Entry and espionage with noisy signals

Games and Economic Behavior 83 (2014) 127–146

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Entry and espionage with noisy signals ✩ Alex Barrachina a , Yair Tauman b,c , Amparo Urbano d,∗ a

Department of Economics, University Carlos III, Madrid, Spain IDC Herzliya, Israel Stony Brook University, USA d Department of Economic Analysis and ERI-CES, University of Valencia, Spain b c

a r t i c l e

i n f o

Article history: Received 27 March 2013 Available online 7 November 2013 JEL classification: C72 D82 L10 L12 Keywords: Espionage Entry Asymmetric information Signal-jamming

a b s t r a c t We analyze the effect of industrial espionage on entry deterrence. We consider a monopoly incumbent who may expand capacity to deter entry, and a potential entrant who owns an Intelligence System. The Intelligence System (IS) generates a noisy signal based on the incumbent’s actions. The potential entrant uses this signal to decide whether or not to enter the market. The incumbent may signal-jam to manipulate the likelihood of the noisy signals and hence affect the entrant’s decisions. If the precision of the IS is commonly known, the incumbent benefits from his rival’s espionage. Actually, he benefits more the higher is the precision of the IS while the spying entrant is worse off with an IS of relatively high quality. When the IS quality is private information of the entrant, the incumbent is better off with an IS of high expected precision while the entrant benefits from one of high quality. In this case espionage makes the market more competitive. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Firms’ acquisition of information about other firms’ production processes and techniques, costs, recipes and formulas, customer datasets, actions, decisions, plans and strategies, and other such sensitive data consumes an important part of their resources and endeavor.1 Recent advances in communication and information technologies have increased firms’ incentives to push the limits of ethical competitive intelligence activities and have led them to acquire information about other firms illegally and unethically, namely by engaging in industrial espionage.2 For instance, according to US State Department and Canadian Security and Intelligence Service Reports from 1997, industrial espionage costs US business over 8.16 billion dollars annually. Moreover, 43% of American firms have suffered at least six incidents of industrial espionage,3 covering a host of actions aimed at achieving cost advantages, maintaining market leadership, and the like. The issue is becoming so important that on February 1st, 2011 the Financial Times wrote, “Taking into account all types of industrial espionage but counting only the cost to American businesses, US intelligence officials put the cost of lost sales due to illicit appropriation ✩ The authors thank Eilon Solan and an anonymous referee for very valuable comments and suggestions. Alex Barrachina and Amparo Urbano wish to thank the Ministry of Science and Technology and the European Feder Funds under projects SEJ2007-66581 and ECO2010-20584, ECO2010-19596 and the Generalitat Valenciana under the Excellence Programs Prometeo 2009/068 and ISIC 2012/021 for their financial support. Corresponding author. E-mail addresses: [email protected] (A. Barrachina), [email protected] (Y. Tauman), [email protected] (A. Urbano). 1 As Business Week reported in 2002, most of the large companies have competitive intelligence staff. Indeed, many large US firms spend more than a million dollars a year on competitive intelligence issues, and multinational firms like Kodak, General Motors and British Petroleum have their own competitive intelligence units. See Billand et al. (2009). 2 Nasheri (2005). 3 See Solan and Yariv (2004, footnote 1).

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of technology and business ideas at $100bn–$250bn a year. General Motors, Ford, General Electric, Intel and Boeing are among the US companies known to have suffered from industrial espionage attacks, though all are wary of discussing the details.” Recently, a leading article in The Economist (February 21st, 2013) analyzed China’s cybercrime under the topic of “China’s cyber-hacking. Getting ugly,” claiming that some criminal tribes operate from within the target multinational firm and misappropriate its resources, while others use purloined property and know-how to start rival businesses after (or even before) leaving the firm. In this paper, we analyze the impact of industrial espionage on entry deterrence. The assumption is that the entrant spies on the incumbent’s actions via some spying technology. The technology is not perfectly accurate and hence the incumbent’s actions generate noisy signals. It is shown that when the spying technology accuracy is common knowledge, only the incumbent benefits from his rival’s espionage activities. The incumbent uses the preemptive capabilities of a Stackelberg leader to signal-jam, or to strategically choose his actions, to manipulate the likelihood of noisy signals and hence the likely inferences drawn by the entrant. In contrast, when the entrant has private information of the precision of the technology, she uses both her private knowledge and her second-mover advantage to benefit from espionage. The incumbent will signal-jam again using the expectation of the precision as a proxy for its true value. However, the entrant cannot be manipulated as before and, for some values of the spying technology precision, she makes equilibrium entry decision independently of the signal received. Although entry deterrence has been extensively studied in the literature,4 the effects of espionage in this context have not yet been analyzed. Market entry is one of the most fundamental decisions for a firm, as is the incumbent’s reaction when faced with the threat of a new entrant in the market.5 One possible response by the incumbent to deter entry is to invest in capacity expansion. Examples of strategic capacity expansion are famously exemplified by the titanium dioxide industry (see Ghemawat, 1984 and Hall, 1990) and the “sleeping patents” of the bio-pharmaceutical industries and/or Airline companies (see Sull, 1999), among others. The classical and stylized model of entry deterrence by capacity expansion assumes the existence of a monopoly incumbent and a potential entrant to the market. Under perfect information and sequential moves, the entrant stays out of the market if the incumbent expands its capacity and enters the market otherwise. However, capacity expansion requires an investment and the entrant usually does not observe ex-ante the incumbent’s decision of whether to invest in capacity. Hence, the entrant enters only if she believes with high probability that the incumbent has not expanded its capacity. To analyze industrial espionage in this context, we extend the entry game to include spying actions. This sort of activity is typically conducted by a market research firm. For example, the case study of Mezzanine Group (2010) deals with an entrant in the energy market that asked a consulting company (The Mezzanine Group) to evaluate the competitive landscape of the Ontario market, whereby the incumbent’s strategies were part of the information given to the entrant. However, when the spy is a decision-maker who can act strategically then double-crossing may take place (see Ho, 2008). To avoid these strategic effects, the assumption is that the entrant operates a costless Intelligence System (IS) set to detect the incumbent’s action.6 This could be the case if a firm already possessed spying technology before encountering a new rival. For instance, the potential entrant may have the ability to place a Trojan horse in the computer system of the monopoly incumbent and obtain information about the action he plans to take. Today, the use of Trojan horse computer viruses for industrial espionage is a reality. For instance, in 2005 three top-tier Israeli firms were suspected of using Trojan horses in Israel to monitor competition in the cable, communications, office equipment, photocopy, and cars and trucks7 industries. The IS sends out one of two signals. One signal, labeled i, indicates that the incumbent invests in new capacity and another signal, labeled ni, indicates the opposite. The IS has a precision, meaning that the signal sent by the IS will be correct with a probability equal to the IS precision. Based on the signal received, the entrant decides whether (or with what probability) to enter the market. Let us start with the benchmark case where the precision of the IS is commonly known. Namely, the probability that the IS sends an accurate signal is commonly known. Since the entrant observes only the signal sent by the IS, the incumbent can advantageously signal-jam or choose his capacity expansion probability to influence the conditional probability that the entrant decides to enter the market, given the signal received. It is assumed8 that the investment cost of the incumbent is relatively low and the incumbent prefers to make this investment if he is sure that the entrant will enter. The entrant, on the other hand, will not enter if she knows that the incumbent expanded his capacity. Since the entrant does not observe the incumbent’s decision but only a signal of his action, if the incumbent does not expand capacity and the precision of the IS is sufficiently high, the entrant will detect the incumbent’s action with high probability and is likely to enter the market. In this case, the incumbent expands his capacity with high probability, and the signal ni is less likely to be emitted. Consequently, when the entrant observes the signal ni, she will no longer rely on its accuracy and rather refrain from entering the market with positive probability. On the other hand, if the entrant observes the signal i she will stay

