Market must be defended: The role of counter-espionage policy in protecting domestic market welfare

Governments of advanced economies are extremely concerned about the illicit acquisition of information on critical technologies employed by their industries, and countering this economic espionage is quickly becoming one of their top priorities. The present paper advances the theoretical analysis of the interaction between economic espionage and counter-espionage, and presents a first approximation to an inquiry into the rationale for the influence of market competition in its dynamics. The proposed model assumes a country with a one-market economy open to international trade whose product is supplied by domestic firms. Moreover, successful economic espionage implying market entry of foreign firms would harm domestic welfare. Considering counter-espionage policy as entry barrier and sufficient efficiency in espionage and counter-espionage efforts, the analysis of the benchmark case characterized by no foreign consumer and one foreign firm suggests that demand characteristics play an important role in the complex influence of competition in espionage. Irrespective of this, optimal counter-espionage effort is always positive although negatively affected by competition. JEL classification: C72; K42; L10


Introduction
The illicit attempts by countries to acquire information on critical manufacturing processes and technologies employed by industries in other countries is considered economic espionage. 1 Although the information technology revolution that the world has undergone since the 1990s has facilitated economic That branch of economic espionage can have extremely important conse- 10 quences not only for the companies affected, but also for the general economy of the country being spied on. Although it is arduous to quantify the magnitude of these consequences (Nasheri, 2005, p In this study, the product commercialized in this international market is Weighing up the potential loss in welfare and the cost of this effort, the aim of C's counter-espionage policy is to thwart the foreign attempts at obtaining the necessary information to replicate the secret technology needed to participate in the market. Therefore, this policy is considered in its role as barrier to 85 entry with the aim of protecting the welfare generated by the market to C's participants. In this context, C decides the counter-espionage effort taking into account that the failure of these attempts also depends on the effort exerted in the espionage activities. Nevertheless, this espionage effort is unobservable to C, similarly as the counter-espionge effort is unobservable to F . 90 As mentioned above, the analysis carried out in the present paper, as a first approximation towards studying the rationale behind the influence of the level of market competition on the dynamics of the interaction between espionage and counter-espionage, aims to highlight the most elemental aspects of this influence. This aim has two important implications. The general theoretical 95 framework outlined above focuses on the simplest possible characterization of the international market and the aspects related to the product value chain such that the higher the initial level of market competition (that is, the larger the initial number of C's firms in the market), the higher the aggregate welfare of all the market participants. However, despite this simplicity, this general 100 theoretical framework still implies some complexities which would obscure the aim of the paper, such as the existence of cases in which competition might not have such a positive effect on the aggregate welfare of C's market participants and the relatively high number of parameters.
So, although the paper develops in some depth the general theoretical frame-105 work, to avoid these complexities the analysis focuses on a benchmark case in which there is only one firm from F interested in participating in the international market and the product commercialized is only demanded by C's consumers. In this benchmark case, the initial level of market competition always has not only a positive effect on the aggregate welfare of C's market partici-110 pants, 6 but also a negative one on the welfare generated by the participation of F 's firm in it. These effects are behind the results obtained which, despite avoiding the complexities mentioned above, suggest that the most elemental influence of the initial level of market competition is characterized by a non-trivial relationship between espionage and this level of competition. 115 More precisely, considering that both C and F are sufficiently efficient in exerting their respective efforts, these results show two basic patterns in their behavior. First, C is always willing to increase the effort in its counter-espionage policy the higher F 's effort espionage, and F is always prepared to decrease this effort the higher the effort exerted by C. Namely, C regards every espionage 120 effort as a strategic complement while F regards every counter-espionage effort as a strategic substitute, and therefore, there is a strategic asymmetry (Tombak, 2006) between C and F . And second, both F and C will exert smaller efforts in their respective espionage and counter-espionage activities given the effort of the rival the higher the initial level of competition in the market. In the case 125 of F , the profits its firm can obtain from participating in the market decrease with this level of competition. In the case of C, the higher the initial level of competition, the smaller the loss in welfare of its domestic participants implied by the participation of F 's firm in the market.
These reductions in efforts, together with the strategic asymmetry between C 130 and F , imply that, on the one hand, a higher initial level of market competition always has a negative effect on the equilibrium counter-espionage effort and, on the other hand, the effect of this higher level of competition on the equilibrium espionage effort depends on the relationship between both reductions, which is influenced by the characteristics of market demand. In particular, an elastic 135 6 As considered in the last section of the paper together with other potential lines of future research, extending the analysis to the general theoretical framework, and therefore, to cases in which market competition might not have such positive effect, would move this study of counter-espionage even closer to the strand of theoretical research analyzing the desirability of regulation of market entry, briefly presented in the next section of the paper. demand would enhance the positive effect of a higher initial level competition on the aggregate welfare of C's market participants, given that the decrease in prices would lead to a proportionally higher increase in sales. An elastic demand would also weaken the negative effect of this higher competition on the profits that F 's firm would obtain from participating in the market. Similar positive 140 effects of market's willingness to pay come through in its influence in when the demand can be considered as sufficiently elastic.
