Software Anti-piracy and Pricing in a Competitive Environment: a Game Theoretic Analysis

Transcription

1 Software Anti-piracy and Pricing in a Competitive Environment: a Game Theoretic Analysis We study a problem of two software firms competing on price in a market where consumers can choose between purchasing and pirating software. Firms invest in anti-piracy effort to prevent software piracy. We study both the monopolistic case and the duopolistic one. We obtain several interesting results under competition. In particular, when the network effect increases, each firm adopts a stronger anti-piracy measure only if the cross-effect of antipiracy effort on the other firm is high. If both cross-effect and network effect are low, each firm invests more in anti-piracy with an increase in network effect. Our results show the importance of considering competition and cross effect when a firm designs an anti-piracy policy. Key words: Software piracy, Anti-piracy effort, Cross effect, Game theory

2 1. Introduction Software piracy, the unauthorized copying, distribution, or use of software products, is by far the biggest problem facing the software industry. According to a global recent survey by The Software Alliance (also known as BSA) (2014), 43% of the PC software was installed without without proper licensing in 2013; the total commercial value of the unlicensed installation approximated to $62.7 billion globally in In addition, the use of pirated software can increase security threats, interfere with existing systems and cause system malfunction and downtime. A recent survey by IDC shows that the annual cost to enterprises for dealing with malware from pirated software can reach $114 billion (IDC 2013). Pirated software can be obtained from various sources. According to the same survey (IDC 2013), 45% pirated software is from online websites and P2P network and 21% from street market. Facing the threat of software piracy to their revenue and business growth, many companies exert great effort against software piracy. Software giant Microsoft has built a worldwide anti-piracy team which works closely with law enforcement agents to support criminal prosecutions. Microsoft is estimated to spend more than $10 million a year on its intelligence-gathering operations for anti-piracy and another $200 million on developing anti-piracy technology (New York Times 2010). To reduce the chances of accessing pirated content intentionally or unintentionally by Internet users, Security software company McAfee has both patented and developed various technologies that detect and block pirated material from any websites when visited by users (Burgess 2013, Ernesto 2013). Also, many software firms and industrial organizations have links on their websites to encourage the public to report piracy against their products. Software & Information Industry Association (SIIA) offers rewards up to $1 million for reporting piracy activities (siia.net 2015). BSA also has such reward program to combat software piracy ( 1.1 Problem Description We study a problem of two software providers who sell competing products in a consumer market. Pirated software products are available on the market for free, but consumers incur cost for using such pirated products. Consumers have heterogeneous valuations for these two products. They choose either a legitimate product or a pirated version to maximize their own utilities which also include the contribution of network effect: the more consumers use 2

3 a particular software product (pirated or purchased), the more valuable this product is to its users. A software provider can exert anti-piracy effort which increase the cost of using pirated software. In our problem, we also consider cross effect of anti-piracy effort. When a provider exerts anti-piracy effort, not only the cost of using its own pirated software increases, but the cost of using the pirate version of its competitor s product will also increase. Cross effect arises for several reasons. First, anti-piracy technology such as the one developed by McAfee usually block the access to pirated products in general, including both its own product and a competitor s product. Also anti-piracy campaign through shutting down illegal websites and user education makes pirated software less available and thus more costly to obtain and use. Software anti-piracy through legal enforcement and raid can increase the chance of users being caught of using any pirate software, leading to a higher cost of using pirate software overall. In the past studies, it is commonly assumed that there is only one software provider in the market which faces the competition from its pirate software (Conner and Rumelt 1991, August and Tunca 2008, Lahiri and Dey 2012). We study a more realistic setting of two competing software providers. In this paper, we study mainly three research questions. First, how will competition among legitimate and pirated products affect the anti-piracy effort? Second, how will anti-piracy effort be affected by the network effect when there is competition among legitimate and pirated products? Third, how will the network effect affect the profit of each firm due to competition? 2. Literature Review Software piracy has long been an important topic in the information system research area. Many papers have studied policies and strategies to combat piracy. Chen and Png (2003) explores how a government sets the fine for copying, tax on copying medium, and subsidy on legitimate purchases, while a monopoly publisher sets price and spending on detection. However, our paper focuses how network externality and other factors affect pricing and anti-piracy decisions in a competitive environment. Chellappa and Shivendu (2005) study the issue of product sampling when there is piracy. They use a two-stage model of piracy for a market where consumers are assumed to be heterogeneous in their marginal valuation for quality and their moral costs. They find that product sampling can internalize any potential 3