4

For a survey of this literature see Wilson (1992). See Milgrom and Roberts (1982, p. 282). 6 The extension to a costly IS which depends on its precision is straightforward once the equilibrium payoff of the entrant is analyzed. 7 See Singer-Heruti (2005). 8 We leave aside the case where the investment cost is sufficiently high so that the incumbent would prefer not to invest even if he knew that the entrant would enter. 5

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out with probability one. As the IS precision increases, the probability of the signal ni and the conditional probability that the entrant enters given ni decreases. Therefore, the incumbent signal-jams by expanding capacity with positive probability that increases as IS precision increases. Alternatively, if the IS precision is less accurate, the incumbent will expand capacity with lower probability, knowing that it is likely that his action will not be detected. Furthermore, he will expand capacity with positive probability that decreases as IS precision increases in order to decrease the unconditional probability that the entrant enters. Thus, the incumbent benefits from the IS more than its owner, the entrant, if the IS is relatively accurate. Actually, the incumbent benefits more the higher is the precision of the IS, while the entrant is worse off for relatively high quality IS. Next we analyze the asymmetric information case where the IS precision is privately known to the entrant. The incumbent only knows the distribution of this precision.9 He will signal-jam at the (Bayesian) equilibrium by following the expectation of the precision as a proxy for its real value. However, the entrant cannot be fooled as easily as before because she knows the true value of the precision and compares this value with its expectation. She plays pure strategies and, for some values of the precision, takes an entry decision independently of the signal received. Namely, if the expected precision of the IS is not too high, the incumbent will believe that the true precision is low and he will not expand capacity with high probability in the belief that the entrant will be less likely to detect him. The entrant, following the more likely signal, ni, will enter the market, irrespective of the actual precision. On the other hand, if the expected precision of the IS is high enough, the incumbent will believe that the true precision is relatively high and the entrant is likely to detect his action. Hence, the incumbent expands capacity with relatively high probability, and the entrant, following the less likely signal, ni, enters the market if the actual precision is relatively high and stays out otherwise. If the signal is i, the entrant does not enter, irrespective of the IS precision. The entrant obtains a positive payoff if the IS precision is sufficiently large and this payoff increases as this precision increases, whereas the incumbent is better off when the expected precision of the IS is high rather than low. Therefore, when the entrant engages in noisy espionage activities, the spied upon incumbent always conducts signaljamming, and he will be more successful in manipulating signals sent to the entrant when he has more knowledge of the spying technology’s accuracy. Conversely, when the IS accuracy is known only to the entrant and this information is very asymmetric, market competition is increased since the entrant is more likely to enter. 1.1. Related literature The paper is related to several research strands. The first has to do with research on industrial espionage. Solan and Yariv (2004) study espionage games, where the precision is costly and is a choice strategic variable. Provan (2008) works along lines more closely related to the ones discussed in the current paper but opts for a more computational-based approach, using linear programming solutions for two-person zero-sum games. Matsui (1989) considers a two-person repeated game where every player has a small probability of perfectly detecting the other player’s action and revises his strategy accordingly. Ho (2008) focuses on double-crossing espionage phenomenon. Here, espionage is used to learn about the rival’s private information. Unlike Ho (2008), espionage in our paper, as well as in Solan and Yariv (2004) and Matsui (1989), yields information about the rival’s action. Whitney and Gaisford (1999) study a duopoly competition (Airbus and Boeing) where one company spies in an attempt to learn its rival’s technology and is able to lower its own marginal cost as a result. The intelligence system may result in either a pure success or a pure failure and does not generate any noise. The outcome of the IS is assumed to be common knowledge. In a recent paper, Billand et al. (2009) study espionage in a Cournot model of several firms with differentiated goods. Their model is of symmetric information, where firms observe the full espionage activity before choosing their quantity levels. Finally, our paper is similar in spirit to Biran and Tauman (2009), who deal with the role of intelligence in nuclear deterrence. The second strand is the signal-jamming literature on oligopolistic models (Riordan, 1985; Aghion et al., 1993; Mirman et al., 1993; Caminal and Vives, 1996; Alepuz and Urbano, 2005, among others). This body of work deals with the way firms internalize how their output decisions affect price and influence the inferences of other firms and hence their subsequent actions. A third limit pricing/signaling strand develops two period models of entry deterrence whereby firms have private information about demand or cost and take actions (limit pricing) that influence the inference of other firms. In the limit pricing models of Milgrom and Roberts (1982), Harrington (1986, 1987), Caminal (1990), and Mailath (1989), among others, either the incumbent’s price or the duopoly prices signal information about their costs to an entrant. Finally, this paper is related to sharing information in oligopoly markets as in Vives (1984), Raith (1996), and Gal-Or (1985, 1986, 1987, 1988), among others, where the equilibrium outcomes depend on the information revealed. The above duopoly models with incomplete information show that the leader can benefit from the follower being fully informed about the environment: with full information, the leader can avoid having to signal his private information, and as a result, increase his profits. Even though the present paper is not about signaling, the results derived when the precision of the IS system is publicly available are similar to those obtained in the papers cited above. Specifically, the incumbent can

9 For instance, the incumbent correctly believes that the entrant introduced a Trojan horse in his computer system, but he is not sure of how accurate is the information the potential entrant obtains through it.

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Fig. 1. The IS.

Fig. 2. The basic game.

benefit from an improved information system available to the entrant because he can reduce the incidence of entry without having to overinvest in capacity. On the other hand, the effects that appear when only the entrant has private information about the IS accuracy, resemble the informational effects in the above papers, where players revise their strategies in an attempt to “fool” rivals but, at a pure strategy equilibrium, are not successful in their attempt. The next section sets out the basic model. In Section 3 we analyze the Nash equilibrium of the entry game with espionage when the IS precision is commonly known. Section 4 is devoted to finding the Bayesian Nash equilibrium strategies when the IS precision is the entrant’s private information. Section 5 concludes the paper. 2. The basic model The model looks at two firms, M and E. The Incumbent Firm, M, is a monopolist and E is a potential entrant.10 In an attempt to deter E from entering, M considers whether to invest in a new capacity. E has an Intelligence System (IS) that monitors the action of M. The IS sends a noisy signal, one of the two signals i or ni. The signal i indicates that M invests and the signal ni indicates that M does not invest. The IS sends the right signal with probability α and the wrong signal with probability 1 − α . For simplicity we assume that the precision, α , of the IS is independent of the action of M. It is assumed without loss of generality that 1/2  α  1. If α = 1/2, the IS is of no relevance and if α = 1, the IS is perfect. Fig. 1 summarizes the above. Based on the signal received, E decides whether to enter the market. Fig. 2 displays the payoffs of the two firms based on their possible actions, where

0 < b < c < 111

(AS1)

Notice that b could be seen as the incumbent’s cost of making a mistake (not investing) when E enters and (1 − c ) as the cost of making a mistake (investing) when E does not enter. 3. The benchmark case: α commonly known We first focus on the case where α is common knowledge. In particular, M knows that E spies on him with an IS of precision α . Let us start with the two extreme cases where α = 1/2 and α = 1. If α = 1/2, then it will be as if E does not operate an IS on M, and the strategic game between M and E is described in Fig. 2. This game has a unique Nash equilibrium in which: M invests with probability 12 and E enters the market with

1−c , which decreases as both b and c increase. Namely, the higher the payoff of M from expanding capacity, probability 1− c +b the lower the probability that E enters.

