the electronic payment instrument.

Lemma 4

The probability that a merchant accepts the EPI decreases with the merchant fee, with the consumer fee and with the level of liability that is borne by merchants or by consumers.

Proof. See Appendix E.

A higher merchant fee lowers the transaction margin that the merchant obtains if he accepts the EPI, whereas it reduces the merchant's incentives to accept the EPI, which is a standard -and thus increases the probability that a consumer wishes to use the EPI. From assumption (A2), the …rst e¤ect dominates in our framework, and therefore, the probability that a merchant accepts the EPI decreases with his liability level.

Stage 1: Prices and liability levels

At the …rst stage, the payment platform choses the prices that maximise its pro…t,

P = (f + m c)V P EL P ,
where V P denotes the transaction volume, as follows:

V P = Z bs c b S h(b S )(1 H B (f + B xL(1 q ))db S ; (5)
EL P denotes the average expected loss, or:

EL P = P xL Z bs c b S (1 q )h(b S )(1 H B (f + B xL(1 q )))db S ; (6)
and q = q(e S ).

If B = 0, as q does not depend on b S , we have

EL P = P xL(1 q )V P : (7)
For all B 2 [0; 1], the transaction volume decreases with the consumer transaction fee and with the merchant fee. While this e¤ect is standard in the literature, another question arises in our framework, that is the impact of the transaction prices and the merchants'liability on the expected fraud loss that is borne by the payment platform.

Variations of the expected loss with the prices

We start by determining how the expected fraud loss is impacted by the choice of transaction fees and by the level of liability that is borne by merchants.

Proposition 1

The expected loss incurred by the payment platform on fraudulent transactions (EL P ) decreases with the consumer transaction fee and with the level of liability that is borne by merchants. EL P decreases with the merchant fee only if the elasticity of the merchant's e¤ ort to the merchant fee is small or if the elasticity of the merchant's demand to the merchant fee is high.

Proof. See Appendix F.

An increase in the consumer fee decreases the number of merchants who accept the EPI, whereas it increases merchants' investments in fraud detection technologies. It follows that a higher consumer fee decreases the expected loss that is incurred by the payment platform.

Moreover, a higher level of liability for merchants decreases the expected loss that is borne by the payment platform, as it decreases merchants'acceptance of the EPI, whereas it increases merchants'investment in fraud detection technologies.

An increase in the merchant fee has two e¤ects on the expected loss that is incurred by the payment platform. The higher the merchant fee, the lower the number of merchants who accept the EPI, and the lower the transaction volume. This e¤ect reduces the expected loss that is incurred by the payment platform. At the same time, a higher merchant fee decreases the merchants' investment in fraud detection technologies, which increases the expected loss borne by the payment platform. The impact of an increase in the merchant fee on the expected loss depends on how both e¤ects compensate each other.

The pro…t maximising price structure under exogenous liability regime

Proposition 2 gives the pro…t maximising price structure for a given level of merchants'liability.

Proposition 2

The pro…t maximising price structure re ‡ects the platform's trade-o¤ between balancing pro…ts between both sides of the market and minimizing the expected loss on fraudulent transactions. The total price is implicitely de…ned by

f + m c f = 1 " V B (f ) + @EL P =@f f @V P =@f ;
and the price structure veri…es

f m = 1 " V S (m ) + @EL P =@m m @V P =@M 1 " V B (f ) + @EL P =@f f @V P =@f ;
where " V B (f ) = (@V P =@f )(f =V P ) and " V S (m) = (@V P =@m)(m=V P ) denote the elasticity of the transaction volume to the consumer fee and the merchant fee respectively.

Proof. We denote by M P = f + m c the payment platform's gross margin. Assume that there is an interior solution. Solving for the …rst-order conditions of pro…t maximisation yields @ P @f = M P @V P @f + V P @EL P @f = 0; and @ P @m = M P @V P @m + V P @EL P @m = 0:

These equations can be rewritten as

f + m c f = V P f @V P =@f + @EL P =@f f @V =@f ; (8)
and

f + m c m = V P m@V P =@m + @EL P =@m m@V P =@m : (9)
Introducing the elasticities " V B (f ) = (@V P =@f )(f =V P ) and " V S (m) = (@V P =@m)(m=V P ) and dividing the …rst equation by the second equation yields the result of Proposition 2. In Appendix G-A, we show that the second-order conditions of pro…t maximisation are veri…ed if B = 0 and we assume that they hold if B 6 = 0.

It is interesting to compare the prices that we …nd in an interior solution with the prices obtained in the standard two-sided market monopoly pricing formula obtained by Rochet and Tirole (2003). Equations ( 8) and (9) show that with respect to the standard price structure in two-sided markets, the price structure that we obtain encompasses an additional term that takes into account the platform's trade-o¤ between maximising its pro…t and minimizing the expected loss on fraudulent transactions.

Notice that if the zero liability rule applies for consumers (that is if B = 0), from (7), the expected loss only depends on the transaction prices through the transaction volume. It follows that, in this case, the price structure is the same as the one obtained by Rochet and Tirole (2003), that is:

f m = " V B (f ) " V S (m)
; and the total price is implicitely de…ned by:

f + m c (1 S )xL(1 q ) f = 1 " V B (f ) : 4.3.

An Example under the zero liability rule for consumers

We consider for example the case of uniforms distribution on [0; 1] for b B and b S , with a cost function C S (e S ) = k S (e S ) 2 =2, a detection probability q(e S ) = e S , where > 0. We also assume that the zero liability rule applies for consumers. We prove in Appendix H that, if S > 0, the pro…t maximising transaction fees are

m = 1 3 (1 + c + xL(1 3 S )) + k S 3 S ( 1 + 2 S ) (e S ) 2 ; (10)
and

f = 1 3 (1 + c + xL) k S 6 S (2 S ) (e S ) 2 ; ( 11
)
where e S = S x L=k S . From ( 10) and ( 11), the consumer fee is higher than the merchant fee if S 6 = 0, as we have

f m = S xL(1 q ) + k S 2 (e S ) 2 0: (12)
If the demands are uniform and symmetric, in Rochet and Tirole (2003)'s model, the pro…t maximising transaction fees are such that f = m. Equation (12) shows that, if S > 0, the payment platform tends to lower the merchant fee to provide merchants with incentives to invest in fraud detection technologies. The price structure changes in favor of merchants. This is not necessarily the case if demands are not symmetric, or if B 6 = 0. If the zero liability rule does not apply for consumers, the payment platform can use the transaction prices on both sides of the market to encourage merchants to invest in fraud detection technologies, because of the transaction volume e¤ect that we highlighted in Lemma 2.

The pro…t maximising level of liability

We proceed by assuming that the payment platform has the opportunity to choose the merchant's level of liability at the same stage as the transaction prices. Thus, we start by determining how the merchant's level of liability impacts the plaform's pro…t. We know from Proposition 1 that the expected loss that is borne by the payment platform decreases with the level of liability borne by merchants. It remains to study how the level of liability borne by merchants impacts the transaction volume. We have

@V P @ S = @ b b S @ S h S ( b b S )(1 H B (f + B xL(1 q ))) | {z } Term I + Z bs c b S h S (b S ) @D B (f + B xL(1 q )) @ S db S | {z } : Term II (13)
The …rst term of ( 13) is negative. It re ‡ects the fact that fewer merchants accept the EPI when the level of liability that is borne by merchants increases. The second term of ( 13) is positive. It

shows that more consumers wish to pay with the EPI when merchants invest in fraud detection technologies. It follows that a higher level of liability for merchants has an ambiguous impact on the transaction volume. Note that if the elasticity of the merchants'demand to their liability level is small (that is, if term I is small), the transaction volume may increase with the merchants' level of liability. Moreover, if the zero liability rule applies for consumers, the second term of ( 13) is null, and the transaction volume decreases with the merchant's level of liability. Proposition 3 gives the pro…t maximising level of liability for merchants.