This implies that if the demand were sufficiently elastic (inelastic), the reduction in C's counter-espionage effort due to a higher initial level of competition would be high (small) relative to F 's reduction. Consequently, given the 145 strategic asymmetry between C and F , the latter would be better off increasing (decreasing) its espionage effort and, as a result, an increase in the initial level of market competition would have a positive (negative) effect on the equilibrium espionage effort. However, there are two crucial aspects related to this influence of elasticity of demand. Firstly, considering the demand as sufficiently 150 elastic/inelastic in this sense also depends on the pair of initial number of firms in the market defining the increase in the initial level of competition. Secondly, there exists a sufficiently high initial number of firms in the market such that, no matter how elastic the demand is, the reduction in C's effort due to an increase in the initial level of competition is always small relative to F 's reduction. 155 Namely, there exists a critical initial number of firms from which the equilibrium espionage effort decreases with the initial level of competition. This role played by price elasticity of demand, influenced by market's willingness to pay and contingent to the initial number of firms in the market, is behind the complex dynamics of the equilibrium espionage effort under variations 160 in the initial level of market competition. According to this role, the smaller the elasticity of demand, the fewer pairs of initial number of firms in the market exist with respect to which the demand can be considered sufficiently elastic and, therefore, the above-mentioned critical number of firms decreases. If the elasticity of demand is low enough, the equilibrium espionage effort decreases 165 with the initial level of competition regardless of the initial number of firms competing in the market. However, when the elasticity of demand increases, the critical number of firms becomes larger and the behavior of the equilibrium espionage effort, for competitive intensities lower than the one defined by this critical number of firms, does not necessarily decrease. Furthermore, it strictly 170 increases when elasticity of demand is sufficiently high.
The remainder of the paper is organized as follows. Section 2 presents the related literature and Section 3 sets out the general model. Section 4 develops this general model and analyzes the equilibrium of the benchmark case. Espionage and counter-espionage efforts in the equilibrium of the benchmark case 175 are discussed in Section 5, while Section 6 is devoted to analyzing how the level of market competition affects these efforts. Finally, Section 7 concludes the paper and suggests several lines for future research.

Related literature
The present paper contributes to the model-based analyses of economic es-180 pionage and counter-espionage activities. This section briefly reviews some of the most important contributions to these analyses and shows that, given the different roles in which economic espionage and counter-espionage feature, these analyses are related to different strands of the theoretical literature.
One of the first theoretical studies of economic espionage, by Whitney and 185 Gaisford (1999), also analyzed its interaction with counter-espionage, but its approach differs in several aspects from the analysis in the present paper. Firstly, Whitney and Gaisford's (1999) model considers a context in which there are only two firms competing in an international market and the objective of economic espionage is to obtain the cost-reducing technology owned by one of the firms.

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Secondly, although the effort exerted in counter-espionage is endogenous and the effects on consumers' surplus of its interaction with espionage are considered, the aim of such counter-espionage is to maximize the expected profit of the domestic firm, not the expected domestic welfare generated by the market. Lastly, they do not study the effects of the level of competition in the market on the dynamics 195 of the interaction between espionage and counter-espionage efforts.
Despite these differences, Whitney and Gaisford (1999), like the present paper, considered successful espionage as an illicit technology transfer (Glitz and Meyersson, 2020). But espionage can also be considered a medium of direct knowledge spillovers across countries (see Lee, 2005, footnote 2, p. 337). This 200 is the perspective of the more recent theoretical analysis by Grabiszewski and Minor (2018), in which a firm in a given country commits a certain effort towards R&D and a foreign firm conducts espionage activities in an attempt to obtain the former's innovation. As the authors pointed out, this perspective on espionage parallels the intellectual property and patent literature. In this 205 sense, it would also be related to the study of secrecy and defensive publishing as alternatives to patenting (for example Johnson, 2014), and the analysis of commercial piracy. Some relatively recent theoretical studies of the latter and its regulatory aspects are Martínez-Sánchez (2010), López-Cuñat and Martínez-Sánchez (2015), Banerjee (2011Banerjee ( , 2013 and Poddar (2012, 2018). 210 Grabiszewski and Minor (2018) also included in their analysis the counterespionage policy of the government in the innovator firm's country, but in its role as a cost-enhancing barrier to the acquisition of the innovation by the foreign firm and, therefore, to the latter's participation in the same market as the innovator firm. Therefore, despite their differences in nature, the exogenous 215 counter-espionage policy in Grabiszewski and Minor's (2018) model shares the entry-barrier spirit of the endogenous counter-espionage policy considered in the present paper. This model-based study of the entry-barrier aspect of counterespionage is also connected to the theoretical literature analyzing the desirability of regulation or deregulation of market entry and, therefore, the convenience of 220 establishing or relaxing barriers to entry.
The theoretical literature studying the desirability of market entry regulation is based on a comparison of the number of firms that would operate in a market under free entry with the number of firms that would maximize the welfare of market participants. This strand of research found market circumstances under 225 which the latter is smaller than the former and, therefore, regulation of entry might be desirable. Two prominent early contributions along these lines are Mankiw and Whinston (1986) and Suzumura and Kiyono (1987). Among the wide array of later studies considering different market environments, a recent contribution by Kang  Model-based studies dealing with market entry deregulation examined its convenience also under several different circumstances. One of the first of these studies is by McCormick et al. (1984), presenting arguments according to 235 which monopoly deregulation might not be as convenient as generally thought.
While some later contributions explicitly highlighted circumstances that limit the scope of these arguments (e.g., Crew and Rowley, 1986;Sutter, 1997, 2000), others found that entry deregulation might not be desirable in contexts in which the market is not necessarily dominated by a monopoly.