10 Slightly abusing the notation, the letter E is used to denote the entrant, the action “enter” and also for expectations later on. However, no confusion should arise. 11 The case where b < 0 is trivial as I is strictly dominated by NI.

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Fig. 3. The game with a perfect IS.

Fig. 4. The general game.

Fig. 5. The general game without dominated strategies.

The payoff of E is zero (E is indifferent between entering and not entering), and the payoff of M is 1−bc +b (which increases with b and c). The second case is α = 1 and M’s action is perfectly detected by E. In this case, E chooses her action based on M’s action. This game can be visualized as in Fig. 3. A perfect signal allows the Incumbent to take advantage of his position as leader in a sequential game and the backward induction is the unique Nash equilibrium. M expands his capacity and E does not enter the market. This outcome yields a higher payoff of M than the equilibrium outcome for α = 1/2. The payoff of E is zero in both cases. Hence, only M benefits from a perfect IS. The entrant, who spies on M and who is able to perfectly monitor M’s action (before taking her action) does not benefit at all from such an IS. This result follows from the assumption that α is commonly known. The general case: 1/2 < α < 1. The entrant has four pure strategies. A pure strategy of E is a pair (x, y ) where both x and y are in { E , N E }, x is the action of E if she observes the signal ni and y is her action if she observes the signal i. Fig. 4 describes the game, G α , between M and E in strategic form (see Figs. 1 and 2). For instance, the strategy ( E , N E ) of E is to enter the market if the signal is ni and not to enter if the signal is i. The strategy ( E , E ) is to enter the market, irrespective of the signal. Note that the strategy ( N E , E ) of E is strictly dominated by her strategy ( E , N E ), since α > 1/2. Therefore, the strategy ( N E , E ) can be removed, and the resulting game is as in Fig. 5. 1−b ¯ = max[ 12 , 1− Let α ]. This parameter plays a central role in our analysis. The value α¯ is a threshold to assess the b+c ¯ = (1 − b)/(1 − precision of the IS and hence the players’ choices of action. This can be observed by noting that when α = α b + c ), E will choose her pure strategy ( E , N E ) (i.e., E will enter if the signal is ni and will stay out if the signal is i), and ¯ = 1/2 and M will be indifferent between investing and not investing. However, when [1 − b/(1 − b + c )] < 1/2, then α α > α¯ , for all α ∈ (1/2, 1). This means that the precision α of the IS is sufficiently large and hence M knows that his action ¯ > 1/2 and hence will be correctly detected with high probability. On the other hand, if [1 − b/(1 − b + c )] > 1/2, then α

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¯ (i.e., the precision of the IS is not too accurate and M will assign significant probability that his action will either α < α ¯ as above. Proposition 1 states the equilibrium signal-jamming of the Incumbent and the not be correctly detected) or α > α Entrant’s equilibrium entry decisions under the different scenarios. All subsequent proofs appear in Appendix A. ¯ . Then G α has a unique Nash equilibrium. Proposition 1. Suppose that α = α (1) The Incumbent randomizes between expanding and not expanding its capacity. The probability that M expands capacity is de¯ , and is increasing in α , for α¯ < α < 1. creasing in α , for 1/2 < α < α ¯ and she will randomize her two actions (2) If the Entrant observes the signal ni, she will enter the market with probability 1 if α < α ¯ . If she observes the signal i she will randomize her two actions if α < α¯ and will stay out with probability 1 if α > α¯ . if α > α ¯ . Then, the Entrant will enter the market with certainty if she observes the signal ni and she will stay out (3) Suppose now that α = α with certainty if she observes the signal i. The Incumbent has a continuum of best reply strategies.

¯ , there is always one column out of the three (see Fig. 5) that is dominated and the Entrant Intuitively, when α = α ¯ , the strategy (N E , N E) is dominated and the randomizes between the remaining two columns. For instance, when α < α ¯ , the strategy (E , E) is dominated and the Entrant randomizes between ( E , E ) and ( E , N E ); and similarly, when α > α randomization is over ( E , N E ) and ( N E , N E ). The different scenarios depend on the comparison of the Incumbent’s cost of making a mistake (not investing) when E enters and that of making a mistake (investing) when E does not enter. ¯ > 1/2. This happens when the Incumbent’s cost Thus, when 1 − c > b (see Fig. 2), then [1 − b/(1 − b + c )] > 1/2 and α of investing if E does not enter is larger than that of not investing if E enters. The equilibrium outcome depends on whether α < α¯ or α > α¯ , which determines the entrant’s strategies. If the precision of the IS is not too accurate (i.e., 1/2 < α < α¯ ) M will assign significant probability that his action will not be accurately detected. Therefore, in an attempt to conceal his action, M mixes his two strategies I and NI, with probabilities 1 − α and α , respectively. Thus both signals i and ni have a reasonable likelihood of occurring. However the signal ni is more likely than the signal i since α > 1 − α . As a result, E will enter with probability 1 if the signal is ni and will randomize between ( E , E ) and ( E , N E ) if the signal is i. ¯ ), M knows that his action will be correctly On the contrary, if the precision, α , of the IS is sufficiently large (α > α detected with high probability. Therefore, he would expect that if he did not expand capacity, E would probably enter. Hence, M expands his capacity with high probability: specifically, with probability α . Consequently, the signal ni is not likely to occur. If E observed this signal, she would not trust it and would randomize over ( E , N E ) and ( N E , N E ). As a result, the probability of E not entering would be nonzero. On the other hand, because E is expecting signal i to be emitted, whenever she observes it she will trust the signal and will not enter with probability 1. When, however, 1 − c  b, the Incumbent’s cost of not investing when E enters is larger than that of investing when ¯ = 1/2, which, in turn, means that α > α¯ for all E does not enter. In this case, [1 − b/(1 − b + c )] < 1/2 and hence α α ∈ (1/2, 1). Since b  1 − c the penalty for M for not expanding capacity if E enters is relatively high. Thus M invests in capacity expansion with relatively high probability. In fact, this probability is shown to be equal to the precision α of the IS and, since α > 1/2, E expects to observe signal i with higher probability than that of signal ni. As a result, E will stay out for sure if she observes the signal i and E will hesitate if she observes the less expected signal ni. Namely, as observed above, she will randomize over ( E , N E ) and ( N E , N E ). The higher the precision α of the IS, the higher the probability that M invests and the higher the probability that E stays out. ¯ = [1 − b/(1 − b + c )], M will be indifferent between investing and not investing and the game will have Finally, if α = α 1−b and E will play the pure strategy multiple equilibrium points: M will invest with probability p˜ where 1−cb+c  p˜  1− b+c ( E , N E ) (i.e., E will enter if the signal is ni and will stay out is the signal is i). An increase in the quality, α , of the IS increases the reliability of the signal generated. Hence, when E observes the signal i, she enters the market with lower probability and M is better off. The fact that as α increases M invests with lower probability and reduces the probability of the signal i is a less intuitive observation. However, we argue in Proposition 2 that this decreases the probability that E enters. Proposition 2. Consider the equilibrium of G α . Then, the probability that E enters the market is positive and decreasing in α ∈ (1/2, 1).

