Rivalry and uncertainty in complementary investments with dynamic market sharing
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Abstract
We study the effects of revenue and investment cost uncertainty, as well nonpreemption duopoly competition, on the timing of investments in two complementary inputs, where either spilloverknowledge is allowed or proprietaryknowledge holds. We find that the exante and expost revenue market shares play a very important role in firms’ behavior. When competition is considered, the leader’s behavior departs from that of the monopolist firm of Smith (Ind Corp Change 14:639–650, 2005). The leader is justified in following the conventional wisdom (i.e., synchronous investments are more likely), whereas, the follower’s behavior departs from that of the conventional wisdom (i.e., asynchronous investments are more likely).
Keywords
Complementary investments Duopoly Investment analysis Nonpreemption Real option game UncertaintyJEL Classification
C70 C72 C73 D81 D92 G311 Introduction
Sometimes it pays to put the (new) cart before the (new) horse. Conventional wisdom says that “when a production process requires two extremely complementary inputs, a firm should upgrade (or replace) them simultaneously”.^{1} When raising the quality of one input, a firm should upgrade its complement at the same time. These guidelines are corroborated by the deterministic models of Milgrom and Roberts (1990, 1995) and Colombo and Mosconi (1995).
However, the above literature neglects the effect of uncertainty. Smith (2005) considers operating cost and investment cost uncertainty, using a real option model, and concludes that the conventional wisdom described above does not necessarily hold if the costs are uncertain and their growth rates differ significantly. However, she neglects the effect of competition.
We provide a somewhat richer model, which considers the effect of uncertainty, as well as duopoly nonpreemption competition, on the timing of the investment in two complementary inputs. We rely on the Siddiqui and Takashima (2012) leader–follower duopoly model setting, which comprises two alternative market structures: first, where spilloverknowledge (SK) is allowed, and second, where there is proprietaryknowledge (PK).
In line with Smith (2005), we find that when the costs of two inputs are uncertain and decreasing at different rates, it may pay, for both firms, to invest first in the input whose cost is falling more slowly and wait to invest in the input whose cost is falling more rapidly. However, this guideline applies more to the follower, and is dependent on the market structures and assumptions on the exante and expost market shares of the two firms. The leader is more prone than the monopolistic firm of Smith (2005) to invest in the two inputs at the same time (follows the behavior suggested by the conventional wisdom), whereas the follower is more likely to invest in the two inputs sequentially (follows the behavior of the monopolist firm of Smith 2005)—see Fig. 13 in “Appendix C”.
Our model setting is for a nonpreemption game, therefore, the leader invests at the time as a monopolist, and gets 100% of the market while operating alone, regardless of with which input(s) she operates (input 1 alone, input 2 alone, or input 1 and input 2 at the same time). Thus, the degree of competition (i.e., how the market share is divided between the two firms when both are active) does not affect the timing of the leader’s investment in the two inputs at the same time. However, we find that it influences significantly the timing of the follower’s investment. The more asymmetric is the expost market share between the two firms, favouring the leader, the later the follower invests, which makes more likely asynchronous inputinvestments. This finding is somewhat surprising, because in a leader–follower investment game as soon as the game ends for the leader, the follower is in a monopolylike position.
Furthermore, we show that: (i) in nonpreemption duopoly SK and PK markets, an increase in the input cost growth rate differences makes both firms more likely to invest in the two inputs sequentially, (ii) for an inactive firm, a decrease in the cost growth rate of one input and an increase in the cost growth rate of another input accelerates the investment in the input whose cost growth rate is increasing, deters the investment in the input whose cost growth rate is decreasing, but may have no effect on the timing of the investment in the two inputs at the same time. We also find that sequential investments are more likely: (iii) when firms are inactive and the cost growth rate of one input is increasing and the cost growth rate of another input is decreasing, so there is a negative change in the cost growth rate of the two inputs together.
Finally, we conclude that, for both the SK and the PK markets: (iv) for simultaneousinput investments, the sensitivity of the follower’s investment threshold to changes in the degree of complementarity is greater than that of the leader, and (v) an increase in the degree of complementarity between input 1 and input 2, accelerates the investments of both firms in the two inputs at the same time, if they are inactive, and the investment in input 2(1) if they are active with 1(2).