Proposition 3 A monopolistic payment platform chooses a level of liability for merchants that re ‡ects a trade-o¤ between minimizing the expected loss on fraudulent transactions and maximising the transaction volume. The interior solution for the pro…t maximising level of liability for merchants solves

(f + m c) @V P @ S (f ;m ) = @EL P @ S (f ;m ) ;
where (f ; m ) denote the pro…t-maximizing prices of Proposition 1. If the transaction volume increases with the liability level that is borne by merchants, there is a corner solution such that the payment platform lets the merchants bear all the losses.

Proof. The payment platform chooses the level of liability that maximises its pro…t. Solving for the …rst-order condition of pro…t maximisation yields @ P @ S = (f + m c) @V P @ S @EL P @ S :

In an interior solution, we have

(f + m c) @V P @ S = @EL P @ S :
From Proposition 1, we know that the expected loss decreases with the level of liability that is borne by merchants. It follows that, if the transaction volume increases with the level of liability borne by merchants, the pro…t maximising liability level is a corner solution, with the merchants bearing the maximum share of the loss.

In Appendix G-B, we show that the second-order conditions of pro…t maximisation are veri…ed if B = 0, and we assume that they hold if B 6 = 0.

Proposition 3 shows that the payment platform has an incentive to share the losses on fraudulent transactions with the merchants, as this encourages merchants to accept the electronic payment instrument, unless merchants' demand is inelastic to the level of liability. However, the choice of a liability regime is also a means for the payment platform to extract rents from merchants if the elasticity of the merchants'demand to the liability level is small. Proposition 4 explains how fraud losses are allocated by a pro…t maximising monopolistic plaform under the zero liability rule for consumers.

Proposition 4 Under the zero liability rule for consumers, if the detection probability is strictly increasing with S , the platform lets the merchants bear the maximum share of fraud losses.

Proof. See Appendix G-B.

Under the zero liability rule for consumers, the payment platform chooses a level of liability that maximises the probability of fraud detection, if there is an interior solution (See Appendix G-B). However, under our assumptions, the probability to detect a fraudulent transaction is strictly increasing with the merchants'investment e¤ort. Since the merchants'e¤ort is strictly increasing with their share of fraud losses, S , the probability of fraud detection is strictly increasing with S . Therefore, in our model, there is a corner solution under the zero liability rule, such that the payment platform lets the merchants bear all the losses. This result does not hold if the merchants'e¤ort vary non monotonically with S .

In for fraud, they pay a share of fraud losses through the transaction fees, which is not explicitely de…ned through a liability regime. This example also shows how transaction prices vary with the fraud rate. If S = 1, from ( 10) and ( 11), we have

@m @x S =1 = 2L 3 (1 q ) 0; and @f @x S =1 = L 3 (1 q ) 0:
The merchant fee decreases with the fraud rate in our example when the merchants'share of fraud losses is high, whereas the consumer fee increases with the fraud rate. The total price decreases with the cost of fraud [START_REF] Sullivan | The Changing Nature of U.S Card Payment Fraud: Industry and Public Policy Options[END_REF] , whereas the distortion in the price structure increases with the fraud rate and the fraud losses. Proposition 5 shows that this result is very general under the zero liability rule for consumers.

Proposition 5 Under the zero liability rule, if the detection probability is strictly increasing with S , the merchant fee and the consumer fee vary in opposite directions with the fraud rate.

Proof. See Appendix G-C.

Welfare maximising liability levels

To study welfare maximizing liability levels, we assume that the merchant's level of liability is decided by a social planner at the …rst stage, who maximises the sum of the platform's pro…t, the consumer surplus and the merchant surplus. Then, the payment platform chooses the transaction fees at the following stage. Our aim is to compare the pro…t maximising level of liability for merchants, which is chosen by the payment platform, to the welfare maximising level of liability for merchants.

The welfare maximising liability level under the zero liability rule for consumers

We start by analyzing the simple case in which consumers bear zero liability on fraudulent transactions. For this purpose, we need to determine how the liability level borne by merchants impacts the transaction fees that are chosen by the payment platform.

Lemma 5 If the zero liability rule applies for consumers, the transaction fees chosen by the payment platform decrease with the level of liability that is borne by merchants.

Proof. See Appendix I.

A higher level of liability for merchants provides the payment platform with incentives to lower its prices on both sides of the market, if the zero liability rule applies for consumers.

The payment platform's pricing strategy re ‡ects a trade-o¤ between increasing its margin and increasing the transaction volume. A higher level of liability for merchants has two e¤ects on this trade-o¤. First, it decreases the expected loss on fraudulent transactions, which amounts to a reduction of the platform's marginal cost. The platform bears a lower share of fraud losses, while merchants obtain higher investment incentives. This marginal cost reduction increases the platform's incentives to lower its prices. Second, a higher level of liability for merchants reduces the transaction volume, as fewer merchants adopt the electronic payment instrument, which reduces the platform's incentives to increase its prices. Therefore, when the level of liability for merchants increases, the transaction fees paid by consumers and merchants fall. The payment platform loses some rents on both sides of the market, but this loss is compensated by higher rent extraction through the liability regime, which provides merchants with higher investment incentives.

We are now able to compare the pro…t maximising level of liability and the welfare maximising level of liability for merchants if consumers do not bear any liability for fraudulent transactions. We assume that social welfare is a concave function of the transaction fees. [START_REF] Verdier | Interchange Fees and Incentives to Invest in Quality of a Payment Card System[END_REF] Proposition 6 Under the zero liability rule for consumers, if social welfare is a concave function of S , the pro…t maximising level of liability for merchants is lower than (or equal to) the welfare maximising level of liability.

Proof. See Appendix J-B.

Corollary 1 If the detection probability is strictly increasing in S , the pro…t maximizing and the welfare maximizing levels of liability are equal under the zero liability rule for consumers.

We showed in Lemma 5 that the transaction fees paid by the users decrease with the level of liability that is borne by merchants. A direct consequence of Lemma 5 is that consumer and merchant surplus increase when merchants' liability increase. It follows that, from the point of view of total user surplus maximisation, it is socially optimal to let the merchants bear the maximum liability on fraudulent transactions. However, if the regulator takes into account the payment platform's pro…t, the welfare maximising level of liability for merchants is not necessarily equal to one, except if consumers are held liable for fraud.

The payment platform does not place enough liability on merchants to maximise social welfare, except in the case where it is maximises its pro…t by letting the merchants bear the maximum liability on fraudulent transactions. This is because the payment platform internalizes imperfectly the impact of the liability regimes on consumer and merchant surplus. In our model, since our assumptions imply that the detection probability is strictly increasing with S , the platform lets the merchant bear all the liability for fraud under the zero liability rule. Therefore, in this case, the welfare maximising level of liability is equal to the pro…t maximising level of liability. However, the welfare maximising level of liability could be strictly higher than the pro…t maximising level of liability if the merchants' e¤ort could vary non monotonically with the share of fraud losses (that is, if Assumption A2 was lifted). Our result is also driven by the assumption that the probability to detect a fraudulent transaction only depends on merchants' investment. In section 6, we prove that the prices may increase with the level of liability borne by merchants if the investments are shared between the payment platform and the merchants.

The welfare maximising level of liability if consumers bear some liability for fraud

If consumers bear some liability for fraud, the plaform's prices do not necessarily decrease with the level of liability borne by merchants. The platform can now use the transaction fees to impact merchants'incentives to invest in fraud detection technologies.

Platform' s investments under the zero liability rule for consumers

We analyze if our welfare result under the zero liability rule holds in an extension of the model that allows the payment platform to invest an amount e P in fraud detection technologies.