α for all

¯ ), M expands capacity with a probability that decreases as If the precision of the IS is not too accurate (1/2 < α < α α increases. This behavior of M implies that the increase in α has two opposite effects on the unconditional probability that E enters. The positive effect is the increase in the probability of the signal ni, which contributes to the increase in the probability that E enters (E enters with probability 1 when she observes the signal ni). On the other hand, it decreases the conditional probability that E enters given i, which contributes to the decrease in the probability that E enters. It turns out that the latter effect outweighs the positive effect and as a result the unconditional probability that E enters decreases in α . ¯ < α < 1), M will expand capacity with probability increasing Similarly, if the precision of the IS is relatively accurate (α in α . Hence, as α increases, the probability of the signal ni and the conditional probability that E enters given ni decreases. This implies that the unconditional probability that E enters decreases as α increases.

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Fig. 6. The entrant’s equilibrium expected payoff.

The literature on entry deterrence where the incumbent actually prefers that the entrant is fully informed (Milgrom and Roberts, 1982) supports this result because, in equilibrium, the established firm practices limit pricing but the entrant is not fooled by this strategy. Similarly, in our model, the Incumbent practices signal-jamming but the probability that entry actually occurs in equilibrium is greater than zero. Proposition 3. Consider the equilibrium of G α . Then, the expected payoff of the Incumbent is increasing in α , for all α , and the expected ¯ and is zero for all α , α¯ < α < 1. payoff of the Entrant is increasing in α for 1/2 < α < α

¯ (see The proposition implies that E is better off when the precision of the IS is higher as long as it is smaller than α Fig. 6). Let Π E be the equilibrium expected payoff of E. ¯ then E will randomize between ( E , E ) and An explanation of Fig. 6 is as follows. Proposition 1 states that if 1/2 < α < α ( E , N E ). Moreover, in equilibrium the Entrant will enter the market with probability 1 if the signal is ni, and with some probability (see Appendix A) if the signal is i. Furthermore, by Proposition 1, at this interval, the Incumbent will randomize between expanding and not expanding its capacity with probability decreasing in α . The effect is to increase the probability of the signal ni, which contributes to the increase of the Entrant’s expected payoff from playing ( E , E ). Since she plays a mixed strategy, her expected payoffs from playing ( E , E ) are the same as those from playing ( E , N E ), and the former are clearly increasing in α . Therefore, even though the probability that E enters the market is decreasing in α , the entrant’s ¯ . Thus, in fact, if the entrant could endogenously change α , expected payoffs are increasing in α , for the interval 1/2 < α < α ¯. she would choose α = α ¯ < α < 1, by Propositions 1 and 2, even if the IS is cost free E has no incentive to use it since her payoff Note that for α is zero, irrespective of the quality α . The expected payoff of the Incumbent is increasing in α , for all α , since the probability that E enters the market is decreasing in α . Hence, M is better off with a perfect IS, even thought that means perfect monitoring of his actions. As in Gal-Or (1987), an immediate consequence of the above analysis is that the incumbent has a strong incentive to directly reveal his action to the follower instead of letting him infer it from the IS signals. In summary, when the Incumbent knows the value of α (namely, α is common knowledge), he signal-jams using the IS precision as a correlation device and benefits from his rival’s espionage. This signal-jamming activity is inefficient for the market, since it may imply excess capacity. Therefore, an important question is whether espionage activities enhance efficient market capacity, as compared with no espionage, when the IS precision is common knowledge. The ¯ ). In this interval, the incumbent signalanswer is affirmative only if the IS precision is small enough (1/2 < α < α jamming strategy forces him to invest less in capacity to decrease the unconditional probability that the Entrant enters. This behavior increases the probability of the signal ni and hence increases the Entrant’s profits. As a consequence, both firms are better off, which is Pareto improving, and the market capacity is more efficient than under no espionage. 4. Asymmetric information about the precision of the IS In this section we assume that the precision α of the IS is the private information of its owner, E. The Incumbent, who does not know α , assigns a continuous probability density function f (α ) > 0 to every α , with 12  α  1 and

1 1 2

f (α ) dα = 1. In other words, E knows the game G α which is actually being played while M does not know which

game is being played. However, M knows that α is chosen according to f (α ), and this is commonly known. The symbol Γ denotes this game. Let u M and u E be the utilities of the two firms from the various outcomes. As in the previous section (see Fig. 2), it is assumed that

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Fig. 7. The decision of E when the signal is ni.

Fig. 8. The decision of E when the signal is i.

u M ( I , E ) = b,

u E ( I , E ) = −1

u M ( I , N E ) = c,

u E (I , N E ) = 0

u M ( N I , E ) = 0,

u E (N I , E ) = 1

u M ( N I , N E ) = 1, u E (N I , N E ) = 0 1 Let E (α ) = 1 α f (α ) dα be the expected value of α . Namely, E (α ) is the expected quality of the IS from the perspective of 2

the uninformed M. The next proposition shows that at the (Bayesian) equilibrium the Incumbent signal-jams again, using the expectation of the precision, and that the Entrant follows a pure strategy.

¯ . Then Γ has a unique perfect Bayesian equilibrium. Proposition 4. Suppose that E (α ) = α ¯ , there exists p¯ 1 , 12 < p¯ 1 < 1, such that M will expand capacity with probability p¯ 1 . If the signal is ni, E will not enter (1) If E (α ) > α the market if α < p¯ 1 and she will enter if α > p¯ 1 . If the signal is i, E will not enter, irrespective of the precision α of the IS. ¯ , there exists p¯ 2 , 0 < p¯ 2 < 12 , such that M will expand capacity with probability p¯ 2 . If the signal is ni, E will enter the (2) If E (α ) < α market, irrespective of the precision α of the IS. If the signal is i, E will not enter if α > 1 − p¯ 2 and she will enter if α < 1 − p¯ 2 . Figs. 7 and 8 illustrate the results of Proposition 4. Note that, unlike the case where α is commonly known, the equilibrium strategy of E of type α is a pure action (enter or not enter the market with probability 1). However, M mixes his two pure actions to signal-jam, as in the common knowledge case. The action of E depends on both the expected and the actual precision of the IS.