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Focusing on a market regulated through licenses, Kang and Lee's (2001) model captures market characteristics under which social welfare is harmed by partial deregulation due to rent-seeking by the incumbents and the potential entrants. According to the generalization of this model by Lee and Cheong (2005), although complete deregulation is unlikely to be obtained, they found a 245 particular case in which partial deregulation is likely to be welfare-improving.
However, Lee and Cheong (2005) agreed with Kang and Lee (2001) in indicating the convenience of carefully designing the deregulation process to reduce the losses due to rent-seeking (for example, auctioning off new licenses).
Returning to the theoretical analysis of espionage, Sakai (1985), Billand et al.

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(2016) and Kozlovskaya (2018) did not consider measures to counter espionage activities, but they paid special attention to the effect of information-gathering activities among firms on social welfare. The effects of such activities on market entry are analyzed by a relatively new strand in this literature, represented by Barrachina et al. (2014Barrachina et al. ( , 2021 and Barrachina (2019). They focused on a 255 context, not necessarily characterized by international free trade, in which an incumbent wishes to deter a potential entrant from entering the market. 7 3. The general model The proposed model considers the interaction between economic espionage and counter-espionage based on ruining measures 8 when the objective of gath-260 ering information on sensitive technologies is the participation of firms from the spying country in a particular market. 9 As mentioned in the Introduction, given that countries define their counter-espionage policy based on their entire economy (in which different markets are interrelated), the model assumes that the economy of the country countering espionage consists of only one market, 265 although it is open to international trade. The aim of this assumption is to allow to isolate the influence of the level of competition in a particular spied market on the dynamics of the interaction between economic espionage and counter-espionage, this being the focus of the present paper.
The model assumes that the product commercialized in this international 270 market is initially only supplied by n firms from this country, denoted by C, 10 the reason being that participation in this market requires a specific and secret technology only known by C's firms. However, there are m firms from a foreign country F interested in participating in this market, with a view to supplying the product from their country. Therefore, the government of F , knowing this 275 requirement to participate in the market, decides what level of espionage effort to exert in an attempt to obtain the necessary information to replicate this 7 See Barrachina et al. (2014Barrachina et al. ( , 2021 and Barrachina (2019) for a more detailed discussion of the sparse theoretical literature analyzing espionage and information-gathering activities not only in economic contexts. 8 The objective of these measures is to thwart the success of economic espionage activties providing programs, protocols and tools at company level. See the Introduction for more details about this category of counter-espionage measures and how it differs from the penaltyenhancing measures. 9 See the Introduction for real examples of this case of interest. 10 The meaning of the term "from" in this sentence is twofold. It means that, on the one hand, these n firms are from C. And on the other hand, they supply the product from C. secret technology. In particular, F decides an espionage effort 0 ≤ e ≤ 1 whose main objective is to maximize the expected economic welfare related to the participation of its economic agents in the market. If the espionage is 280 successful, which happens with probability e, there is a chance for F to acquire the information to replicate the secret technology. In this case, its firms could start supplying the product, competing with the n companies from C.
Weighing up the possibility of espionage, the government of C decides on an effort 0 ≤ c ≤ 1 in deploying a counter-espionage policy (focused on ruining 285 measures, as stated above), which is successful with probability c. A successful outcome of this counter-espionage policy means that C was able to prevent F from acquiring the necessary information to replicate the secret technology needed to participate in the market. For instance, c = 1 would imply that F is prevented from obtaining such information regardless of the effort exerted 290 in espionage. 11 The concern of C's government when deploying this counterespionage policy is the protection of the aggregate economic welfare of its domestic market participants. 12 Therefore, analysis of this interaction between economic espionage and counter-espionage must focus on contexts in which the participation of F 's firms in the international market is harmful for the aggre-295 gate welfare of C's participants. These contexts in the present framework are 11 Exerting effort in espionage or counter-espionage activities implies, on the one hand, funds and time to obtain the required agents and infrastructures, and on the other hand, funds and time spent by them carrying out the activity. 12 As stated in Section 2, C's counter-espionage shares, despite differences in nature, the the present study, a crucial difference with all these models (which actually makes results incomparable) is that a counter-espionage framework such as the one considered by the present model must be based on contexts where the ideal situation would be that no firms from a spying country were willing to participate in the market.
described in detail in the next section.
It could be considered that information leakages with respect to the secret technology may occur more easily when the initial number of C's firms in the market is high (for example, the higher the number of firms, the more likely 300 the existence of an insider willing to accept a certain bribe for providing the information about the secret technology). 13 However, since the theoretical characterization of the present model is not wanted to compromise too much the analytical tractability, it abstracts from the complexities implied by this aspect 14 (continuing with the example, finding such an insider among a higher 305 number of firms would imply devoting more time, this is, exerting higher effort in espionage) assuming that, as explained above, the effectiveness of both espionage and counter-espionage in their interaction only depends on the efforts exerted in them.
We specify that such interaction is modeled as a two-stage game of complete 310 but imperfect information, G (e, c), whose timing is the following. In Stage 1, both F and C decide simultaneously and independently their respective espionage and counter-espionage efforts. Then, F decides an espionage effort knowing that the probability of acquiring the necessary information to replicate the secret technology needed by its firms to participate in the market not only 315 depends on this espionage effort, but also on the effort C may exert to prevent the acquisition of this information, which is unobservable to F . C decides how much effort to exert in this counter-espionage policy knowing that the probability of F 's failure to acquire the information also depends on its espionage effort, which is unobservable to C.