The real options literature for monopoly markets neglects the effect of competition on firms’ investment behavior, but it is very extensive and diverse in terms of practical applications (Martzoukos 2000; Smith 2005; Koussis et al. 2007; BastianPinto et al. 2010; Franklin 2015; Chronopoulos and Siddiqui 2015; Farzan et al. 2015). Smets (1993) initiated a new branch of literature, now named “real option game” models, which study firms’ investment behavior considering uncertainty and (duopoly) competition (Dixit and Pindyck 1994, Ch. 9; Grenadier 1996; Huisman 2001; Weeds 2002; Huisman and Kort 2003, 2004; Paxson and Pinto 2005; Pawlina and Kort 2006; Hsu and Lambrecht 2007; Moretto 2008; Thijssen 2010; Femminis and Martini 2011; Leung and Kwok 2012; Pereira and Rodrigues 2014).^{2}
Yet, the above literature neglects the existence of complementarity between investments. Furthermore, with few exceptions (Huisman 2001, Ch. 9; Smith 2005; Decamps et al. 2006; Nishihara 2012; Siddiqui and Takashima 2012), firms are usually assumed to hold (exante) only one option to invest.
However, firms often use inputs whose qualities are complements. Therefore, investment decisions on upgrades or replacements must consider the degree of complementarity between the inputs and the possibility of sequentialinput investments. In this article “complementarity” exists if the investment in one input increases the marginal or incremental return to another input in terms of “net cost savings”. More generally, in industrial organization contexts, complementarity exists if the implementation of one practice increases the marginal return to another practice (see, e.g., Carree et al. 2011). When the implementation of a technology/practice decreases the marginal return to the other technology/practice, there is “substitutability” (or subadditivity).^{3}
Notice that, due to technological progress, input costs are usually uncertain and might follow different evolution patterns. For instance, according to a joint report by the U.S. Solar Energy Industries Association (SEIA) and GTM Research, the cost of solar power (technology) in the U.S. is now 60% cheaper than in early 2011, but the cost of the solar panel sites may have increased.^{4} Therefore, it might be optimal to invest in the solar panel sites first and defer the investment in the solar panels. Similar guidelines apply to wind energy investments, if the cost growth rates of the wind towers and wind farm sites differ significantly. Furthermore, wind towers comprise several components (e.g., the tower, rotor hub, blades, etc.) whose cost growth rates may differ significantly. Therefore, it might be optimal to replace the components of old wind farms sequentially, starting with those whose cost is decreasing more slowly.^{5}
There are also industries whose production is organized in subindustries, with each subindustry corresponding to a production stage of the overall industry. This is the case of the textile industry, which is organized in four main subindustries: spinning, weaving, finishing and making up, where there is high (production quality/efficiency) complementarity. The adoption of a more advanced spinning technology increases not only the efficiency of the spinning mill but also the efficiency of the weaving mills which use yarns produced by the spinning mill whose technology was upgraded. But asynchronous investments might be possible if there is a high cost growth rate asymmetry between the technologies used in each of these subindustries. Thus, we can see firms operating with highly advanced spinning technologies and relatively obsolete weaving mills, or viceversa (Griffiths et al. 1992; Conrad and James 1995).
The concept of complementarity is also used to study economic decisions in other contexts. For instance, when planning R&D activities, firms make strategic decisions regarding the degree of complementarity (sometimes called compatibility) between the incumbent products and the new products they aim to launch in the future since the diffusion of an innovation depends, to some extent, on the diffusion of complement innovations which amplify its value.^{6}
It has been also argued that the pace of modernization of industries is often influenced by the degree of technological complementarity between the technologies adopted in the industries. Milgrom and Roberts (1994) study the Japanese economy between 1940 and 1995 to interpret the characteristic features of Japanese economic organization in terms of the complementarity between some of the most important elements of its economic structure. Colombo and Mosconi (1995) examine the diffusion of flexible automation production and design/engineering technologies in the Italian metalworking industry, giving particular attention to the role of the technological complementarity and the learning effects associated with the experience of previously available technologies. From Milgrom and Roberts (1990, 1995) models, we infer that it is relatively unprofitable to adopt only one part of the modern manufacturing technologies. Milgrom and Roberts (1990, p. 524) suggest that “we should not see an extended period of time during which there are substantial volumes of both highly flexible and highly specialized equipment being used sidebyside”.
The rest of the article is organized as follows. In Sect. 2, we outline the model assumptions and describe the two market structures. In Sect. 3, we derive the value functions and investment thresholds for the two firms and each of the market structures, and provide some illustrative sensitivity analysis. In Sect. 4, we show some further results. In Sect. 5, we conclude and offer guidelines for future research.