The platform incurs a cost C P (e P ) per transaction, where C P is a convex cost function. The probability to detect a fraudulent transaction, which we now denote by q(e S ; e P ), increases with the platform's investments, that is @q=@e P 0. The platform chooses its level of investment at the same stage as the prices, and merchants are able to observe this decision before deciding whether or not to accept the electronic payment instrument. In a supplementary note, which is available upon authors' request, we show that the welfare result obtained under the zero liability rule does not hold when the platform's investments are taken into account. [START_REF] Verdier | Interchange Fees in Payment Card Systems: a Survey of the literature[END_REF] In this situation, the prices chosen by the payment platform do not necessarily decrease with the level of liability borne by merchants.

The intuition of this result is the following. The platform now trades o¤ between providing merchants with incentives to invest in fraud detection technologies and choosing to make itself the fraud prevention e¤ort. The result of this trade-o¤ is impacted by the relative cost of investment for the platform and the merchants, and by the fact that their technological choices may be either independent or may in ‡uence each other.

An example with independent investments

We start by analyzing the case in which the merchants'investments and the platform's investment are independent. This case can be illustrated by assuming for instance that q(e S ; e P ) is linear and separable in e S and e P , that is q(e S ; e P ) = de S + he P ;

where d 0 and h 0. To understand better the impact of the platform's investments on our welfare result under the zero liability rule for consumers, we specify quadratic investment cost functions for the merchants and the platform, such that C S (e S ) = k S (e S )2 =2 and C P (e P ) = k P (e P ) 2 =2, where k S > 0 and k P > 0. We also assume uniform distributions on [0; 1] for b S and b B . Under these assumptions, at the equilibrium of stage 2, each merchant invests an amount e S = dLpx S =k S in fraud detection technologies. At stage 1, if S > 0, the prices chosen by the platform are

f = 1 3 (1 + c + Lx) k P 6 (e P ) 2 k S 6 (e S ) 2 2 S S ; m = 1 3 (1 + c + Lx(1 3 S )) k P 6 (e P ) 2 (1 6 S ) + k S 3 (e S ) 2 2 S 1 S ;
where (e P ) = hLp=k P denotes the pro…t maximising investment of the platform. [START_REF]Optimal Payment Card Systems[END_REF] This illustration shows that the consumer fee is not necessarily higher than the merchant fee, unlike our previous example with uniform distributions under the zero liability rule. We have

@f @ S = d 2 L 2 x 2 (1 S ) 3k S ; (14)
and

@m @ S = Lx + (xL) 2 3k P k S d 2 k P (4 S 1) + 3h 2 k S : (15)
From ( 14), the consumer fee decreases with the level of liability borne by merchants, whereas from (15), the merchant fee increases with the level of liability borne by merchants if the investment cost of the platform is su¢ ciently low and if the platform's contribution to increase the detection probability is high enough (through the parameter h). This result can be explained as follows. A higher level of liability for merchants has two e¤ects on the platform's incentives to invest in fraud detection technologies. First, it decreases merchants'acceptance, which reduces the marginal bene…ts of investing in fraud detection technologies for the payment platform.

Second, it reduces fraud losses, which amounts to a reduction of the platform's marginal cost.

This e¤ect impacts the platform's investments in two opposite directions. On the one hand, it decreases the platform's incentives to invest, as the platform bears a lower share of fraud losses. On the other hand, it increases the platform's margin per transaction, which can result in higher investment incentives.

From the point of view of merchants, an increase in their liability raises the value of the platform's investments, as this improves the quality of service provided by the platform. The payment platform trades o¤ between extracting this additional surplus from the merchants through the merchant fee and increasing the transaction volume through lower fees. The variation of the merchant fee with the merchants'share of fraud losses re ‡ects this trade-o¤, which is not present on the consumer side.

Related investments

We now analyze the case in which the platform's decision to invest in fraud detection technologies impacts positively the merchant's investment e¤ort. This case can be illustrated by assuming for instance that q(e S ; e P ) is a product of the merchant's investment e¤ort and the platform's investment e¤ort, that is q(e S ; e P ) = de S e P + he P ;

where d 0 and h 0. At the equilibrium of stage 2, the merchant's investments in fraud detection technologies are positively related to the platform's prevention e¤ort, and we have e S = de P xL S =k S . Therefore, the platform takes into account this e¤ect in its trade-o¤ between providing merchants with investments incentives and choosing to bear itself the fraud prevention e¤ort. At stage 1, the platform chooses the transaction fees

f = 1 3 (1 + c + Lx) hLxe P 6 ; m = 1 3 (1 + c + Lx(1 3 S )) + k S 2 (e S ) 2 hLx 3 e P (1 3 S ) + hLxe P 6 ;
where e P = hk S Lx=(k

P k S d 2 L 2 x 2 S (2 S )) if k P k S d 2 L 2 x 2 S (2 S ) > 0.
The platform's level of investment increases with the share of liability borne by merchants. This result can be explained as follows. An increase in the level of liability borne by merchants amounts to a reduction of the platform's marginal cost, which results in higher investment incentives for the platform. A higher level of liability also raises the impact of the platform's e¤ort on merchants' investment incentives, which provides the platform with an additional incentive to increase its level of prevention e¤ort. As in our previous example, the consumer fee decreases with the level of liability borne by merchants, whereas the merchant fee varies non-monotonically with the share of fraud losses.

Risk averse merchants

If merchants are risk averse, they do not necessarily invest more in fraud prevention, because they trade o¤ between reducing the probability that a fraud occurs and reducing their investment cost. The result that risk averse agents do not exert more e¤ort for self protection is standard in the literature on insurance markets (see for instance Dionne and Eeckhoudt, 1985). [START_REF]Determinants of Optimal Interchange Fees in Payment Systems[END_REF] On the contrary, risk averse agents who invest in self insurance increase their level of e¤ort. However, we do not study this possibility in our setting, as merchants cannot reduce the amount of fraud losses.

Lifting the assumption that merchants are risk neutral impacts our results under the zero liability rule for consumers. As in the case where B 6 = 0, merchants care about the volume of transactions paid with the EPI. Therefore, merchant's investment are related to the price paid by consumers. Furthermore, merchants'investments may either increase or decrease with the share of fraud losses. For instance, with a CARA utility function, merchant's investments increase with their share of fraud losses if their coe¢ cient of absolute risk aversion is su¢ ciently small, whereas they may decrease with their share of fraud losses if their coe¢ cient of absolute risk aversion is high, and if the fraud prevention e¤ort has a limited impact on the probability to detect a fraudulent transaction (See Appendix K). However, with other utility functions, the incentive to invest in fraud prevention may be as strong for risk preferrers as for risk avoiders. 26 Therefore, our welfare result under the zero liability rule for consumers does not hold if we assume that merchants are risk averse. The payment platform does not necessarily reduce the probability that a fraud occurs by increasing merchants' share of fraud losses, as merchants may react by reducing their investment e¤ort. In this situation, as in Proposition 3, the pro…t maximising level of liability re ‡ects a trade-o¤ between maximising the transaction volume and minimising the cost of fraud. However, unlike in Proposition 1, the expected loss varies non monotonically with the share of fraud losses borne by merchants.

2 5 As Becker and Ehrlich (1972) note: "Self insurance and market insurance both redistribute income toward hazardous states, whereas self-protection reduces the probability of these states. Unlike insurance, self protection does not redistribute income because the amount spent reducing the probability of a loss decreases income in all states equally, leaving unchanged the absolute size of the loss".

2 6 Bryis and Schlesinger (1990) show that it is not possible to …nd a subclass of risk-averse utility functions to yield a monotonic relationship between risk aversion and the level of self-protection.