4 benefits of piracy. Sundararajan (2004) investigates how a monopolistic seller should choose the optimal pricing schedules and technological deterrence levels when digital piracy exists in the market and the degree of piracy can be influenced by implementing digital rights management (DRM) systems. Wu and Chen (2007) study the versioning strategy in the presence of piracy. They find that when there is no piracy, a single version is the optimal strategy for an information goods provider. However, when piracy exists, firms tend offer more than one quality through versioning, an effective and profitable instrument to fight piracy under some conditions. August and Tunca (2008) consider the decision of providing security patches to users. They find that in the case that piracy tendency is low, only when the piracy enforcement level is high, vendor s software security patch restrictions is optimal. When patching costs are sufficiently low, a vendor should provide patches to pirated software as well. Johar et al (2012) investigate a publisher who gain profit through advertisement when providing content to consumers who have heterogeneous valuation. The publisher needs to determine two dimensions, the content quality and content distribution delay, in its content provision strategy. They find that that when piracy exists, the publisher should improve at least one dimension of content provision. Another stream of literature on the other hand, studies the potential positive effect of piracy on software producers. Before Conner and Rumelt (1991), it was believed that software piracy would decrease software firms profits and increase prices, which would harm the social welfare. Conner and Rumelt (1991) incorporated the network externality into their model and examine piracy s effect on software firms profit. When more people use one software, whether legitimate or pirated, consumers can gain higher utility through network effect, making consumers willing to pay more for the product. Therefore if network effect is large, then firms can benefit from piracy. Shy and Thisse (1999) extend the monopolistic results of Conner and Rumelt (1991) to a duopolistic framework. They show that firms will allow piracy in order to increase the market size. Similarly, they show that if network effects are strong, then firms can benefit from not exerting anti-piracy effort. Gu and Mahajan (2005) also investigate piracy s effect on firms profits when competition exist. They show that piracy can reduce price competition, and can be beneficial to firms when markets have high wealth gaps. Jain (2008) find that strong network effects may lead to higher levels of copyright protection in some case. They show that when network effect is strong, stronger copyright enforcement can reduce price competition. Lahiri and Dey (2012) find when piracy enforcement is low, the monopolist have more incentive to invest in quality. Tunca and 4

5 Wu (2013) show that information goods industry, by suing file sharing networks or users who share copyrighted material on peer-to-peer networks, can sometimes hurt legitimate publishers of information goods. In our paper, we assume that a consumer chooses to buy or pirate software based on his or her utility. One contribution of our paper is to consider anti-piracy s cross effect which is not considered in other papers. We obtain several interesting results. For example, different from previous works (Conner and Rumelt 1991, Jain 2008), we show anti-piracy effort increases with network effect only if cross effect is high enough. 3. The Model Consider two competing firms (denoted by 1 and 2) located at the endpoints of a Hotelling line [0,1]. Each firm produces one software product with quality q i, i = 1, 2. Using a common assumption about the cost of information goods, we let the cost of producing an extra copy of software be 0. Let p 1 and p 2 be the price of legitimate software products that firms and 2 sell. Both software products have pirated version. We assume the pirated versions are costless to obtain since nowadays consumers can download pirated software products easily from many software pirating websites. Table 1 in the appendix contains a summary of key variables used in this paper. Consumers are heterogeneous in their taste for a particular software product and we assume they are uniformly located on the Hotelling line between 0 and 1. Denote t to be the mismatch cost parameter. Therefore a consumer located at x will incur the mismatch cost tx for using product 1 and t(1 x) for using product 2. Furthermore we assume the quality of both software products are large enough so that the market is fully covered (otherwise the problem becomes less interesting since each firm could act as a local monopoly). Then each consumer (for simplicity we use he to refer to a consumer later on) faces four options: buying software 1, buying software 2, pirating software 1, pirating software 2. Figure 1 depicts consumers choices. If a consumer located at x chooses to buy, his utility U i of buying and using a legitimate software product 1 or 2 is given by U 1 = q 1 + k(d 1 + D 3 ) tx p 1 (1) or U 2 = q 2 + k(d 2 + D 4 ) t(1 x) p 2 (2) 5