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135

¯. Fig. 9. The expected payoff of E when E (α ) > α

¯ , M will believe that the true precision is relatively high and E will be likely If the expected precision of the IS exceeds α to detect his action. Hence, M will expand his capacity with relatively high probability, and E, following the less likely signal ni, will enter the market if the actual precision is relatively high (α > p¯ 1 ) and will not enter otherwise. If the signal is i, the entrant will not enter, irrespective of the IS precision. ¯ , M will believe that the true IS precision is low If, on the other hand, the expected precision of IS does not exceed α and the probability will be high that he will not expand capacity in the belief that E will probably not detect him. Then E, following the more likely signal ni, will enter the market, irrespective of the actual precision. Furthermore, E will enter the market even if she receives the signal i and if the actual precision α is relatively small (α < 1 − p¯ 2 ), otherwise she will not enter. Next we analyze the expected payoff of the entrant. Let π E (α ) be the equilibrium expected payoff of E when the precision of the IS is α . The next proposition provides a significant change from the symmetric information case. Proposition 5. Consider the equilibrium of Γ . Then:

¯ . Then for all α in the interval (1/2, p¯ 1 ) E does not enter the market, irrespective of the signal received, and (1) Suppose that E (α ) > α π E (α ) is zero in this interval. For all α in( p¯ 1 , 1), π E (α ) is strictly increasing. ¯ . Then for all α in the interval (1/2, 1 − p¯ 2 ) E enters the market, irrespective of the signal sent by the IS, (2) Suppose that E (α ) < α and π E (α ) is a positive constant in this interval. On the other hand, for all α in (1 − p¯ 2 , 1), π E (α ) is strictly increasing. 1 (3) Let E (Π E ) = 1 Π E (α ) f (α ) dα be the unconditional expected payoff of E. Then 2

E¯ (Π E ) < E (Π E )

¯ ) and E (Π E ) = E (Π E (α ) | E (α ) < α¯ ). where E¯ (Π E ) = E (Π E (α ) | E (α ) > α In contrast to the common knowledge case, Proposition 5 shows that in the asymmetric case E is always better off with a perfect IS. The payoff of E as function of α is constant up to a certain level since her equilibrium strategy does ¯ and E enters the market if E (α ) < α¯ ). (See not depend on the signal generated by the IS (E does not enter if E (α ) > α Figs. 9, 10.) Thereafter, the equilibrium expected payoff of E is strictly increasing because she detects M’s action with a relatively accurate precision and choose her strategy accordingly. Given α , the expected equilibrium payoff of E when ¯ is greater than when E (α ) > α¯ because, in the former case, M invests with low probability believing that E is not E (α ) < α likely to detect him. Finally, part (3) of Proposition 5 asserts that also a priori E, the owner of the IS, is better off with a random selection of α from a distribution f which yields on average smaller rather than higher values of α provided that f is commonly known. ¯ , M’s expected payoff as a function of Next let us consider the Incumbent’s expected payoff. Note that when E (α ) > α the probability p of expanding capacity is





1

E Π M ( p ) = p c + (b − c )



(1 − α ) f (α ) dα + (1 − p ) 1 − p



1

α f (α ) dα p

(see (A.7) in the proof of Proposition 4). By Proposition 4(1) it is easy to verify that given ¯ is case E (α ) > α



   ¯ ¯ πM α  E (α ) > α¯ = p 1 c + (1 − p 1 ),

< α < p¯ 1 [ p¯ 1 (1 − b + c ) − 1]α + 1 − p¯ 1 (1 − b), p¯ 1 < α < 1 1 2

α , the expected payoff of M in

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A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

¯. Fig. 10. The expected payoff of E when E (α ) < α

Note that, when α < p¯ 1 , E does not enter the market independently of the precision α . Since M expands capacity with probability p¯ 1 , his expected payoff is p¯ 1 c + (1 − p¯ 1 ) (see Fig. 2). When, however, α > p¯ 1 , E enters if the signal is ni and does not enter otherwise. Therefore, when M expands capacity, E obtains the signal i with probability α and the signal ni with the complementary probability, hence M’s expected payoff is p¯ 1 α c + p¯ 1 (1 − α )b. When M does not expand capacity (this happens with probability 1 − p¯ 1 ), E obtains the signal i with probability 1 − α and the signal ni with the complementary probability. In this case, M’s expected payoff is (1 − p¯ 1 )(1 − α ). Putting these two expected payoffs together yields the above ¯ ). expression for π M (α | E (α ) > α ¯ , M’s expected payoff is given by Similarly, when E (α ) < α



 1  α f (α ) dα + (1 − p ) (1 − α ) f (α ) dα 

1

E Π M ( p ) = p b + (c − b) 1− p

1− p

(see (A.11) in the proof of Proposition 4). By Proposition 4(2) it is easy to show that given

   πM α  E (α ) < α¯ =



α,

p¯ 2 b, < α < 1 − p¯ 2 [ p¯ 2 (1 − b + c ) − 1]α + 1 − p¯ 2 (1 − b), 1 − p¯ 2 < α < 1 1 2

More intuitively, one can obtain this expression by taking into account the equilibrium strategies given in Proposition 4 and the payoffs in the table of Fig. 2. The following proposition summarizes this result, states the unconditional expected payoff of M and shows that the incumbent is better off with an IS of high expected precision than with one of low expected precision (see Appendix A). Proposition 6. Consider the equilibrium of Γ . Then:

¯ , πM (α ) is constant for (1) If E (α ) > α

1 2

< α < p¯ 1 . For p¯ 1 < α < 1, πM (α ) is strictly decreasing if

1 2

< p¯ 1 <

< p¯ 1 < 1. ¯ , πM (α ) is constant for 12 < α < 1 − p¯ 2 and is strictly decreasing for 1 − p¯ 2 < α < 1. (2) If E (α ) < α 1 (3) Let E (Π M ) = 1 Π M (α ) f (α ) dα be the unconditional expected payoff of M. Then increasing if

1 1−b+c

1 , 1−b+c

and it is strictly

2

E¯ (Π M ) > 1 − E (α ),

E (Π M ) < 1 − E (α )

and

E¯ (Π M ) > E (Π M )

¯ ) and E (Π M ) = E (Π M (α ) | E (α ) < α¯ ). where E¯ (Π M ) = E (Π M (α ) | E (α ) > α In the common knowledge case M is always better off when E perfectly detects his action. Results differ under asymmetric information. Up to a certain value of α the ex-post expected payoff of M is constant because, as already mentioned, the equilibrium strategy of E does not depend on the signal she receives. Thereafter, the payoff of M may be increasing or decreasing. (See Figs. 11–13.) If the expected precision of the IS is relatively high and M invests with high probability ( 1−1b+c < p¯ 1 < 1), his payoff will be increasing because a more accurate IS is more likely to generate the signal i, provoking E to stay out of the market. However, if he invests with relatively low probability ( 12 < p¯ 1 < 1−1b+c ), the IS is unlikely to generate the signal i and his payoff decreases. However, even when M invests with high probability, his expected equilibrium payoff is greater when 1 < α < p¯ 1 than when p¯ 1 < α < 1. In the former case E stays out for sure while in the latter case there is a positive 2 probability that she enters the market. Hence, M is better off when E does not spy on him or when the IS has low accuracy.

A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

< p¯ 1 <

¯ and Fig. 11. The expected payoff of M when E (α ) > α

1 2

¯ and Fig. 12. The expected payoff of M when E (α ) > α

1 1−b+c

1 1−b+c

137

.

< p¯ 1 < 1.

¯. Fig. 13. The expected payoff of M when E (α ) < α

If, however, the IS expected precision is low while the IS is quite accurate (1 − p¯ 2 < α < 1), the equilibrium payoff of M is decreasing in α since in this case M has a high probability of not expanding capacity in the belief that E is unlikely to detect his action, and as α becomes larger, E is more likely to detect M’s action and therefore will enter. Nevertheless, even though the payoff of M is decreasing in this region, it is still greater when the IS is less accurate ( 12 < α < 1 − p¯ 2 ) because in the latter case E enters for sure while in the former case with positive probability E stays out. Hence, M prefers E to spy on him, but with an imperfect IS. In fact, the closer α is to 1 − p¯ 2 (from above) the better is M since this precision increases the likelihood that the IS generates the signal i, inducing E to stay out. As for the unconditional profit of M, quite surprisingly, M who does not know α , prefers an IS with higher expected precision rather than one with lower expected precision. This is in contrast to E’s preference, who prefers a random selection of α from a distribution where E (α ) is smaller rather than higher.