The role of interchange fees

In this section, we examine an important regulatory challenge, which is the impact of merchant liability on the level of interchange fees. 27 This issue has been examined in the United-States after the vote of the Dodd-Frank act in July 2010, which gives to the Federal Reserve Board the power to regulate interchange fees on debit card transactions. Among the regulatory rules, the "fraud adjustment rulemaking" provides the Board with the opportunity to assess how card networks' authorization choices and fraud procedures may burden the merchant community and potentially increase the volume of debit card fraud. The rulemaking also gives the Board the opportunity to promote the use of the fraud adjustment mechanism as a means of creating incentives for banks and merchants to migrate to more e¤ective fraud detection technologies.

To study this issue, we modify our model setting, by making the standard assumption that the payment platform is now composed of imperfectly competitive issuers and perfectly competitive acquirers. 28 We also assume for simplicity of the model that consumers bear no liability on fraudulent transactions ( B = 0). The issuers charge a fee f (c I a) to the consumers, whereas the acquirers charge merchants with their perceived marginal cost, that is m = a + c A .

As in the literature, we make the standard assumption that f is decreasing with a, and that the pass-through rate is lower than one, that is @f =@a 1. At the …rst stage of the game, the payment platform chooses the level of interchange fee that maximises banks' joint pro…t.

Then banks choose the transaction prices, merchants invest in fraud detection technologies and consumers make their payments decisions. We denote the pro…t maximising interchange fee by a P , and study how the pro…t maximising interchange fee is impacted by the level of liability that is borne by merchants.

Proposition 7 If the issuers are imperfectly competitive and if the acquirers are perfectly competitive, the pro…t maximising interchange fee decreases with the level of liability that is borne by merchants.

Proof. See Appendix L.

Proposition 4 has important implications for regulatory decisions about interchange fees.

It means that, if merchants bear a higher share of the loss on fraudulent transactions, the pro…t maximising interchange fee becomes lower. The result of Proposition 7 may change if consumers are held liable for fraudulent transactions. In this case, merchants' investments are impacted by the transaction fees and by the interchange fee that is chosen by the payment 2 7 Interchange fees are paid by the acquiring bank to the issuing bank each time a consumer makes a transaction. 2 8 For instance, this assumption is also made in Rochet and Tirole (2002).

platform. The payment platform may decide either to lower or to increase the interchange fee to provide merchants with incentives to increase their investment in fraud detection technologies, depending on the relative importance of the expected loss e¤ect and the transaction volume e¤ect that we highlighted in Lemma 2.

Another interesting aspect of the problem is that regulators may wish to …x a maximum level for the interchange fee, but the payment platform can react by adjusting the level of liability that is borne by merchants for fraudulent transactions. In Appendix L, we show in a simple example that, if the regulator chooses a low level for the interchange fee, the payment platform reacts by choosing a high level of liability for merchants, which may not be desirable from the point of view of social welfare.

Conclusion and discussion

Our results highlight the fact that liability regimes can be used by monopolistic payment platforms to extract rents from merchants. From the point of view of a social planner, payment platforms do not place enough liability on merchants for investments that only depend on the merchants'side under the zero liability rule. This result changes if the platform shares the cost of investments with merchants.

Another issue that deserves further research is the problem of compliance in payment systems. This paper has considered only prices and liability regimes as an incentive to encourage merchant investment. However, we think that it would be interesting to compare the impact of di¤erent measures on investments and fraud losses such as compliance rules, price incentives and liability shifts.

If he sets p > v, the merchant attracts only EPI users. In this case, he makes pro…t

= (p d + b S S x(1 q)L m C S (e S ))D B (f + p v + B (1 q)xL): (16)
We now show that the merchant always makes more pro…t by setting p = v. For this purpose, we prove that

lim p !v d dp < 0,
and that for any p > v, we have d dp < 0:

From ( 16), we have:

d dp = D B (f +p v+ B (1 q)xL) h B (f +p v+ B (1 q)xL)(p d+b S S x(1 q)L m C S (e S )):
We also have

lim p !v d dp = D B (f + B (1 q)xL) h B (f + B (1 q)xL)(v d + b S S x(1 q)L m C S (e S )):
This quantity is negative if and only if

v d + b S S x(1 m C S (e S )
1

H B (f + B (1 q)xL) h B (f + B (1 q)xL) : (17)
As the merchant accepts the EPI, we have that b S S x(1 q)L m C S (e S ) 0. It follows that (A2) is a su¢ cient condition for (17) to hold.

We can now prove that for any p > v, d dp < 0: To simplify the notations, we denote by

g D B = D B (f + p v + B (1 q)xL): We have d dp = g D B h B (f + p v + B (1 q)xL)(p d + b S S x(1 q)L m C S (e S )) < g D B h B (f + p v + B (1 q)xL)(v d + b S S x(1 q)L m C S (e S )) = g D B 1 h B (f + p v + B (1 q)xL) 1 H B (f + p v + B (1 q)xL) (v d + b S S x(1 q)L m C S (e S )) :
We have g D B 0. Therefore, a su¢ cient condition for d dp < 0 to hold is that the term into bracket is negative. The term into brackets is negative if and only if

v d + b S S x(1 q)L m C S (e S ) 1 H B (f + p v + B (1 q)xL) h B (f + p v + B (1 q)xL) :
As by assumption (A1) the hazard rate is increasing, we have 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), we have that

v d + b S S x(1 q)L m C S (e S ) 1 H B (f + B (1 q)xL) h B (f + B (1 q)xL) :
It follows that

v d + b S S x(1 q)L m C S (e S ) 1 H B (f + p v + B (1 q)xL) h B (f + p v + B (1 q)xL) .
Therefore, we have that, for any p > v, d dp < 0. It follows that the merchant makes more pro…t by setting p = v, which enables him to attract cash-users and EPI users. We can conclude that all merchants choose a price such that p = v.

Appendix C: proof of Lemma 2. We proceed in two steps. First, we determine the pro…t maximising level of investment of a merchant who accepts the EPI. Second, we prove that, if the merchant fee is not too high, some merchants accept the EPI. We start by the …rst step. A merchant who accepts the EPI chooses the level of investment in fraud detection technologies that maximises its pro…t,

= p d + D B (f + B (1 q)xL)(b S S x(1 q)L m C S (e S )):
Solving for the …rst-order condition of pro…t-maximisation yields

" S xL dq de S e S C 0 S (e S ) # D B + [b S S x(1 q)L m C S (e S )] dD B de S e S = 0. (18)
We The second-order condition must be veri…ed at e S , that is,

where M S = b S S x(1 q)L m C S (e S ).

Under the assumption that C S is convex and q is concave, the …rst term of this inequality is negative. To determine the sign of the second term, we use equation (18). Since B j e S is negative and since the merchant's margin is positive if he accepts the EPI, we conclude that S xL dq de S e S C 0 S (e S ) 0 at the pro…t maximising level of investment. We have

@D B @e S = h B ( B xL(1 q ) + f ) B xL dq de S ; (20)
and

@ 2 D B @e 2 S = h 0 B ( B xL(1 q ) + f )( B xL dq de S ) 2 + h B ( B xL(1 q ) + f ) B xL d 2 q de 2 S : (21)
From ( 20 Appendix D: proof of Lemma 3. From the envelop theorem, we have that, for any z 2 f B ; S ; f; m; b S g @e S @z = @ 2 @ 2 e S e S ! 1 @ 2 @e S @z e S ! :

As from the second-order condition @ 2 =@ 2 e S 0, it follows that @e S =@z has the same sign as @ 2 @e S @z e S .

Let us study the variation of the merchant's investments with the merchant fee. We have that @ 2 @e S @m = h B (f + B (1 q)xL) B xL dq de S 0:

From the envelop theorem, @e S =@m has the same sign as @ 2 =@e S @m. It follows that the merchant's investment always decreases with the merchant fee. Similarly, we have that

@ 2 @e S @b S = h B (f + B (1 q)xL) B xL dq de S 0: (23)
It follows that the merchant's investments in fraud detection technologies increase with the merchant's transactional bene…t.