6 Consumer choice: Buy 1 Pirate 1 Pirate 2 Buy 2 0 Figure 1: Consumer demands in the duopolistic case 1 where D i (D 2+i ) is the demand for legitimate (pirated) software i, i = 1, 2. If a consumer chooses to pirate, he could get software for free by downloading or copying it. However, he may not get the fully quality q i by using pirated software. For instance, software firms may not provide patches for pirated software (August and Tunca 2008), so that computers using pirated version could be vulnerable for viruses, and a consumer s utility could decrease. Also, many software firms provide customer support only to those who own legitimate software products. Therefore a consumer naturally discounts the quality of pirated software. Consequently both the network benefit and mismatch cost of using pirated software decrease as a consumer does not value pirated software so high as legitimate one. Let θ i, i = 1, 2 be such discount factor for product 1 and 2. Then the utility of a consumer who uses a pirated version of software product 1 can be written in the following form: U 3 = θ 1 [q 1 + k(d 1 + D 3 ) tx] c 1 (e 1, e 2 ). (3) Penalty cost term c 1 (e 1, e 2 ) in (3) depends on the probability of a consumer being caught of using pirated software and subsequently required to pay penalty. As anti-piracy efforts e 1 exerted by firm 1 increases, a consumer is more likely to be caught using pirated software 1. Therefore we expect c 1 to increase with e 1, i.e., c 1 / e 1 > 0. Also there is a cross effect by anti-piracy effort e 2 from the other firm and we assume c 1 / e 2 > 0 as well. Similarly, the utility a consumer obtains by using a pirated software product 2 is given by: U 4 = θ 2 [q 2 + k(d 2 + D 4 ) t(1 x)] c 2 (e 1, e 2 ). (4) To proceed, we use the commonly used linear-quadratic framework to simplify the analysis. That is, we use a linear form of penalty cost c i (e 1, e 2 ) = a i e 1 + b i e 2, i = 1, 2 (5) and a quadratic cost of anti-piracy effort: r i e 2 i, i = 1, 2. 6

7 4. Analysis of the Model and Results 4.1 Benchmark: Monopoly Case We first analyze the case when there is only one software provider in the market. This analysis serves as a benchmark for the subsequent duopolistic analysis. Without the loss of generality, we assume that the monopolist is firm 1 and located at 0 of a Hotelling line where consumers are uniformly distributed. Penalty cost c 1 (e 1, e 2 ) becomes c 1 (e 1 ) = a 1 e 1 as e 1 is the only anti-piracy effort in the monopolistic case. The time sequence is the following. At the first stage, the monopolistic firm decides the anti piracy effort e 1. At the second stage, the firm decide the price of the software product p 1. At the third stage, consumers make their purchase or pirating decisions. We use backward induction to solve this problem. It can be shown that in the monopolistic case, the second and third stages can be merged into one stage and the final decisions will still be the same. For comparison with the duopoly s case, we keep two stages separate. At the third stage, a consumer can obtain a legitimate version, gaining utility U 1 given by Equation (1), or acquire a pirated one, gaining utility U 3 given by (3). Let x 1 be the point on the Hotelling line where a consumer is indifferent between purchasing and pirating and x 2 be the point where a consumer is indifferent between pirating or not using the software. In other words, the total demand is simply given by x 2. Then we have U 1 x=x1 = U 3 x=x1 (6) and That is and U 3 x=x2 = 0 (7) q 1 tx 1 + kx 2 p 1 = θ 1 (q 1 tx 1 + kx 2 ) a 1 e 1 (8) θ 1 (q 1 tx 2 + kx 2 ) a 1 e 1 = 0 (9) From (9), we can solve for x 2 : Plugging (10) into (8), we can get x 1 : x 2 = θ 1q 1 a 1 e 1 θ 1 (t k) x 1 = q 1 t + k q 1 t (t k) p 1 (1 θ 1 )t + 7 ( (10) 1 (1 θ 1 )t k ) 1 a 1 e 1. (11) t θ 1 (t k)