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In Stage 2 there are two possible scenarios. If the espionage conducted by F is not successful (probability 1 − e), or if it is successful but so is the counterespionage policy of C (with probability ec), F does not obtain the necessary information to replicate the secret technology and its firms cannot participate 13 We thank an anonymous referee for this comment. 14 This is an interesting line for future research, as detailed in the last section of the paper.
in the market (Scenario 1 ). The probability of this scenario is 1 − e + ec. If the espionage carried out by F is successful (probability e) and the counterespionage policy of C is not (probability 1 − c), F acquires the information to replicate the secret technology and the m firms from F can participate in the market, competing with the n firms from C (Scenario 2 ). The probability of this second scenario is e(1 − c). policy. 16 Nevertheless, the analysis carried out in the present paper will consider the whole cost spectrum, and how these particular cost structures contribute to facilitating the achievement of the paper's main objective.
A second consideration is the international market. As already stated, the 345 product commercialized in the market is initially only supplied by n identical 15 Assuming that the application by C's firms of the programs, protocols and tools provided by C's counter-espionage policy is costless also contributes to this aim, allowing the attention to be focused on the case of interest in which such an application is affordable for every firm. 16 Convexity in the cost of these efforts captures the limited nature of the resources, funds and time, devoted by each country's government to espionage or counter-espionage activities.
Given the limited nature of these resources, the higher the amount of resources devoted to one of these activities, the higher the marginal value of other governmental activities. In other words, the marginal opportunity cost of funds and time devoted to one of these activities is increasing and, therefore, so must be the marginal cost of the effort exerted.
firms from C (where n ∈ Z + and defines the initial level of market competition).
Moreover, the product is homogeneous and firms competeà la Cournot. With respect to the demand, it is assumed that all the consumers have identical preferences and α, β ∈ [0, 1], such that α+β ≤ 1, are, respectively, the proportions of 350 C's and F 's consumers in the international market. Therefore, consumers from other countries represent a proportion 1 − α − β of the international demand, which is given by the following inverse demand function: where Q = n i=1 q i is the total amount of product in the market; q i is the amount of product supplied by firm i, i = 1, ...., n; and a, b are strictly positive 355 parameters (a, b > 0) characterizing the demand for the product. 17 In particular, it is well known that price elasticity of demand is inversely related to parameter b, and a represents the market's willingness to pay for the product.
As shown later, these two parameters play an important role in the most elemental effect of the initial level of market competition on the dynamics of the 360 interaction between F 's espionage effort and C's effort in his counter-espionage policy.
Lastly, we consider the aspects related to the value chain of the product. In this respect, production and export costs are assumed to be zero to simplify the analysis as explained above. The cost of the initial investment to participate in 365 the international market is also assumed to be equal to zero not only to simplify the analysis but also to focus it on the case of interest in which this cost is 17 This market characterization shares some aspects with those considered in some of the previously mentioned theoretical studies of market entry regulation/deregulation. For example, this is the same characterization as in Kang and Lee (2001), except for its international conceptualization in the present model. Note that, according to this conceptualization, model sufficiently low that the m firms from F have incentives to participate in the market.
As discussed later, although under the present framework C's firms are al-

Equilibrium analysis
As stated in the previous section, the main concern of C and F when deciding their respective counter-espionage and espionage efforts is the welfare 380 generated by the international market for their own economic agents participating in it. According to G (e, c), C and F can decide their efforts in the first stage of the game anticipating the welfare for their respective economic agents in each possible scenario of the second stage, and therefore, G (e, c) can be solved backwards. More precisely, by applying backward induction the two-stage game 385 G (e, c) can be analyzed as a one-shot game of imperfect information defining the appropriate objective (payoffs) functions for C and F including the welfare generated by the market. The following subsection deals with the definition of these objective functions. As stated above, C and F decide their counter-espionage and espionage efforts in the first stage of G (e, c), taking into account the welfare generated by 18 The willingness of every firm from C to carry out a perfect application of the programs, protocols and tools provided by the counter-espionage policy is ensured in this framework by the assumed costeless nature of this application (considered before) and the negative effect of the participation of F 's firms in the market. the international market for their own economic agents in the second stage of the game. This welfare is represented by the surpluses that economic agents obtain from participating in the market. Note that the assumption of the general model 395 that all the firms' expenses related to the product are equal to zero implies that firms (considered as sellers) are the only agents involved in the product value chain and, therefore, suppliers' surplus represents welfare on that side of the market. 19

Welfare generated by the international market and the objective functions
The surpluses that economic agents obtain from participating in the inter-400 national market in the two possible scenarios considered in the second stage of G (e, c) are summarized in the following lemma, whose proof is presented in the Appendix.
Lemma 1. Consider the second stage of G (e, c). The surpluses generated by the international market for its participants in Scenario 1 are: where SS n is the suppliers' surplus of C's domestic firms and CS n is the surplus of all the consumers in the international market. The surpluses in Scenario 2 are: where SS F n+m and SS C n+m are the suppliers' surpluses of F 's and C's firms respectively, and CS n+m is consumers' surplus.