We now study the variation of the merchant's investments with the consumer transaction fee. Using the same reasoning, we know that @e S =@f has the same sign as @ 2 =@e S @f . We have

@ 2 @e S @f = S xL dq de S C 0 S (e S ) dD B df +[b S S x(1 q)L m C S (e S )] B xL dq de S e S h 0 B (f + B (1 q)xL):
From the …rst-order condition, we have that It follows that @ 2 =@e S @f 0 since h merchant's investment increases with the transaction fee that is paid by the consumer.

We determine the variation of the merchant's investments with the consumer liability. Using the same reasoning, @e S =@ B has the same sign as @ 2 =@e S @ B . We have

@ 2 @e S @ B = S xL dq de S C 0 S (e S ) dD B d B +[b S S x(1 q)L m C S (e S )] B (1 q)x 2 L 2 dq de S e S h 0 B :
Exactly like in the previous proof, we have that @ 2 =@e S @ B 0 since h 0 B is positive. It follows that the merchant's investment increases with the consumer liability.

Let us study the variation of the merchant's investments with his level of liability. From the reasoning above, @e S =@ S has the same sign as @ 2 =@e S @ S . We have that From assumption (A2), we have that

@ 2 @e S @ S
h B (f + B (1 q(e S ))xL) D B j e S 1 xL B :
As 1 q(e S ) 2 [0; 1], it follows that

h B (f + B (1 q(e S ))xL) D B j e S 1 xL B (1 q(e S ))
:

Therefore, we have that

1 xL B (1 q(e S )) h B (f + B (1 q(e S ))xL) D B j e S 0.
As dq de S e S 0 and xL D B j e S 0, we can conclude that @ 2 @e S @ S e S 0:

It follows that, from assumption (A2), the merchant's investments in fraud detection technologies increase with his liability level. From ( 18), we have that C 0 S (e S ) S xL dq de S e S 0. From Lemma 3, we know that de S =d S 0.

It follows that the right-hand side of the equality is positive.

Let us now determine the sign of the left-hand side of the equality. From Lemma 3, we know that de S db S = @ 2 @ 2 e S e S ! 1 @ 2 @e S @b S e S ! :

(E-2)

Replacing for de S =db S in (E-1), we obtain that

1+
It follows from (E-2) that @ b b S @ S 0.
Therefore, the probability that a merchant accepts the EPI decreases with the level of liability that is borne by the merchants.

Impact of the transaction bene…t received by merchants on EPI acceptance:

Similarly, we have that As de S =df 0, it follows that @ b b S =@f 0. It can be also proved in a similar way that @ b b S =@ B 0.

Impact of the transaction fees on EPI acceptance: We also have that

@ b b S @m " 1

0:

It follows that @ b b S =@m 0.

Appendix F: Proof of Proposition 1. We start by determining the variation of the expected loss with the consumer fee. To that end, we denote by (b S ; f; m; S ; B ) the function de…ned by

(b S ; f; m; S ; B ) = (1 q )h(b S )(1 H B (f + B xL(1 q )):
From (6),

EL P = P xL Z bs c b S (b S ; f; m; S ; B )db S :
Hence, we have that 29

@EL P @f = P xL " @ b b S @f ( b b S ; f; m; S ; B ) + Z bs c b S @ (b S ; f; m; S ; B ) @f db S # ; 2 9
The conditions to use the Leibniz rule apply.

where

@ (b S ; f; m; S ; B ) @f = dq de S @e S @f (1 H B (f + B xL(1 q ))h(b S ) 1 (1 q ) B xLh B 1 H B + (1 q )h(b S )h B (f + B xL(1 q )):
From Lemma 3, we have @e S =@f 0. We also have that dq=de S 0. From assumption (A2), we have

1 (1 q ) B xLh B 1 H B 0:
It follows that @ (b S ; f; m; S ; B )=@f 0. Since, from Lemma 4, @ b b S =@f 0, we conclude that @EL=@f 0.

We now determine the variations of the expected loss with the merchant fee. We have

@EL P @m = P xL 2 6 6 6 4 @ b b S @m ( b b S ; f; m; S ; B ) | {z } T ermA + Z bs c b S @ (b S ; f; m; S ; B ) @m db S | {z } T ermB 3 7 7 7 5 ;
where

@ (b S ; f; m; S ; B ) @m = dq de S @e S @m (1 H B (f + B xL(1 q ))h(b S ) 1 (1 q ) B xLh B 1 H B 0.
From Lemma 3, we have @e S =@m 0. From Lemma 4, @ b b S =@m 0. From Assumption (A2), we have

1 (1 q ) B xLh B 1 H B 0:
It follows that term A is negative, whereas term B is positive. Therefore, an increase in the merchant fee has an ambiguous impact on the expected loss that is borne by the payment platform.

Let us now study how the level of liability that is borne by merchants impacts the expected loss. We have

@EL P @ S = EL P P + P xL 2 6 6 6 4 @ b b S @ S ( b b S ; f; m; S ; B ) | {z } T ermC + Z bs c b S @ (b S ; f; m; S ; B ) @ S db S | {z } T ermD 3 7 7 7 5 ;
where @ (b S ; f; m; S ; B ) @ S = dq de S @e S @ S

(1

H B (f + B xL(1 q ))h(b S ) 1 (1 q ) B xLh B 1 H B 0:
From Assumption (A2),

1 (1 q ) B xLh B 1 H B 0:
As dq=de S 0, and since @ b b S =@ S 0 from Lemma 4, it follows that @EL P @ S 0:

Appendix G: Second-order conditions if B = 0.

Appendix G-A: second-order conditions if the payment platform chooses the transaction prices. We provide here the second-order conditions of pro…t maximisation if B = 0. The …rst-order conditions of pro…t maximisation are

@ P @m = D B (f ) [D S (b m S ) M P h S (b m S )] = 0; (24)
and

@ P @f = D S (b m S ) [D B (f ) M P h B (f )] = 0: (25)
The second derivatives of the platform's pro…t with respect to the prices and the liability level are

@ 2 P @m 2 = 2h S D B h 0 S D B M P ; (26)
@ 2 P @f 2 = 2h B D S h 0 B D S M P ; @ 2 P @m@f = h B D S h S D B + M P h S h B ; @ 2 P @m@ S = xL(1 q ) @ 2 P @m 2 (1 S )xL @q @ S h S D B ; @ 2 P @f @ S = xL(1 q ) @ 2 P @m@f (1 S )xL @q @ S h B D S ; @ 2 P @ 2 S = 2xLD B h S xL(1 q ) 1 q + (1 S ) @q @ S + M P D B xLh S @q @ S (xL(1 q )) 2 h 0 S +xLV P " 2 @q @ S + (1 S ) ( @ 2 q @e 2 S @e S @ S 2 + @q @e S @ 2 e S @ 2

S

)#

:

We denote by det M the determinant of the Hessian matrix at the pro…t maximising transaction fees. It can be checked that the second-order conditions of pro…t maximisation are veri…ed as h 0 S 0 and h 0 B 0. From ( 24) and ( 25), we have that, at the pro…t maximising prices,

D S = M P h S and D B = M P h B . Therefore, we have det M j (f ;m ) = 2h S h B D S D B + 2h 0 B D B D 2 S + 2h 0 S D S D 2 B + h 0 S h 0 B D B D S M 2 P + h 2 S D 2 B > 0; (27) and @ 2 @m 2 (f ;m ) < 0,
which proves that the conditions for a maximum to exist at (f ; m ) hold.

Appendix G-B: second-order conditions if the payment platform chooses the transaction prices and the level of liability for merchants. We provide here the conditions under which the second-order conditions are veri…ed at y = (f ; m ; S ) by computing the coe¢ cients of the Hessian matrix.