8 At the second stage, the firm chooses the selling price p 2 to maximize its profit: π 1 (p 1, e 1 ) = x 1 p 1 r 1 e 2 1 (12) Plugging (11) into the profit function (12) and differentiating the profit with price p 1, we can get the first order condition with respect to p 1 : π 1 (p 1, e 1 ) p 1 = q 1 t + k t from which we can solve for p 1 : θ 1 q 1 θ 1 (t k) 2p 1 (1 θ 1 )t + ( 1 (1 θ 1 )t k t 1 θ 1 (t k) )a 1e 1 (13) p 1 = 1 2 (1 θ 1)[q 1 + kq 1 (t k) + ( 1 (1 θ 1 ) k θ 1 (t k) )a 1e 1 ] (14) At the first stage, after one substitutes (14) into the profit function (12), the profit function becomes only a function of decision variable e 1. The firm needs to choose the effort to maximize its profit: π 1 = (1 θ 1)a 1 [q 1 + kq 1 e 1 4t (t k) + ( 1 k 1 θ 1 θ 1 (t k) )a 1 k 1e 1 ][ 1 θ 1 θ 1 (t k) ] 2r 1e 1 (15) We can get the optimal effort of anti-piracy as: e 1 = and plugging e 1 into (14), we can get p 1. q 1 + k q 1 t k 8r 1 t + ka (1 θ 1 )[ ka 1 θ 1 (t k) + a 1 a 1 1 ] θ 1 (t k) (1 θ 1 ) 1 θ 1 By using comparative statics analysis, we can study how different parameters can affect the optimal decision variables and the firm s profit. The results are summarized in Theorem 1 (Proofs of all theorems are omitted due to space constraint): Theorem 1. In the monopoly s case, we have the following results: 1. As network effect increases, the optimal anti-piracy effort decreases, the price increases and the firm s profit increases. That is, e 1/ k < 0, p 1/ k > 0, and π 1/ k > As software quality increases, the optimal anti-piracy effort increases, the price increases and the firm s profit increases. That is, e 1/ q 1 > 0, p 1/ q 1 > 0, and π 1/ q 1 > As perceived quality for using pirated software increases, the optimal anti-piracy effort increases, the price decreases and the profit decreases. That is, e 1/ θ 1 > 0, p 1/ θ 1 < 0 and π 1/ θ 1 < 0. 8 (16)

9 From Theorem 1.1, as the network effect increases, it is optimal for a software company to decrease its anti-piracy effort. This is similar to the one found in Conner and Rumelt (1991). The intuition is the following: as the network effect increases, a software company has incentive to reduce anti-piracy effort and allow a higher level of piracy so that more users will join the network even though they might choose to pirate. As a result, the software company can charge legitimate users a higher price due to increased network size and its profit increases. Therefore, when the network effect is strong, it is important for a monopolistic software provider to have a larger user base. On the other hand, as software quality increases, it becomes more attractive for both legitimate and pirating users. In this case, it is optimal for a software company to increase anti-piracy effort to reduce software piracy. At the same time, the company is able to charge legitimate users a higher price and increase its profit, as shown in Theorem 1.2. From Theorems 1.3, as quality of pirated software increases, it becomes more attractive to use pirated software. That means the competition between legitimate and pirated software becomes stronger. Then a software company s optimal response is to increase its anti-piracy effort to discourage the use of pirated software on the one hand, and to decrease its price to make legitimate software more competitive on the other hand. As a result, the monopolist s profit decreases as pirated software becomes attractive. Comparing Theorems 1.2 with 1.3, it is interesting to see that although anti-piracy effort always increases for an increase in either q 1 or θ 1, the behavior of profit is different. A monopolist s profit increases for a higher q 1 even with a higher anti-piracy effort since it becomes more attractive to purchase its product. However for a higher θ 1, it becomes more attractive to pirate and a firm puts in more anti-piracy effort and gets less profit. Next we explore a duopolistic case to see whether the same results hold in a competitive environment. 4.2 Duopoly Case We now study the setting that two firm are in the market.they lie at each end of the Hotelling line. That is, firm 1 lies at point of 0, and firm 2 at 1. The time sequence is similar to the monopoly case: At the first stage, both firms determine their anti-piracy effort e 1 and e 2 ; at the second stage, both firms set prices p 1 and p 2 ; at the third stage, consumers choose which type of software to use and whether to pirate or not. 9