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A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

5. Concluding remarks In spite of the importance for a potential entrant of obtaining information about incumbent firm(s), the effects of espionage on entry deterrence models have not been studied before in the literature. Our paper takes an initial step in this direction but leaves plenty of opportunity for further research. In this paper we have considered a model where a potential entrant uses an inaccurate Intelligence System (IS), of a given precision, to spy upon a monopoly incumbent and detect his decision of whether to expand capacity. Using this information, the entrant makes entry decisions. It may seem that more precise information would always benefit the entrant, but our results show that, if the precision of the spying technology is commonly known to both firms, then only the incumbent will benefit from a perfect IS while the entrant will prefer a less accurate one. However, when the IS precision is the Entrant’s private information, she can avoid being completely manipulated by the incumbent’s signal-jamming, and, in fact, for some values of the precision, she will make equilibrium entry decisions independently of the signal. The Incumbent’s signal-jamming behavior under both information structures is similar to the one for the oligopoly games expounded by Riordan (1985), Aghion et al. (1993), Mirman et al. (1993), Caminal and Vives (1996), Milgrom and Roberts (1982), Alepuz and Urbano (2005), among others. Also, the incumbent’s manipulation of information is related to results of Vives (1984), Raith (1996), Gal-Or (1985, 1986, 1987, 1988), about sharing information in oligopoly markets, where allowing rivals to acquire better information of their profit functions leads to a higher correlation of strategies. In our model, how does the correlation among strategies affect the incentives to share information? Note that the Incumbent and the Entrant’s actions are correlated through the IS signals, whose accuracy depends on the IS precision. Consequently, a more accurate IS will have a different effect on such correlation depending on whether the information about the IS precision is common knowledge or privately known. Under common knowledge, the Incumbent will use the preemptive capabilities of a Stackelberg leader and will correlate his investment strategy with the IS precision in an attempt to manipulate the signal occurrence likelihood. Namely, the Incumbent will expand capacity with probability either increasing as IS precision increases (whenever the IS is relatively accurate) or decreasing as IS precision increases (whenever the IS is not too accurate), implying that the unconditional probability that the Entrant enters decreases when the IS precision increases. An increase in the IS accuracy increases the reliability of the signal generated. Hence, when the Entrant observes the signal i she enters the market with lower probability and the Incumbent is better off. Therefore, in a common knowledge scenario, the higher the IS precision, the lower the probability that the Entrant enters and the higher both the Incumbent’s expected profits and his overinvestment in capacity. As already mentioned and reflected in Gal-Or (1987), the Incumbent benefits from an improved information system available to the entrant because he can reduce the incidence of entry without having to overinvest in capacity. The results obtained under common knowledge do not necessarily apply under asymmetric information, where the Entrant is always better off with a perfect information system. Given that the IS precision is now private knowledge of the Entrant, and unless the IS is very accurate, the Entrant may find it profitable to correlate her pure strategy with the IS precision, and reduce the correlation with the signals generated by the Incumbent’s actions. Thus, as in the literature of oligopoly models of incomplete information, the Incumbent’s preemptive capabilities of a Stackelberg leader are reduced. Consequently, the Incumbent will prefer a situation whereby the Entrant does not spy on him or that the IS accuracy is low enough. The Entrant’s payoff as a function of the IS precision is constant up to a certain value and strictly increasing above that value. Therefore, under private knowledge of the IS precision, the Entrant prefers a perfect IS and incorrect beliefs on the part of the Incumbent (an IS with a low expected precision), because she would enter in the majority of cases. Even though in the present paper we assume that the IS accuracy is exogenously stated, extensions to the analysis would allow the Entrant to determine endogenously how precisely to conduct her espionage. Industrial espionage has become an increasingly common business practice. Although espionage is considered an illegal and unethical activity, it has the potential to yield desirable strategic effects and/or profit-shifting effects in markets with barriers to entry. Do espionage activities enhance market competition? The answer depends on the asymmetry of the information about the accuracy of the spying activity. If the IS precision is common knowledge, there will be small room for spying activities. Only if this precision is small enough will the Incumbent’s probability of expanding capacity be lower than in the case of not spying, leading to greater efficiency in the market. When the IS accuracy is high enough, and the information is available only to the Entrant and is very asymmetric, market competition grows because the Entrant enters in the majority of cases. Spying will also be beneficial to consumers because, in general, the expected market output will rise, the expected price will fall and the expected consumer surplus will increase. What are the main policy implications? Should espionage be completely banned? From the above results, whether espionage is a virtue or a vice (Whitney and Gaisford, 1999) is still an open question which requires further research. Appendix A Proof of Proposition 1. It is easy to verify that the unique Nash equilibrium strategy of M is to invest with probability p, such that

A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

139

Fig. 14. The signal conditional probabilities.

p (α ) =

1 − α,

α,

¯ 1 /2 < α < α α¯ < α < 1

¯ , she will play ( E , E ) with probability q∗ , where As for E, if 1/2 < α < α q∗ =

1 − b − α (1 + c − b)

(A.1)

1 − α (1 + c − b)

¯ < α < 1, E will play ( E , N E ) with probability qˆ , where and ( E , N E ) with probability 1 − q∗ . If α qˆ =

1−c

(A.2)

α (1 − b + c ) + b − c

and ( N E , N E ) with probability 1 − qˆ . ¯ , the game will have multiple equilibrium points: M will invest with probability p˜ where If α = α and E will play ( E , N E ) purely. 2

c 1−b+c

 p˜ 

1−b , 1−b+c

¯ . In equilibrium the Entrant will enter with probability 1 if the Proof of Proposition 2. Consider first the case 1/2 < α < α signal is ni, and with probability q∗ if the signal is i. Hence Prob( E ) = Prob(ni ) + q∗ Prob(i ) where p = 1 − α . By Fig. 14,

Prob(ni ) = p (1 − α ) + (1 − p )α = (1 − α )2 + α 2 and

Prob(i ) = p α + (1 − p )(1 − α ) = 2(1 − α )α By (A.1)

Prob( E ) =

1 + 2bα 2 − α (1 + c + b) 1 − α (1 + c − b)

Since [2 − (1 + c − b)α ]α − 1 = −(α − 1)2 + (b − c )α 2 < 0,

∂ Prob( E ) 2b[[2 − (1 + c − b)α ]α − 1] = <0 ∂α [1 − α (1 + c − b)]2 ¯ < α < 1. In equilibrium the Entrant will not enter if the signal is i, and she will enter with probability Next assume that α qˆ if the signal is ni. Hence Prob( E ) = qˆ Prob(ni ) + 0 Prob(i ) Using Fig. 14 (but now p = α ) we have

Prob(ni ) = p (1 − α ) + (1 − p )α = 2(1 − α )α By (A.2)

Prob( E ) =

2(1 − c )(1 − α )α

α (1 − b + c ) + b − c

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A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

and it is decreasing in

α since

∂ Prob( E ) −2(1 − c )[(α − 1)2 (c − b) + α 2 ] = <0 ∂α [α (1 − b + c ) + b − c ]2

2

Proof of Proposition 3. It can be easily verified that

ΠM =

(1−α )b , 1−α (1+c −b) α (2c −b)+b−c α (1−b+c )+b−c ,

¯ 1 /2 < α < α

α¯ < α < 1

and

ΠE =

¯ 1 /2 < α < α α¯ < α < 1

2α − 1, 0,

and the proof follows immediately. b−c ¯ , Π˜ M = 1−cb+c and Π˜ E ∈ [0, 11− If α = α −b+c ].