As shown in the proof of the following lemma (presented in the Appendix), the appropriate payoff functions for C and F such that G (e, c) can be analyzed as a one-shot game of imperfect information can be easily obtained taking into  Lemma 2. The appropriate payoff functions for F and C, denoted by U F and U C respectively, such that G (e, c) can be analyzed as a one-shot game of imperfect information are the following: Let us next study in some depth these payoff functions. In this regard, note that (7) and (8) can be written as where ∆SS C = SS C n+m − SS n is the variation in suppliers' surplus of C's 430 domestic firms and ∆CS = CS n+m − CS n the variation in the surplus of all the consumers in the international market, both due to F 's firms participation in it. Namely, as shown in (11) and (12), without considering the cost of their respective efforts, the payoff functions of F and C can be divided in two parts.
The first part is determined by surpluses of each country's participants in the 435 market in the scenario in which F 's espionage activities are not successful. The second part is the expected variation in these surpluses under F 's potential acquisition of the information to replicate the secret technology and its firms' participation in the market. It is easy to see that and In the case of F , the expected variation in welfare following the potential success of its espionage activities is always positive because all the consumers benefit from F 's firms participation in the market (∆CS is always strictly positive) since it will imply a lower market price and higher quantities sold. However, firms from C are always adversely affected by such participation (∆SS C 445 is always strictly negative) because of the decrease not only in the market price but also in the quantity of product sold by each of them. Thus, the success of F 's espionage activities has two opposite effects on the aggregate welfare of C's participants in the market: positive for consumers and negative for firms. It is relatively easy to see that the net variation in this welfare due to such success, Therefore, although the payoff functions of F and C given by (9) and ( (15), is non-negative only ifᾱ < α ≤ 1 and m ≥m, whereᾱ Following Lemma 3, there are two contexts in the framework considered 470 by G(e, c) that imply a negative expected variation in the aggregate welfare of C's participants in the market under F 's potential success in its espionage activities 20 and, therefore, that justify C's counter-espionage policy. The first one is when C's consumers do not participate in the international market, α = 0, or when they represent a relatively small proportion of the market demand, 475 α ∈ ]0,ᾱ]. In the first case, C is only concerned about the surplus obtained by its domestic firms which, as stated above, is always negatively affected by the participation of F 's firms in the market. In the second case, C also takes into account the positive effect of such participation on the surplus of its consumers.
However, the proportion of the international demand they represent is so small that this positive effect never compensates the negative one seen on the surplus of C's firms.
This negative effect, furthermore, is not always compensated when C's consumers represent a relatively high proportion of the international demand, α ∈ ∆W < 0, also depend on this initial level of market competition, since the thresholdsᾱ andm defining these contexts, given by (16) and (17)  license that would characterize the market deregulation process. They found that such partial deregulation would harm market welfare regardless of the initial level of market competition due to rent-seeking by the incumbents and the potential entrants. However, although this benchmark case implies that entry also means a loss in welfare no matter the initial number of C's firms in the market, it is due to the foreign origin of the firm and represents an adequate context for analyzing counter-espionage.
and m = 1 in (9) and (10), This benchmark case has two relevant implications. Firstly, the aggregate surpluses of C's market participants taken into account by C in its payoff func-530 tion, given by (19), are not only increasing (as stated above) but also concave with respect to the initial level of market competition. Secondly, the surplus taken into account by F in its payoff function, given by (18), namely the profit its firm will obtain from participating in the market if espionage activities are successful, is decreasing and convex with respect to this level of market compe- can be analyzed as the one-shot game of imperfect information defined by the payoff functions of F and C given, respectively, by (18) And the critical point of U F by: In addition, on the one hand, . Note that γ > 0 and δ < 1 for all a, b > 0 and n ∈ Z + .

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The above discussion leads to the following corollary, which defines the bestresponse function of each country.
Corollary. The unique best effort of each country in response to the effort of the other country is given by the following best-response functions. The bestresponse function of C is: And the best-response function of F is: Consequently, and as to be expected, in spite of the necessary constraints on In the case of F , δ < 0 implies a characterization of the market in which exerting the maximum effort in the espionage activities is never justified (not even as a response to a nonexistent counter-espionage policy) by the economic welfare generated through the participation of F 's firm in the market. It is clear 590 that such characterization of the market is defined, according to the market's willingness to pay for the product, by a < a 1 , where: Only the participation of F 's firm in a market characterized by a ≥ a 1 generates a sufficiently high enough economic welfare to justify exerting the maximum effort in espionage activities as a response to sufficiently low enough 595 efforts in the counter-espionage policy. 23 Consistently, δ < 1 is the upper bound for the range of C's counter-espionage efforts for which F is prepared to exert the maximum effort in espionage (0 ≤ c ≤ δ), and its increasing behavior with respect to a reflects that the higher the welfare implied by the characteristics of the market, the higher the range.
Therefore, C's maximum counter-espionage effort is only justified as a response to high enough efforts exerted in espionage 25 if the loss in welfare implied by the participation of F 's firm in the market is sufficiently high enough, which 610 is the case in a market in which a ≥ a 2 . Consistently, γ > 0 is the lower bound for the range of strictly positive F 's espionage efforts for which C is willing to exert the maximum effort in counter-espionage (γ ≤ e ≤ 1), and its decreasing behavior with respect to a reflects that the higher the loss is in the aggregate welfare of C's market participants, the higher is this range. 615 23 Remember that exerting the maximum effort in the counter-espionage policy would imply that F 's firm is prevented for sure from participating in the market, and therefore, F 's best response is to exert zero effort in espionage, as refelected in its best-response function given by (23). 24 Remember that, as stated in the previous subsection and according to Lemma 3, the benchmark case G b (e, c) ensures that this effect is strictly negative regardless of the initial level of market competition. Section 6 analyzes this negative effect in more detail. 25 As specified in C's best-response function given by (22), the best response for C when there is no espionage activity is to deploy no counter-espionage policy.