Denoting the Hessian matrix at y = (f ; m ; S ) by

H = 0 B B B @ a 1 b c b a 2 d c d a 3 1 C C C A , the second-order conditions are veri…ed 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
det H 0 (See hereafter). If these conditions are veri…ed, this proves that the Hessian matrix is semi-de…nite negative at y = (f ; m ; S ).

Let us start by the case in which there is an interior solution. From (26), as h 0 S and h 0 B are positive, we have that a 1 0 and a 2 0. We already proved in Appendix G-A that a 1 a 2 b 2 0. We now prove that a 1 a 3 c 2 0 and that a 3 a 2 d 2 0.

At y = (f ; m ; S ), if the solution is interior, from ( 24) and ( 25), we have that D S = M P h S and D B = M P h B : The …rst-order condition of pro…t maximisation with respect to S is @ P @ S = (f + m c) @V P @ S @EL P @ S = 0: From (7), we have that

@EL P @ S = xL(1 q )V P + (1 S )xLq @V P @ S .
From (5), we have that @V =@ S = D B (f )h S ( b b S )xL(1 q ). Therefore, the …rst-order condition with respect to S writes

M P h S D B xL(1 q ) = xLD B D S 1 q + (1 S ) dq d S :
As at y = (f ; m ; S ) we have D S = M P h S , in interior solution, we have that

xL(1 q ) = xL 1 q + (1 S ) dq d S ;
that is dq d S x = 0.

It follows that, in an interior solution, the payment platform chooses the level of liability that maximises the probability of fraud detection.

We denote by

= 2 @q @ S + (1 S ) ( @ 2 q @e 2 S @e S @ S 2 + @q @e S @ 2 e S @ 2

S

) :

(GB-1)

Lemma 6 We have @ 2 e S =@ 2 S 0:

Proof. From the implicit function theorem, we have @e S @ S = @ 2 @ 2 e S e S ! 1 @ 2 @e S @ S e S S (e S ) 0, we have N 0. Therefore, we can conclude that @ 2 e S =@ 2 S 0.

Since @ 2 e S =@ 2 S 0; @q=@e S 0 and @ 2 q=@e 2 S 0, from (GB-1), we have 0. As dq d S x = 0 and from (26), we have that @ 2 P @ 2 S x = (xL(1 q )) 2 2h S + M P h 0 S D B + xLV P ; @ 2 P @ S @f x = xL(1 q )h S D B ; and @ 2 P @ S @m x = xL(1 q )D B (2h S + h 0 S M P ):

We now compute a 1 a 3 c 2 and a 3 a 2 d 2 at y = (f ; m ; S ). We have We also have Appendix J-B: The social welfare maximising level of liability if B = 0. We start by proving that the payment platform'pro…t is concave in S at the pro…t maximising prices (f ; m ), which are chosen at stage 2 (after a benevolent social planer chooses the liability level for merchants). From (26), we have

a 1 a 3 c 2 = @ 2 P @f 2 x @ 2 P @ 2 S x @ 2 P @ S @f x 2 = D 2 B [xL(1 q )] 2 (3h
@ 2 P @ 2 S = 2xLD B h S xL(1 q ) 1 q + (1 S )
@q @ S + M P D B xLh S @q @ S (xL(1 q )) 2 h 0 S +xLV P " 2 @q @ S + (1 S ) ( @ 2 q @e 2 S @e S @ S 2 + @q @e S @ 2 e S @ 2

S

)#

:

From Appendix J-A, at the pro…t maximising prices, we have that M P h B = D B and M P h S = D S . It follows that

@ 2 P @ 2 S (f ;m ) = 2(xL) 2 D B h S (1 q ) 1 q + (1 S )
@q @ S M P D B (xL(1 q )) 2 h 0 S +xLV P " @q @ S + (1 S ) ( @ 2 q @e 2 S @e S @ S 2 + @q @e S @ 2 e S @ 2

S

)#

:

Since h 0 S 0, @ 2 q=@e 2 S 0, and @ 2 e S =@ 2 S 0 from Lemma 6, we conclude that

@ 2 P @ 2
S (f ;m ) 0.

We now study the concavity of the total user surplus. For this purpose we need to determine the sign of @ 2 f =@ 2 S and @ 2 b b S =@ 2 S . With uniform distributions for b B and b S on [0; 1], this sign is positive, and, from Appendix J-B, we have

@ 2 f @ 2 S = @ 2 b b S @ 2 S = xL 3
@q @ S (1 S ) @ 2 q @ 2 S 0.

In general, the total user surplus is not necessarily a concave function of S . We have We now prove that a su¢ cient condition for total user surplus to be concave in S is that The total user surplus is a concave function of S if and only if @ 2 S S @ 2 S + @ 2 S B @ 2 S 0:

@ 2 S B @ 2 S = 2 4 @ 2 b b S @ 2 S h S ( b b S ) @ b b S @ S ! 2 h 0 S ( b b S ) 3 5 E(b B f B xL(1 q )=b B c b B ) +2 @ b b S @ S @f @ S h S ( b b S )D B (f ) @ 2 f @
As at the pro…t maximising prices, with uniform distributions, D B = D S , we have that, at the pro…t maximising prices,

@ 2 S S @ 2 S + @ 2 S B @ 2 S = 5

:

As D B belongs to [0; 1], a su¢ cient condition for the total surplus to be concave in S is that

@ 2 b b S @ 2 S 6 5 @ b b S @ S ! 2 :
From (32) and (33), this condition is equivalent to @q @ S (1 S ) @ 2 q @ 2 S

(1 S ) 2 3 xL @q @ S 2 ; that is @q @ S 1 (1 S ) 2 3 xL @q @ S (1 S ) @ 2 q @ 2 S 0:

volume by encouraging merchants to accept the EPI and maximising the margin per transaction. As h 0 S ( b b S ) 0, @f =@a 0, and @e S =@ S 0, we have @ 2 I @a@ S 0:

It follows that the interchange fee decreases with the level of liability that is borne by merchants.

If consumers could be held liable for fraudulent transactions, the merchants'fraud prevention e¤ort could depend on the interchange fee. In that case, the payment platform could decide to lower the interchange fee to encourage merchants to invest in fraud prevention technologies.

We now show in an example that if the interchange fee is chosen by a regulator at stage 1, the payment platform can react at stage 2 by adjusting the level of merchant liability. For instance, assume that b B and b S are uniformely distributed on [0; 1] and that the issuer is a monopolist. In this case, we have that f = (1 a + c I )=2. The payment plaform chooses the level of liability for merchants that maximises the issuer'pro…t, where r = 1 q . The second-order condition writes

(1 + xL)D S (b m S ) 2(xL) 2 (1 q )(1 q + (1 S )) (1 S )(xL) 2 (1 q ) 0.

From the implicit function theorem, @ S =@a has the same sign as @ 2 I =@a@ S . We have @ 2 I @a@ S = 2xL(1 q ) (1 S )xL 0:

We …nd that the liability level that is chosen by the payment platform decreases with the level of interchange fee. This example shows that if a regulator chooses a level of interchange fee that is quite low, the platform can react by increasing the level of liability that is borne by merchants.

  and decide whether or not to accept the EPI. If he decides to accept the EPI, a merchant may invest an amount e S in fraud detection technologies. Investment in "self-protecting" measures to improve fraud detection costs C S (e S ) to the merchant, where C S (e S ) is paid per transaction, C 0 S (e S )

4. 1

 1 Stage 3: consumer payment decisionsWe start by determining the probability that a consumer wishes to use the EPI. Consider a consumer whose transaction bene…t is b B 2 b B ; b B . This consumer is randomly matched to one merchant, who may or may not accept the EPI. If the merchant accepts the EPI, the consumer chooses his payment method by comparing his expected utility if he pays cash and if he pays with the EPI.