10 At the third stage, a consumer located at point x 1 will get the same utility whether he uses pirated software product 1 or legitimate software product 1, as shown in Figure 1. Then U 1 in Equation (1) equals U 3 in (3) at x = x 1. That is, q 1 + k(d 1 + D 3 ) tx 1 p 1 = θ 1 [q 1 + k(d 1 + D 3 ) tx 1 ] c 1 (e 1, e 2 ) (17) Similarly, U 2 in Equation (2) equals U 4 in (4) at x = x 3 : q 2 + k(d 2 + D 4 ) t(1 x 3 ) p 2 = θ 2 [q 2 + k(d 2 + D 4 ) t(1 x 3 )] c 2 (e 1, e 2 ) (18) and a consumer at point x 2 will get the same utility whether using pirated product 1 or 2. Then, θ 1 [q 1 + k(d 1 + D 3 ) tx 2 ] c 1 (e 1, e 2 ) = θ 2 [q 2 + k(d 2 + D 4 ) t(1 x 2 )] c 2 (e 1, e 2 ) (19) As shown in Figure 1, the total demand for software product 1 is x 2, including pirated version and legitimate version and the total demand for software product 2 is 1 x 2. We shall have: D 1 + D 3 = x 2 (20) D 2 + D 4 = 1 x 2 (21) From (19), (20) and (21), we can solve for x 2 x 2 = θ 1q 1 θ 2 q 2 + (t k)θ 2 c 1 (e 1, e 2 ) + c 2 (e 1, e 2 ) (θ 1 + θ 2 )(t k) (22) From (17), we can get x 1 which is the demand for legitimate product 1 D 1, D 1 = x 1 = (1 θ 1)q 1 + (1 θ 1 )kx 2 p 1 + c 1 (e 1, e 2 ) (1 θ 1 )t (23) and from (18), we can get x 3 : x 3 = 1 D 2 = 1 (1 θ 2)q 2 + (1 θ 2 )k(1 x 2 ) p 2 + c 2 (e 1, e 2 ) (1 θ 2 )t (24) from which we can get the demand for legitimate product 2: D 2 = (1 θ 2)q 2 + (1 θ 2 )k(1 x 2 ) p 2 + c 2 (e 1, e 2 ). (25) (1 θ 2 )t The profits of firms 1 and 2 are given by: π 1 (p 1, e 1 ) = D 1 p 1 r 1 e 2 1 = (1 θ 1)q 1 + (1 θ 1 )kx 2 p 1 + c 1 (e 1, e 2 ) p 1 r 1 e 2 1 (26) (1 θ 1 )t 10

11 π 2 (p 2, e 2 ) = D 2 p 2 r 2 e 2 2 = (1 θ 2)q 2 + (1 θ 2 )k(1 x 2 ) p 2 + c 2 (e 1, e 2 ) p 2 r 2 e 2 2 (27) (1 θ 2 )t By solving p 1 and p 2 from π 1 (p 1, e 1 )/ p 1 = 0 and π 2 (p 2, e 2 )/ p 2 = 0, We can get prices of two software products at the Nash equilibrium: p 1 = (1 θ 1)q 1 + (1 θ 1 )kx 2 + c 1 (e 1, e 2 ) 2 (28) p 2 = (1 θ 2)q 2 + (1 θ 2 )k(1 x 2 ) + c 2 (e 1, e 2 ) 2 After plugging (28) and c i (e 1, e 2 ) = a i e 1 + b i e 2 into (26), we get the first order conditions of π 1 (p 1 (e 1, e 2 ), e 1 ) with respect to e 1 : π 1 = 2[(1 θ (1 θ 1)q 1 + (1 θ 1 )kx 2 + a 1 e 1 + b 1 e 2 ][ 1 )k( a 1 +b 1 ) (θ 1 +θ 2 + a )(t k) 1 ] 2r 1 e 1 = 0 (30) e 1 4(1 θ 1 )t Similarly, we have π 2 = 2[(1 θ (1 θ 2)q 2 + (1 θ 2 )k(1 x 2 ) + a 2 e 1 + b 2 e 2 ][ 2 )k(a 2 b 2 ) (θ 1 +θ 2 + b )(t k) 2 ] 2r 2 e 2 = 0 (31) e 2 4(1 θ 2 )t We can jointly solve the equilibrium anti-piracy efforts e 1 and e 2 from the first-order conditions (30) and (31). However, the expressions are too complicated to gain any analytical insights. To gain further insights, we consider the following two cases: a symmetric case where both firms parameters are the same and an asymmetric case where both firms parameters are the same except that software quality is different between two products Duoplistic Competition with Symmetric Firms When two firms are identical, we can get each firm s equilibrium effort as (29) e sym = (2q + k)(1 θ 1 ) 8r 1 t k a 1 +a 2 2θ 1 (t k) + a 1 1 θ 1 2(a 1 + a 2 ) (32) and the equilibrium price as p sym = (1 θ 1)q 1 + (1 θ 1)k 2 + (a 1 + a 2 )e sym 2 (33) Then we can show the following result from comparative statics analysis: Theorem 2. In the symmetric monopolistic case, we have the following results: 11