Note that a 2 > a 1 . This means that, even though the maximum effort in C's counter-espionage policy ensures F 's firm is prevented from participating in the market and generating a loss in welfare to C's participants while F 's maximum effort in espionage does not necessarily guarantee its firm's participation in the market, market characteristics justifying the former are more demanding than 620 the ones justifying the latter. This reflects that, although the participation of (2) Both c, given by (24), and e, given by (25) Proof. See Appendix 650 Therefore, although the specification of the best-response functions given by (22) and (23)  As shown in Figure 1(a), when 0 < a <ā 1 (which implies that δ < 0 and γ > 1) exerting the maximum effort is never cost justified either in F 's espionage activities or in C's counter-espionage policy. Figure 1(b) shows that, under the first market characterization, just the participation of F 's firm in a 660 market characterized by a =ā 1 generates a high enough economic welfare to justify exertion of maximum effort in espionage activities, but only if C exerts zero effort in its counter-espionage policy. Accordingly, the upper bound for the range of C's counter-espionage efforts for which F is prepared to exert the maximum espionage effort (0 ≤ c ≤ δ) is equal to zero (δ = 0) when a =ā 1 .

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However, such participation does not imply a high enough loss in the aggregate welfare of C's participants to justify the cost of exerting the maximum effort in the counter-espionage policy, not even as a response to maximum effort in espionage activities.
The higher the welfare generated by the participation of F 's firm in the mar-670 ket, the higher the range of C's counter-espionage efforts for which F is prepared to shoulder the cost of exerting the maximum effort in its espionage activities.
Nevertheless, maximum counter-espionage effort is not included in this range because it implies that F 's firm is prevented for sure from participating in the market and, therefore, F 's best response is to exert zero effort in espionage. As 675 stated above, consistently, δ is increasing in a but always smaller than 1, as shown by Figures 2 and 3.  So, on the one hand, as shown by Figure 2(a), an intermediate characterization of the market exists (a 1 < a < a 2 ) in which F is prepared to exert the maximum effort in espionage even in the context of an active counter-espionage 690 policy, but exerting the maximum effort in this counter-espionage policy is still never cost justified. On the other hand, as Figure 3 shows, only under the third market characterization (a > a 2 ) does the participation of F 's firm imply a high enough loss in the aggregate welfare of C's participants such that C is willing to exert its maximum effort when the effort exerted in the espionage activities is 695 not necessarily the maximum but high enough. As stated above, the decreasing behavior of γ with respect to a captures the increasing loss in welfare mentioned previously, and the fact that it is always strictly positive is consistent with C's best response to no espionage activity, to deploy no counter-espionage policy.
A Nash equilibrium of G b (e, c) is given by the espionage and counter-espionage 700 efforts such that each country's effort is the best response to the effort of the other country. Technically, this Nash equilibrium is given by the intersection of the best-response functions defined by (22) and (23). Therefore, lemmas 4 and 5 enable characterization of the equilibrium espionage and counter-espionage efforts in G b (e, c), which is done through the following proposition. has a unique Nash equilibrium in pure strategies and is characterized by e * and c * such that e * = e and c * = c, where c and e are given by (24) and (25) respectively.
(2) This equilibrium exists for all the parameters that characterize G b (e, c),  As stated by Proposition 1 in the previous section, the unique stable situation in the strategic interaction between F and C described by G b (e, c) is one in which both countries exert some strictly positive effort in their respective espionage and counter-espionage activities, but smaller than the maximum. This 720 is true regardless of the specific market characteristics in terms of consumers' maximum willingness to pay for the product, price-elasticity of demand and initial level of competition.
In response to these results, the explanation is clear as to why there exists no particular specification of the benchmark case represented by G b (e, c) such that 725 F and C exert zero effort in their respective espionage and counter-espionage activities in equilibrium. In a hypothetical situation in which C exerted zero effort in counter-espionage, and therefore, there was no external impediment for F to obtain the information to replicate the secret technology, F would always be able to find a strictly positive effort for its espionage activities such that the expected economic welfare from the participation of its firm in the market compensated for the cost of that effort. 26 As discussed in the previous section, only a market characterized by a ≥ a 1 justifies exerting the maximum effort in the espionage activities in this hypothetical situation of no counter-espionage policy, as shown by the corresponding 735 specifications of F 's best-response function in Figures 1(b), 2 and 3. However, scenarios characterized by no counter-espionage policy by C but F exerting some positive effort in its espionage activities (including the maximum one) are not stable either. Note that, in such a situation, C would always be able to define an active counter-espionage policy (even one characterized by a relatively small 740 effort) whose costs would be justified by the decrease in the expected loss in welfare of its domestic market participants implied by the potential participation of F 's firm in the market. 27 Also, as Figures 2 and 3 show, if a > a 1 , F is willing to exert its maximum effort as a response to strictly positive but sufficiently low enough efforts in  Proof. It follows from Proposition 1, according to which c * = c where c is given by (24), and the fact that the degree in n of the polynomial 8b 2 (1 + n) 2 (2 + n) 4 is higher than the degree in n of a 4 (−1 + 2n(1 + n)).