Lemma 2

 2 If the merchant fee is not too high, all merchants such that b S b b S ( S ; B ; x; L; m; f ) accept the electronic payment instrument, where b b S ( S ; B ; x; L; m; f ) 2 b S ; b S . The pro…t maximising investment for a merchant who accepts the EPI solves:

  our example with uniform distributions and quadratic cost functions, the transaction fees under full merchant liability ( S = 1) are f = (1 + c + xL (x L) 2 =2k S )=3 and m = (1 + c 2xL + (x L) 2 =k S )=3. The total price is f + m = 2(1 + c xL)=3 + (x L) 2 =6k S , and the price di¤erence is f m = (xL (x L) 2 =2k S )=3. Even if consumers bear zero liability

0 S

 0 de…ne B = dD B =de S D B =e S the elasticity of the consumers' demand to the fraud detection prevention e¤ort. The merchant's investment in fraud detection technologies is implicitely de…ned by S xL dq de S e S C (e S ) = [b S S x(1 q)L m C S (e S )] B j e S e S :

M

  S (0) 0 and that the sign of M S (y), where y = b S m, depends on m. There are three cases. Let us start by the …rst case, in which the merchant fee m is su¢ ciently high, such that M S (y) < 0. As M S is increasing in y, for all b S 2 b S ; b S and for all y = b S m, we have M S (y) < 0. It follows that no merchant accepts the EPI. In the second case, the merchant fee m is su¢ ciently low, such that b S m > 0 and M S (y) 0; where y = b S m. As M S is increasing in y, for all b S 2 b S ; b S and for all y = b S m, we have M S (y) 0. It follows that all merchants accept the EPI. In the third case, the merchant fee is such that M S (y) > 0 and M S (y) < 0. As M S is increasing in y, from the bijection theorem, there exists a threshold that we denote by b b S ( S ; B ; x; L; m; f ) such that merchants accept the EPI for all b S b b S ( S ; B ; x; L; m; f ).

0 S

 0 (e S ) # 0 . We also have dD B =df 0.

  Appendix E: Proof of Lemma 4.Impact of the level of liability borne by merchants on EPI acceptance: From(2), the threshold above which merchants accept the EPI solves b b S m S xL(1 q ) C S (e S ) = 0:Di¤erentiating this equation with respect to S , we obtain that

a 3 a 2 d 2 = @ 2 P 2 P@ S @m x 2 = D 2 B 0 := 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 : 1 (a 3 a 2 d 2 ) = D 2 B D 2 S

 22220311231222 D S xL 2h S + M P h 0SWe now show that det H 0 at y = (f ; m ; S ). From the rule of Sarrus, we havedet H At y = (f ; m ; S ), since a 1 = D S (2h B + M P h 0 B ), we have a xL 2h S + M P h 0 S 2h B + M P h 0 B :We also have2bdc = 2(xL) 2 (1 q ) 2 h 2 S D 3 B 2h S + M P h 0 S ;andc 2 a 2 b 2 a 3 = 2(xL) 2 (1 q ) 2 h 2 S D 3 B 2h S + M P h 0 S xLD 3 B D S h 2 S :Using the fact that, at y = (f ; m ; S ), we have M P h S = D S and M P h B = D B , we obtain thatdet H = D 2 B D S xLM P 3h B h 2 S + h 0 S h 0 B M 2 P h S + 2h S h B h 0 S M P + 2h2 S h 0 B M P : (28) Second case: B 6 = 0 [TO DO]

2 SD 2 S= @ 2 b b S @ 2 S 2 h@ 2 2 D

 222222 B (f )D S ( b b S ); and @ 2 S S @ D B (f )D S ( b b S ) + @ b b S @ S ! S ( b b S )D B (f ): With uniform distributions on [0; 1] for b B and b S , since D B (f ) = D S ( b b S ), we have that B (f ):

2 @ 2 2 D

 222 B (f ):It follows that the total user surplus is a concave function of S if and only if

For 2 I

 2 instance, with uniforms distributions on [0; 1] for b B and b S and if the issuer is a monopolist, we havea P = c I c A S xL(1 q ) C S (e S @a@ S = D B xL(1 q ) df da a P + 1 h S ( b b S ) + (f + a P c I (1 S )xL(1 q ))h 0 S ( b b S ) D B (f )h S ( b b S ) xL(1 q ) + (1 S )dq de S de S d S :

1 S

 1 c A S xL(1 q ) C S (e S )): With a linear probability such that q(e S ) = e S , where 0 < 1, with a cost fonction such that C S (e S ) = ke 2 S =2, we have e S = S xL k : Solving for the …rst-order condition of pro…t maximisation with respect to S , from the envelop theorem, we obtain that (xLr + (1 S )xL ) [1 a c A S xLr C S (e S )] = 1 + a c I 2 ()xLr xLr ;

  S (e S ), where y = b S m. Note that(22) does not hold if y < 0, which happens if the merchant fee is too high. We have that It follows that M S is increasing in y for all y 0. Note that

	), we have since M S 0 and since dD B de S e S d 2 D B d 2 e S e S 0. It follows that the second term of (19) is negative. Finally, 0 from (21), the last term of (19) is negative. It follows that the second-order condition is always veri…ed at e S .
		We now show that merchants accept the EPI if their transactional bene…t b S is such that
	b S	b b S ( S ; B ; x; L; m; f ), which is the second step of our proof. A merchant accepts the EPI
	if and only if
			b S m	S x(1 q)L C S (e S ) 0.	(22)
	Let us consider the function M S (y) = y S x(1 q)L C M 0 S (y) = 1 + de S dy e S S xL dq de S e S	C	! 0 (e S )	:
	From (18), we have that S xL	dq de S e S	C	0 (e S )	0. We can also prove, using (18) and the
	envelop theorem that	de S dy e S	0.

  2 S + (M P h 0 S ) 2 + 4M P h 0 S h S ) D 2 B D S xL 2h S + M P h 0

	S	0:

e¤ect in the literature on payment cards. Moreover, in our model, the probability that a merchant accepts the EPI also depends on the consumer fee, because merchants exert a positive externality on consumers when they choose to invest in fraud detection technologies. Indeed, this interaction, which is novel in the literature on payment platforms, arises when B 6 = 0 and this is speci…c to our model setting. Finally, a higher consumer fee decreases the probability that a consumer wishes to use the EPI, which reduces the marginal bene…ts of investing in fraud detection technologies and the bene…ts of accepting the EPI for the merchant. Therefore, the probability that a merchant accepts the EPI decreases with the consumer fee.Most importantly, our model is the …rst to highlight the impact of liability regimes on merchants' acceptance of payment media. We show in Appendix E that the level of liability has an ambiguous impact on merchants' choice to accept the electronic payment instrument.On the one hand, a higher liability level increases the loss in case of a fraudulent use of the EPI, which discourages merchants to accept the EPI. On the other hand, it increases the level of e¤ort made by merchants, which reduces the probability that the EPI is fraudulently used

1 Indeed, we have @(f + m )=@x = ( 2L=3)(1 q =2) < 0.

2 W is concave in S for instance if bS and bB are uniformely distributed on [0; 1] under some assumptions about the cost of fraud prevention and the sensitivity of the detection probability which are precised in Appendix J. In general, it is possible to prove that P is concave in S , however, the total user surplus is not necessarily concave in S .

[START_REF] Belleflamme | Platform Competition and Seller Investment Incentives[END_REF] Except in the case where the platform's cost function is linear and if the detection probability is linear in the platform's investment e¤ort.

[START_REF] Bryis | Risk Aversion and the Propensities for Self-Insurance and Self-Protection[END_REF] If e P = 0, we obtain the prices of our previous example, which did not take into account the platform's investments (See[START_REF] Douglas | An examination of the fraud liability shift in consumer cardbased payment systems[END_REF] and (11)).