12 1. As network effect increases, both the equilibrium anti-piracy effort and price increase when the cross effect of anti-piracy effort is high. That is, e sym / k > 0 and p sym / k < 0 when the cross effect parameter a 2 is large enough. 2. As software quality increases, the equilibrium anti-piracy effort the price and the firm s profit all increase. That is, e sym / q 1 > 0, p sym / q 1 > 0, and π sym / q 1 > As perceived quality for using pirated software increases, the equilibrium anti-piracy effort increases; both the price and the profit decrease. That is, e sym / θ > 0, p sym / θ < 0, and π sym / θ < 0. Some results remain the same even in the duopolistic setting: Theorem 3.2 and 3 are the same as Theorem 1.2 and 1.3 in the monopolistic case. However, the result in Theorem 3.1 is quite interesting and different from Theorem 1.1 in the monopolistic case. On the one hand, a higher network effect makes anti-piracy effort less desirable, as one could see in the monopolistic case. On the other hand, under competition, a higher network effect could lead to a higher anti-piracy effort which could reduce the demand for the competitor s product (both legitimate and pirated). when the cross effect of anti-piracy effort is high, the second effect dominates the first effect and anti-piracy effort increases with network effect. 0.4 Anti-piracy effort k Figure 2: Anti-piracy effort without cross effect Even if the cross effect does not exist, i.e, coefficients of a2 = b2 = 0 in c i, we still obtain a result (shown in Figure 2) that is different from what one might have expected under the 12

13 monopolistic case. The result in Figure 2 is quite interesting. In the small network effect (k) region, when k increases, anti-piracy effort increases even though there is no cross effect from anti-piracy effort. Then such increase in anti-piracy effort is purely due to competition. When k further increases, network effect dominates the competition effect. Then anti-piracy effort decreases and each firm tries to increase its network by adopting a more lenient antipiracy policy. Furthermore, we also study how the change in the cross effect of anti-piracy effort influences the equilibrium results. Theorem 3. In the symmetric monopolistic case, an increase in the cross effect of the antipiracy effort leads to an increase in the equilibrium anti-piracy effort, the equilibrium price and each firm s profit. That is, e sym / a 2 > 0, p sym / a 2 < 0, and π sym / a 2 < 0. The result in Theorem 3 seems simple but is not that straightforward. When the cross effect of anti-piracy effort increases, anti-piracy effort becomes more effective in increasing the cost of using competitor s pirated software, not its own pirated software. Then why does a firm still have incentive to increase anti-piracy effort? Also how can a firm even increase its price? The reason is due to the competitive effect. As the competitor s pirated software becomes less attractive, the competitor can increase its price; in turn the first firm can increase price as well. So anti-piracy effort indirectly benefits the firm itself when the cross effect increases. Then each firm is motivated to increase anti-piracy effort even if the direct benefit goes to its competitor, knowing that its competitor is also motivated to do the same; ultimately, both firms benefit. Theorem 3 discovers the interesting role that the cross effect plays an increase in the cross effect can weaken the competition between two software firms. In the next subsection, we explore how relaxing the symmetric assumption can produce richer results Duoplistic Competition with Asymmetric Quality We first study how two companies with software of different qualities react by focusing on anti-piracy effort. Theorem 4. In an asymmetric case with different software quality q i, 1. both firm 1 s anti-piracy effort e 1 and firm 2 s effort e 2 increase with either software 1 s quality q 1 or software 2 s quality q 2 ; That is, e 1 / q i > 0 and e 2 / q i > 0, i=1,2. 13