According to Proposition 2, regardless of the parameters a and b defining the demand in the market, the higher the initial number, n, of firms competing in focusing on how it determines the sign of the variation in the equilibrium espi-onage effort. Note that, given the strategic asymmetry between C and F , the important aspect in this relationship is whether the reduction in the espionage effort exerted by F given the counter-espionage effort of C is small or high in 905 terms of the reduction in the counter-espionage effort exerted by C given F 's espionage effort. Here, the role of price elasticity of demand in each reduction considered individually is behind its influence on their relationship.
In the case of C, the increase in the welfare of its domestic market participants due to a higher initial level of competition is potentiated by a sufficiently In the case of F , the reduction in its firm's profits from participating in 920 the market due to a higher initial level of competition will be smaller under a sufficiently elastic demand than under a not so elastic one. The reason is the same as stated above: under a relatively elastic demand the decrease in the market price due to a higher level of competition implies a proportionally higher increase in the quantities sold in the market. Consequently, F will reduce the 925 espionage effort less if the market demand is sufficiently elastic than if it is not.
Therefore, the influence of price elasticity of demand on the relationship between the reductions in C's and F 's levels of effort, given the effort of the rival, due to an increase in the initial level of competition, and consequently on the sign of the variation in the equilibrium espionage effort, is clear. If the 930 market demand is sufficiently elastic (b sufficiently small), the reduction in F 's espionage effort given the counter-espionage effort exerted by C will be relatively small, in the sense that the necessary reduction in the counter-espionage effort in order for F to keep its previous equilibrium espionage effort is smaller than C's reduction in the counter-espionage effort given the previous equilibrium 935 espionage effort. In this situation, the counter-espionage effort that C would exert in response to the previous equilibrium espionage under the higher initial level of competition is so low that F is better off increasing the espionage effort, and in the new equilibrium with a higher initial level of competition F will exert a higher espionage effort than before.

940
Thus, a sufficiently elastic demand can more than compensate for the negative effect of a higher initial level of competition in F 's espionage effort. However, a sufficiently inelastic market demand potentiates this negative effect. In particular, if market demand is sufficiently inelastic (b sufficiently high) the reduction in F 's espionage effort given the counter-espionage effort exerted by C 945 will be relatively high, in the sense that the necessary reduction in the counterespionage effort for F to keep its previous equilibrium espionage effort is higher than C's reduction in the counter-espionage effort given the previous equilibrium espionage effort. In this situation, under this relatively small reduction in C's counter-espionage effort in response to the previous equilibrium espionage given 950 the higher initial level of competition, F is better off decreasing the espionage effort and, in the new equilibrium with a higher initial level of competition, F will exert a smaller espionage effort than before.
Therefore, in between these two characterizations, there exists an intermediate level of price elasticity of demand (an intermediate value of b) such that 955 the reduction in F 's espionage effort given the counter-espionage effort exerted by C will imply that the necessary reduction in the counter-espionage effort for F to keep its previous equilibrium espionage effort is exactly C's reduction in the counter-espionage effort given the previous equilibrium espionage effort.
In other words, the previous equilibrium espionage effort is F 's best response, 960 under the higher initial level of competition, to C's counter-espionage effort in response to the previous equilibrium espionage given this higher initial level of competition. Consequently, in the equilibrium under a higher initial level of competition in a market characterized by this intermediate level of price elas-ticity, F will exert the same espionage effort as before.
The pair of initial number of C's firms competing in the market {y, z}, where y, z ∈ Z + and y < z, considered to define the increase in the initial level of market competition, determines the threshold, in terms of the value of b and denoted by b y,z , to define the demand as sufficiently elastic/inelastic.
More precisely, b y,z is the value of b such that e * y = e * z , where e * y and e * z are, 970 respectively, the equilibrium espionage efforts under the initial number y and z of C's firms competing in the market. The relationship among these thresholds b y,z , together with the above-discussed influence of price elasticity of demand, is behind the behavior of the equilibrium espionage effort with respect to the initial level of competition.

975
Note that, according to Proposition 1, the espionage effort exerted by F in equilibrium, e * , satisfies e * = e, where e is given by (25). The proposition below summarizes the main aspects in the behavior of e * with respect to the initial level of market competition, abstracting from some specific cases which are considered in more detail in the discussion that follows. According to the (1) Let e * n and e * n be the equilibrium espionage efforts under the initial number of C's firms competing in the marketn andñ respectively, wheren <ñ.
(4) If b 1,2 < b < b 2,3 , then n * ∈ {3, 4}, and e * (4.1) decreases from n = 1 to n = 2, in it due to a higher initial level of competition and, therefore, the smaller the reductions will be in the efforts exerted by C and F given the effort of the rival. Following the discussion above regarding the influence of elasticity of demand, the implication is that the decreasing behavior of e * for n ≥ n * reflects that there always exists some initial number n * of C's firms in the market such that, no matter how elastic the demand is, the reduction in C's effort due to an increase in the initial level of competition is never high relative to the reduction 1020 in F 's effort. In other words, a critical initial number n * of firms exists such that there is no sufficiently elastic demand which more than compensates 29 the negative effect of a higher initial level of competition in F 's espionage effort. 29 In the sense considered when discussing above the influence of price elasticity of demand.
More technically, no matter how small and close to zero b is, no pair of initial number of firms {n,ñ} exists, where n * ≤n <ñ, such that, according to its 1025 corresponding threshold bn ,ñ , the demand can be considered sufficiently elastic (that is, there is no bn ,ñ such that b < bn ,ñ ).