2

Proof of Proposition 4. Suppose that M chooses I with probability p and NI with probability 1 − p. Using Fig. 14 above,

αp α p + (1 − α )(1 − p ) (1 − α )(1 − p ) Prob E ( N I | α , i ) = α p + (1 − α )(1 − p ) (1 − α ) p Prob E ( I | α , ni ) = (1 − α ) p + α (1 − p ) α (1 − p ) Prob E ( N I | α , ni ) = (1 − α ) p + α (1 − p ) Prob E ( I | α , i ) =

Let Π E ( E | α , i ) be the expected payoff of E if the signal is i and she enters the market. Then,

Π E ( E | α , i ) = Prob E ( I | α , i )u E ( I , E ) + Prob E ( N I | α , i )u E ( N I , E ) =

1− p−α

α p + (1 − α )(1 − p )

(A.3)

Similarly

Π E ( E | α , ni ) =

α−p (1 − α ) p + α (1 − p )

ΠE (N E | α, i) = 0 Π E ( N E | α , ni ) = 0

(A.4)

Given p, by (A.3) and (A.4), if E receives the signal i she will prefer E on NE if and only if

1− p−α

α p + (1 − α )(1 − p )

>0

or equivalent if and only if α < 1 − p. That is, if E receives the signal i she will enter if α < 1 − p and she will not enter if indifferent between entering and not entering the market. Similarly, if E receives the signal ni she will enter the market if and only if

α−p >0 (1 − α ) p + α (1 − p ) or equivalently if and only if

α > p.

α > 1 − p. If α = 1 − p E will be

A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

141

Fig. 15. s E (i | α , p ).

Fig. 16. s E (ni | α , p ).

We can now write the best reply strategy of E as a function of p and the signal she receives (see Figs. 15 and 16):

⎧ N E, ⎪ ⎪ ⎪ ⎨ E, s E (i | α , p ) = ⎪ N E, ⎪ ⎪ ⎩

p > 12 , p p

any strategy,

and

s E (ni | α , p ) =

p

1 , 2 1 , 2 1 , 2

1 2 1 2

<α1 <α <1− p 1− p<α 1 α =1− p

⎧ ⎪ ⎪ E, ⎪ ⎨ N E,

p < 12 ,

⎪ E, ⎪ ⎪ ⎩

p

p

any strategy,

p

(A.5)

1 2 1 1 , 2 2 1 , p 2 1 , 2

<α1 <α< p <α1 α=p

(A.6)

1−b ¯  1/2, then E (α ) > α¯ would be ¯ where, without loss of generality, α¯ = 1− (1) Suppose that E (α ) > α (note that if α b+c the only case). We first consider the case where M expands his capacity with probability 12 < p < 1. Let E Π M ( p ) be the expected payoff of M. By (A.5) and (A.6)

 1 E ΠM ( p ) = p

p

α u M ( I , N E ) f (α ) dα + 1 2

 p

+ (1 − p )

(1 − α )u M ( I , N E ) f (α ) dα +

(1 − α )u M ( I , E ) f (α ) dα p

1 2

1

α u M ( N I , N E ) f (α ) dα + 1 2



1



1

α u M ( N I , E ) f (α ) dα + p

Since u M ( I , E ) = b, u M ( I , N E ) = c , u M ( N I , E ) = 0 and u M ( N I , N E ) = 1

(1 − α )u M ( N I , N E ) f (α ) dα 1 2

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A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

 1 E ΠM ( p ) = p c

p

α f (α ) dα + c 1 2

(1 − α ) f (α ) dα + b

 p

1 1 2



1

α f (α ) dα + 1 2

Since

(1 − α ) f (α ) dα p

1 2

+ (1 − p )



1

(1 − α ) f (α ) dα 1 2

f (α ) dα = 1

 E Π M ( p ) = p c + (b − c )





1

(1 − α ) f (α ) dα + (1 − p ) 1 − p



1

α f (α ) dα

(A.7)

p

 Let E Π M ( pˆ | p ) be the expected payoff of M when he chooses to invest with probability pˆ while E believes that M invests with probability p:

    1 1  E Π M ( pˆ | p ) = pˆ c + (b − c ) (1 − α ) f (α ) dα + (1 − pˆ ) 1 − α f (α ) dα p

(A.8)

p

Note that E observes neither the mixed strategy of M nor his actual action. She only observes the signal sent by the IS. Hence, if M unilaterally deviates from his mixed strategy ( p , 1 − p ) to any other strategy, the strategy of E (as a function of her type α and the signal observed) will not change, but the probabilities of the signals will. Assuming that the entrant still expects the incumbent to follow the mixed strategy ( p , 1 − p ), in equilibrium, M should be indifferent between choosing that mixed strategy or either I ( pˆ = 1), or NI ( pˆ = 0). Therefore

 E Π M (1 | p ) = E Π M (0 | p )

(A.9)

By (A.8),

 E Π M (1 | p ) = c + (b − c )

1 (1 − α ) f (α ) dα p

and

 E Π M (0 | p ) = 1 −

1

α f (α ) dα p

Hence by (A.9)

1 c + (b − c )

1 (1 − α ) f (α ) dα = 1 −

p

α f (α ) dα

(A.10)

p

Let

1 g ( p ) ≡ c − 1 + (b − c )

1 (1 − α ) f (α ) dα +

p

be defined for all

1 2

α f (α ) dα p

 p  1. Since f (α ) is continuous in α , g ( p ) is continuously differentiable in p. Also

  g

1 2

= −1 + b + (1 − b + c ) E (α )

1−b By our assumption E (α ) > 1− , hence g ( 12 ) > 0. Since g (1) = c − 1 < 0 by the Mean Value Theorem there is a p¯ 1 , b+c 1 < p¯ 1 < 1, such that g ( p¯ 1 ) = 0. 2

A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

143

Next

g  ( p ) = −(b − c ) f ( p ) − (1 − b + c ) p f ( p )

  = −(1 − b + c ) p − (b − c ) f ( p )

Since f ( p ) > 0

g  ( p ) > 0 if and only if c −b 1−b+c

p<

c−b 1−b+c

1 , 2

Since < g ( p ) is decreasing for 12  p < 1. Since g ( 12 ) > 0 and g (1) < 0 then g crosses the p-axis only once. Therefore, p¯ 1 is the unique solution of g ( p ) = 0 and therefore of (A.10). 1−b Next observe that M has no equilibrium strategy ( p¯ , 1 − p¯ ) such that 0 < p¯  12 and E (α ) > 1− . Otherwise (A.10) b+c should be replaced by

1 c + (b − c )

1 (1 − α ) f (α ) dα = 1 −

1 2

α f (α ) dα 1 2

This implies that

E (α ) =

1−b 1−b+c

1−b which is a contradiction. We conclude that whenever E (α ) > 1− the game Γ has a unique equilibrium. b+c 1−b (2) Suppose next that E (α ) < 1−b+c . Consider first the case where M expands his capacity with probability p, 0 < p < 12 . Similarly to the previous case

 1− p E ΠM ( p ) = p

1

α u M ( I , E ) f (α ) dα +

α u M ( I , N E ) f (α ) dα +

1− p

1 2

 1 + (1 − p )