14 2. the firm with higher quality software exerts more anti-piracy effort; The difference in effort increases with quality difference. When firm 1 s software quality increases q 1, it becomes more tempting to pirate the software. Firm 1 has an incentive to increase anti-piracy effort to reduce piracy and make purchasing the software more desirable. When firm 1 exert more anti-piracy effort, it is more costly to using pirated software product 1, leading to more users using pirated software 2. In turn, firm 2 will exert more effort in anti-piracy to shift some pirate users to legitimate users. In other words, each firm s anti-piracy effort increases even if only one firm s software quality increases, as Theorem 4.1 shows. Then which firm will spend more effort in preventing piracy? As Theorem 4.2 shows, the higher quality firm should put more effort since its product is more likely to be the target for piracy. This matches our daily experience: larger software firms are more actively involved in anti-piracy lawsuits, as their software has more functionalities and is more likely to be pirated. Theorem 5. In an asymmetric case with different effort cost r i, 1. both firm 1 s effort and firm 2 s effort will decrease with either firm s effort cost. That is, e 1 / r i < 0 and e 2 / r i < 0, i=1,2. 2. the firm with higher effort cost exerts less anti-piracy effort; the difference in effort increases with cost difference. The result that a firm s anti-piracy effort decreases with its effort cost is obvious. When its cost of anti-piracy is high, a firm will choose to decrease anti-piracy. However, it is not immediately clear why the other firm will decrease its effort as well, as Theorem 5.2 shows. The explanation is the following. When one firm s anti-piracy effort decreases, more users tend to pirate this firm s product. The pressure of piracy for the other firm will decrease, so the other firm is able to decrease its anti-piracy effort. 5. Conclusion In this paper, we analyzed the problem how software firms should price their products and invest in anti-piracy measures to prevent piracy in a competitive environment. 14

15 In a symmetric duopolistic setting, we show that when cross effect exists, the anti-piracy effort by a firm involved in price competition may increase or decrease with the network effect, which is different from previous results. Even without cross effect, anti-piracy effort may increase or decrease with the network effect in a competitive setting. Another interesting result is that when cross effect increases, each firm increases its anti-piracy effort. Although such increase benefits a competitor directly, this competitor can then increase price since there is less pressure from piracy; the first firm in turn can do the same, benefiting from cross effect ultimately. In an asymmetric duopolistic setting, we find that an increase in one product s quality causes two firms to increase anti-piracy effort, with a higher quality firm putting more effort. On the other hand, when anti-piracy cost becomes more expensive for one firm, then interestingly both firms will decrease anti-piracy effort due to the effect of competition. Our contributions are in two ways. First, we introduced the cross effect concept which commonly exists in reality and has not attracted enough interests in software piracy literature. Second, we find that anti-piracy effort may increase when network effect increases in some circumstances. This result is theoretically interesting, because, it contradicts the previous belief that firms can take advantage of high network effect and exert less anti-piracy effort. The implication to software companies is that when competition exists, they may need to exert more effort if the cross effect of anti-piracy effort is high. There are several extensions possible to our paper. First, we use a Hotellting model to study competition between two software products. Although many papers use this model, in reality, vertical competition on quality also widely exists. Second, it might be interesting to extend our study to include software versioning which is another strategy to combat piracy. 15

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18 Appendix Notation Table 1: Summary of notation Description Parameters q i quality of software product i, where i {1, 2} θ i discounted quality of pirated software relative to a legitimate version, where i {1, 2} k coefficient of a software product s network externality x a consumer s location on the Hotelling line t coefficient of a consumer s mismatch cost of using a legitimate version t coefficient of a consumer s mismatch cost of using a pirated version π i profit of software firm i, where i {1, 2} U i utility a consumer gets from using ith legitimate software product i 1, 2 U 2+i utility a consumer gets from using pirated software product i, where i 1, 2 D i demand of legitimate software product i, where i {1, 2} D 2+i demand of pirated software product i, where i {1, 2} Decision Variables p i price of software product i, where i {1, 2} e i anti-piracy effort exerted by software company i, where i {1, 2} 18