Therefore, and coherently with the influence of elasticity of demand discussed above, the behavior of e * with respect to the initial level of competition for 1 ≤ n < n * is not necessarily decreasing. Moreover, as shown by Figure 4, when 1030 the elasticity of demand increases, this behavior becomes strictly increasing at the same time as n * becomes larger (as according to part (2) of Proposition 3).
The reason for this is that the smaller b and the closer it is to zero, the more thresholds there are associated with pairs of initial number of firms with respect to which the demand is sufficiently elastic. As stated by part (3) of Proposition 1035 3, when the demand is elastic enough (0 < b < b 1,2 ), the critical number of firms n * is relatively high and e * is strictly increasing for 1 ≤ n < n * . Namely, there is no pair of initial number of firms in [1, n * ] such that, according to its corresponding threshold, the demand can be considered sufficiently inelastic.
Consistent with this, b 1,2 < b 3,4 < b 2,3 as stated above, and therefore n * > 3 1040 when 0 < b < b 1,2 . More precisely, given that b 4,5 < b 1,2 , n * = 4 when b 4,5 < b < b 1,2 and, coherently with the increasing behavior of n * with respect to price elasticity of demand, n * > 4 when 0 < b < b 4,5 (a particular example of this last case is represented by Figure 4 with respect to these considerations in the case b > b 2,3 is that e * 2 = e * 3 . When the level of price elasticity of demand is in between the above two 1060 extreme cases, b 1,2 < b < b 2,3 , the behavior of e * for 1 ≤ n < n * , as stated by part (4) Let us denote by Π * n the equilibrium profit of each firm in this scenario in which only the n firms of C are competing in the market. Such equilibrium profit is given by Given that production and export costs are assumed to be zero, the sum of the n equilibrium profits coincides with the suppliers' surplus of the C's The surplus of the m firms from F is SS F n+m , which is given by (4), and F 's consumers obtain the surplus βCS n+m , where CS n+m is given by (6). Therefore: Note that the last element, e 2 , is the cost of espionage effort. Equivalently: Regarding U C , this takes into account the surplus obtained by C's consumers, who represent the proportion α of the whole demand in the international 1385 market, and the surplus of the n firms from C in these two possible scenarios with their respective occurrence probabilities. Specifically: U C = (1 − e + ec) (SS n + αCS n ) + e(1 − c) SS C n+m + αCS n+m − c 2 (A4) where the last element, c 2 , is the cost of counter-espionage effort. Substituting in (A4), on the one hand, SS n and CS n by their expressions given by (2) and (3) respectively in the main text, and on the other hand, SS C n+m and CS n+m by (5) and (6), respectively, also in the main text and simplifying we have: na 2 (αn + 2) 2(n + 1) 2 b + e(1 − c) 2n + α(n + m) 2 a 2 2(n + m + 1) 2 b − c 2 Similarly, it can be easily proved that e > 0 given that both the numerator and the denominator in (25), in the main text, are strictly positive for all n ∈ Z + 1400 and a, b > 0. However, the proof of e < 1 is not that straightforward.
Proof of Lemma 5: In the first market characterization (0 < a ≤ a 1 ), in which δ ≤ 0 and γ > 1, the proof is straightforward.
In the second market characterization (a 1 < a ≤ a 2 ), in which 0 < δ < 1 and 1410 γ ≥ 1, the proof follows from the fact that the intersection of both best-response functions would not be given by the intersection of c(e) and e(c) only if e ≥ 1, which is a contradiction according to the second part of Lemma 4.
Assuming next that x ∈ R, then, according to the Fundamental Theorem of Algebra, the polynomial given by (A7) would have, counted with multiplicity, seven roots and, since (A7) has real coefficients and odd degree, at least one root 1445 is real. This, together with the fact that the domain of e(x) is [1, ∞[, directly leads to κ ∈ {0, . . . , 7}.
Lemma A1 directly leads to the following corollary.
(1) Suppose that κ ∈ {1, . . . , 7} and letx * andx * be the smallest and the 1450 largest critical points of e(x) respectively. Consider thatx * > 1, which includes the case in which κ = 1 andx * > 1. Then, (1.1) following part (1) of Lemma A1, x * =x * and e(x) is decreasing for x ∈]x * , ∞[; (1.2) following part (2) of Lemma A1, it is not possible to know analytically 1455 whether a particular critical point of e(x) is a maximum, a minimum or an inflection point, and therefore, the behavior of e(x) for x ∈ [1,x * [ cannot be analytically characterized.
The analytical intractability of e(x) highlighted by part (2)   In particular, Figure A1 depicts the behavior of e(x) for different values of η, showing that the local maximum of e(x) moves to the left as η increases. But this is only one aspect of the results obtained from the numerical study of e(x).
The following corollary summarizes more precisely its most relevant conclusions.
The following claim is required in order to use Corollary A2 to study the behavior of e * with respect to n ∈ Z + . according to part (1) of Claim A1, e * 1 > e * 2 in this case and e(x) decreases for x >x * (see part (1) of Corollary A2), implies that n * = 1 and e * decreases in n for all n ∈ Z + . Note that the only particularity of the case η = η 2,3 with respect to these considerations is that e * 2 = e * 3 , as implied by part (2) of Claim A1. The analysis of these four cases, together with the fact that b = ηa 2 , proves parts (2)-(5) of the proposition.