B is positive. We can conclude that the

Appendix A: Consumer Protection Laws in Various Countries. The following table

provides some examples of consumer protection laws in various countries. The common feature of consumer protection laws is that consumer bear hardly meaningful responsibility for fraudulent use of cards in all countries.

Country Name of the Law Consumer Protection USA TILA/Reg Z for credit cards Capped at $50 for all unauthorized transactions.

Debit Cards

If the cardholder fails to notify the card issuer within 2 days, the cardholder's maximum liability is $500, of which only $50 can be attributed to fraud occurring during the …rst 2 days after the cardholder learnt the loss or theft.

Europe Payment Service Directive

The cardholder has 13 months to contest an unauthorized transaction. The cardholder's liability is capped at 150 euros if he has failed to keep the personnalized security measures safe.

If the cardholder was a victim from an identity theft, he cannot be held liable. No liability in all cases after the fraud is reported. Right for payment service users to enjoy immediate refund of unauthorized transactions following the establishment of the proof.

Appendix B: Proof of Lemma 1. We prove in Lemma 1 than the merchants who accept the EPI and the merchants who do not accept the EPI set the same price p = v. There are two cases: either a merchant refuses the EPI or he accepts it. Let us start by the …rst case. If a merchant refuses the EPI, all consumers pay cash, and he makes pro…t

In this case, the merchant's pro…t is maximised when he sets p = v, and we have that

In the second case, the merchant accepts the EPI. If he sets p v, he attracts both EPI and cash users. In this case, he makes pro…t

This pro…t is maximised at p = v.

Since 0, we can conclude that det H 0 at y = (f ; m ; S ). Therefore, the Hessian matrix is semi-de…nite negative at y = (f ; m ; S ) and the second-order conditions are veri…ed at y = (f ; m ; S ).

Appendix G-C: Variation of the equilibrium prices with the fraud rate. We di¤erentiate the …rst-order conditions of pro…t maximisation with respect to f and m, which are evaluated at S = 1. Since @b m S =@x = L(1 q ), we have

and

Since h S (b m S ) 0, L(1 q ) 0 and h 0 S (b m S ) 0, @f =@x and @m =@x have opposite signs.

Therefore, the consumer fee and the merchant fee vary in opposite directions with the fraud rate.

Appendix H: An illustration of Proposition 2. We make the following assumptions: where S xL 2 =k 1. The merchant's e¤ort increases with the liability level, the probability that there is a fraudulent transaction, the losses borne by the participants, and the marginal impact of investments on the probability of fraud detection. The probability that a merchant detects a fraudulent transaction is implicitely de…ned by q(e S ) = ( S xL 2 )=k:

In this case, the demands are D B = 1 f , and

: Using the standard price structure/ratio formula, we …nd that the prices verify

Solving for f and m, we obtain that

;

and

We can compute the marginal merchant

The merchant demand is

If S is chosen by the payment platform (at the same stage as the prices), we have that

As M P = 1 f = D B = D S at the optimal prices, we have

In this case, we …nd that the platform's pro…t is maximised by choosing S = 1.

Appendix I: Impact of the merchants'liability on transaction prices under the zero liability rule In this Appendix, we examine how the level of liability borne by merchants 44 impacts the transaction fees that are chosen by the payment platform, if the zero liability rule applies for consumers. By di¤erentiating equations ( 24) and ( 25) that de…ne the …rst-order conditions with respect to S , we obtain that

and

Solving for @m =@ S and @f =@ S in ( 29) and (30), we obtain that

where

and

We proved in Appendix G-A that det M 0. We now prove that R 0 and T 0. We have

and

Using the …rst-order condition, we have that, at the pro…t maximising prices, M P h S = D S and M P h B = D B . It follows that, at the pro…t maximising prices,

and

Since h 0 S and h 0 B are positive, we have R 0 and T 0. As @q =@ S 0 and DetM > 0, it follows that @m =@ S 0 and @f =@ S 0. Hence, Lemma 5 is veri…ed.

Note that, since b b S = m + S xL(1 q ) + C S (e S ), we have

It follows from (32) that

From Lemma 6, we have @ 2 q =@ 2 S 0. Therefore, we can conclude that d obtain the agregate consumer surplus, that is

where

)) denotes the mathematical expectancy conditional on b B f + B xL(1 q ). We have

where

where from the Leibniz rule,

First case B = 0. In this case, from proposition 5, the transaction fees decrease with the level of liability that is borne by merchants. Therefore, Y is positive. Term X is also positive,

From the Leibniz rule, we have

First case B = 0. In this case, from proposition 5, the transaction fees decrease with the level of liability that is borne by merchants. From (31), we have @m =@ S + xL(1 q ) 0. It follows that the merchant surplus increases with the level of liability that is borne by merchants.

As @ 2 q =@ 2 S 0 from Lemma 6, a su¢ cient condition for this inequality to hold is that

If B = 0, we have

:

It follows that (34) holds if

We denote by P S the level of liability that maximises the platform's pro…t and by W S the level of liability that maximises social welfare. We have

Appendix L: risk averse merchants We assume that merchants are risk averse and that their expected pro…t takes the form of a von Neumann Morgenstern utility function which we denote by u S . We assume that u S is concave and that the zero liability rule applies for consumers. The probability that a fraud occurs is denoted by = x(1 q(e S )). We denote by

the merchant's pro…t if a fraud occurs (provided that the merchant accepts the EPI), and by

the merchant's pro…t if a fraud does not occur. The merchant's expected pro…t is

If the merchant accepts the EPI, he invests an amount e S in self protection such that @E( )=@e S = 0, that is

The second-order condition holds as u S is concave. This equation explains how Lemma 4 changes when merchants are risk averse. Except in special cases, the merchant's e¤ort now depends on b S , m, and f , unlike the situation where the merchants are risk neutral and where the zero liability rule applies for consumers. From the implicit function theorem, @e S =@ S has the same sign as @ 2 E( ) @ S @e S e S =e S = C 0 (e S )D 2 B Lu 00

:

Therefore, unlike in the risk neutral case, the merchant's investment e¤ort does not necessarily increase with his share of fraud losses S .

For instance, assume that the merchant's utility function u S is a CARA function, and that his risk aversion index is denoted by . Let q(e S ) = e S , where > 0 and C S (e S ) = ke S . Since 0 (e S ) = x, we have

If the merchant's risk aversion index is su¢ ciently low compared to the impact of the merchant's investments on fraud detection, the merchant's investments increase with its level of liability.

If the merchant's aversion index is high and the impact on fraud detection is small ( ), the merchant's investments may decrease with its level of liability. Compared to the risk neutral case, the merchant is reluctant to invest in fraud detection because investments can be assimilated to a loss in case a fraud does not occur.

In our CARA example, if f 6 = 1, with uniform distributions on [0; 1] for b B and b S , the merchant's e¤ort is

In this special case, the merchant's e¤ort does not depend on b S or m, but it depends on f .

Appendix K: The role of interchange fees. In this section, we look at the impact of merchants' liability on pro…t-maximising and welfare maximising interchange fees. As consumers bear no liability on fraudulent transactions, merchants'investments in fraud detection technologies do not depend on the transaction fees that are paid by the users. We have

where e S solves (4). As m = a + c A , and from the envelop theorem, we have

As the acquirers make zero pro…t, banks'joint pro…t is equal to the issuers'pro…t, Note that the level of investment that is chosen by the merchants depends neither on the transaction fees nor on the interchange fee. From the envelop theorem, as @ I @f f = 0, we have d I da = @ I @a + @ I @m @m @a Solving for the …rst-order condition of pro…t maximisation yields The second-order condition is The pro…t maximising interchange fee re ‡ects a trade-o¤ between increasing the transaction