1 (1 − α )u M ( I , E ) f (α ) dα 1 2

1 − p

α u M ( N I , E ) f (α ) dα + 1 2



1

(1 − α )u M ( N I , E ) f (α ) dα +

(1 − α )u M ( N I , N E ) f (α ) dα

1− p

1 2

Since u M ( I , E ) = b, u M ( I , N E ) = c, u M ( N I , E ) = 0 and u M ( N I , N E ) = 1



1

E Π M ( p ) = p b + (c − b)

 1  α f (α ) dα + (1 − p ) (1 − α ) f (α ) dα 

1− p

(A.11)

1− p

In equilibrium where 0 < p < 1 we have

 E Π M (0 | p ) = E Π M (1 | p )

(A.12)

By (A.11) and (A.12) we have

1 b + (c − b)

1

α f (α ) dα =

1− p

(1 − α ) f (α ) dα

(A.13)

1− p

Let

1 m(x) ≡ b + (c − b)

α f (α ) dα − x

be defined for all

  m

1 2

1 2

1 (1 − α ) f (α ) dα x

 x  1. Clearly m(x) is continuous and differentiable. Since E (α ) <

= (1 − b + c ) E (α ) + (b − 1) < 0

1−b , 1−b+c

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A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

Also

m(1) = b > 0 In addition





m (x) = 1 − (1 − b + c )x f (x) Since f (x) > 0

m (x) > 0 if and only if

1 1−b+c

and decreases for 1−1b+c < x  1. Since m( 12 ) < 0 and m(1) > 0 then m intersects the x-axis only once. Namely, there is a unique x¯ such that m(¯x) = 0 and 12 < x¯ < 1. Thus there exists a p¯ 2 , 0 < p¯ 2 < 1, such

Hence m increases for

1 2

x<

x<

1 1−b+c

that p¯ 2 is the unique solution of (A.13), and p¯ 2 <

1 2

which is consistent with our assumption.

Next it is easy to verify (similarly to the previous case) that there is no equilibrium where p  1−b 1−b+c

the game Γ has a unique equilibrium. conclude that whenever E (α ) < It is also easy to verify that there is no equilibrium where M is playing a pure strategy.

1 2

1−b and E (α ) < 1− . We b+c

2

Proof of Proposition 5. By (A.5), (A.6) and Proposition 4 it is easy to verify that

⎧ 1 − 2 p¯ 2 , ⎪ ⎪ ⎨ ¯ π E (α ) = α − p 2 , ⎪ 0, ⎪ ⎩ α − p¯ 1 ,

¯ , 12 < α < 1 − p¯ 2 E (α ) < α ¯ , 1 − p¯ 2 < α < 1 E (α ) < α ¯ , 12 < α < p¯ 1 E (α ) > α ¯ , p¯ 1 < α < 1 E (α ) > α

and (1) and (2) follow immediately. ¯ . Let To prove part (3) of the proposition, suppose first that E (α ) > α

1 g¯ (x) =

(α − x) f (α ) dα x

Then E¯ (Π E ) = g¯ ( p¯ 1 ). Since g¯ is decreasing in x ( g¯  (x) < 0) and p¯ 1 > 12 ,

E¯ (Π E ) < g¯

  1 2

1  =

α−

1

 f (α ) dα

2

(A.14)

1 2

¯ . Let Suppose next that E (α ) < α g (x) =

1−x 1 (1 − 2x) f (α ) dα + (α − x) f (α ) dα 1− x

1 2

It can be easily verified that g  (x) < 0. Since p¯ 2 <

  E (Π E ) = g ( p¯ 2 ) > g

1 2

  1

= g¯

2

1 2

and by (A.14),

> E¯ (Π E )

2

¯ . By the proof of Proposition 4 g ( p¯ 1 ) = 0, where Proof of Proposition 6(3). (i) Suppose first that E (α ) > α

1 g ( p ) = −1 + c + (b − c )

1 (1 − α ) f (α ) dα +

p

p

1 = −1 + c + (1 − b + c )





α f (α ) dα + (b − c ) 1 − F ( p ) p

Hence,

α f (α ) dα

A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

145

1

α f (α ) dα = 1 − b − (c − b) F ( p¯ 1 )

(1 − b + c )

(A.15)

p¯ 1

Denote

    G ( p ) = ( pc + 1 − p ) F ( p ) + 1 − p (1 − b) 1 − F ( p ) + p (1 − b + c ) − 1 

1

α f (α ) dα p

Then E¯ (Π M ) = G ( p¯ 1 ). Using (A.15) we have







E¯ (Π M ) = ( p¯ 1 c + 1 − p¯ 1 ) F ( p¯ 1 ) + 1 − p¯ 1 (1 − b) 1 − F ( p¯ 1 )

  + p¯ 1 1 − b + c − (c − b) F ( p¯ 1 ) −

1

α f (α ) dα p¯ 1

After rearranging terms we have

E¯ (Π M ) = 1 −

1

1

α f (α ) dα > 1 − p¯ 1

α f (α ) dα = 1 − E (α ) 1 2

¯ . Again, by the proof of Proposition 4 m(1 − p¯ 2 ) = 0, where (ii) Suppose next that E (α ) < α

1 m(x) = b + (c − b)

1

α f (α ) dα − x

(1 − α ) f (α ) dα x

1 = b + (1 − b + c )

α f (α ) dα − 1 + F (x) x

This together with m(1 − p¯ 2 ) = 0 implies

1 (1 − b + c )

α f (α ) dα = 1 − b − F (1 − p¯ 2 )

(A.16)

1− p¯ 2

Note that E (Π M ) = M ( p¯ 2 ), where (using (A.16))

     M ( p¯ 2 ) = p¯ 2 b F (1 − p¯ 2 ) + 1 − p¯ 2 (1 − b) 1 − F (1 − p¯ 2 ) + p¯ 2 1 − b − F (1 − p¯ 2 ) −

1

1− p¯ 2

Rearranging terms implies

1 E (Π M ) = 1 − F (1 − p¯ 2 ) −

α f (α ) dα

1− p¯ 2

Hence,

∂ E (Π M ) = p¯ 2 f (1 − p¯ 2 ) > 0 ∂ p¯ 2 Since p¯ 2 < 12 , we have

  E (Π M ) < 1 − F

1 2

1 −

α f (α ) dα = 1 − E (α ) 1 2

as claimed. Finally, let us prove that E (Π M ) < E¯ (Π M ).

α f (α ) dα

146

A. Barrachina et al. / Games and Economic Behavior 83 (2014) 127–146

The last inequality is equivalent to

1

1

α f (α ) dα > 1 − F (1 − p¯ 2 ) −

1− p¯ 1

α f (α ) dα

1− p¯ 2

or

1 F (1 − p¯ 2 ) >

1

α f (α ) dα − p¯ 1

α f (α ) dα

(A.17)

1− p¯ 2

The last inequality certainly holds if p¯ 1  1 − p¯ 2 . Suppose that p¯ 1 < 1 − p¯ 2 . Then (A.17) is equivalent to 1− p¯ 2

F (1 − p¯ 2 ) >

α f (α ) dα p¯ 1

But since

α  1,

1− p¯ 2

1− p¯ 2

f (α ) dα = F (1 − p¯ 2 ) − F ( p¯ 1 )  F (1 − p¯ 2 )

α f (α ) dα < p¯ 1

p¯ 1

and the proof is complete.

2

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