2 The model
2.1 Industry settings
We formulate a leader–follower investment problem for two specific industry scenarios, following Siddiqui and Takashima (2012, p. 585): (i) symmetric nonpreemptive duopoly with “spilloverknowledge” (SK); and (ii) symmetric nonpreemptive duopoly with “proprietaryknowledge” (PK). The difference between these two scenarios is that, in the former, due to a weak patentprotection, the follower is allowed to proceed with his firststage investment (in input 1) immediately after the leader’s entry (with input 1) and, in the latter, due to a strong patentprotection, the leader invests in the two inputs sequentially (first in input 1 and then in input 2) with the follower inactive.
2.1.1 Nonpreemptive duopoly with SK
This industry setting considers a duopoly where the leader cannot be preempted by the follower in the first move. After the leader’s first move, the follower is allowed to proceed, since he obtains knowledge of the leader investment (input 1) via spilloverknowledge. The diagram in Fig. 1 indicates both investment approaches for the two firms: sequentialinput investment (solid lines) and simultaneousinput investment (dotted lines). From state (1,1) the competition for establishing a dominant position remains sequential until the investment cycle is completed in state (2,2). Similarly, in the direct approach (simultaneousinput investmentdotted line), the leader invests first, before the follower is allowed to proceed. We add to the Siddiqui and Takashima (2012) investment problem the effects of both the investment cost uncertainty and the complementarity between the inputs in the two stages.
2.1.2 Nonpreemptive duopoly with PK
In this scenario the leader is allowed to invest in inputs 1 and 2, sequentially or simultaneously, with the follower inactive, due to proprietaryknowledge.
3 Analytical results
3.1 Simultaneousinput investments
3.1.1 SK market
In this section we consider that both firms are inactive at stage (0,0) and invest, one after the other, in the two inputs at the same time if optimal to do so. \(I_{12} \) is the investment cost if firms invest in the two inputs at the same time.
Follower
Notice that as \(\phi _{12} \) approaches 0, the value of the option to invest in input 1 and input 2 at the same time becomes worthless. Therefore, in (15) \(B_{12} =0\).^{16}
Leader
3.1.2 PK market
“Simultaneousinput” investments are “oneshot” games for the two firms in both the PK and the SK markets. Therefore, the investment behavior of the leader and the follower is the same for the two markets and the following proposition holds:^{22}
Proposition 1
3.1.3 Illustrative results (base parameter values are in Tables 1, 2 and 3)
Market variables
X  \(I_1 \)  \(I_2 \)  \(I_{12} \)  \(\mu _{X} \)  \(\mu _{I_1 } \)  \(\mu _{I_2 } \)  \(\mu _{I_{12} } \)  r  \(\sigma _{X} \)  \(\sigma _{I_1 }\)  \(\sigma _{I_2 } \)  \(\sigma _{I_{12} } \)  \(\rho _{XI_1 } \)  \(\rho _{XI_2 } \)  \(\rho _{XI_{12} } \) 

10  5  5  10  0.02  \(\) 0.05  \(\) 0.10  \(\) 0.075  0.05  0.2  0.2  0.2  0.2  0  0  0 
Firms’ market revenue share, \(D_{k_i k_j } \)
Leader  Follower  

\(D_{1_L 0_F } \)  \(D_{12_L 0_F } \)  \(D_{12_L 1_F } \)  \(D_{12_L 12_F } \)  \(D_{1_L 1_F } \)  \(D_{2_L 2_F } \)  \(D_{12_F 12_L } \)  \(D_{1_F 1_L } \)  \(D_{2_F 2_L } \)  \(D_{1_F 12_L } \) 
1.0  1.0  0.6  0.5  0.5  0.5  0.5  0.5  0.5  0.4 
Complementarity related factors, \(\xi \) and \(\gamma _k \)
\(\gamma _2 \)  \(\gamma _2 \)  \(\gamma _{12} \)  \(\xi =\gamma _{12} (\gamma _1 +\gamma _2 )\) 

0.10  0.10  0.30  0.10 
However, input cost uncertainty can make such behavior unlikely for the follower (in Sect. 4.1 we discuss this finding further). We also find that an increase in the revenue volatility or the investment cost volatility delays the investment for both firms.^{23}
3.2 “Sequentialinput” investments
3.2.1 SK market
SK market, terminalstate: follower
Proposition 2
two inactive firms in a nonpreemption duopoly (SK or PK) market invest in two complementary inputs (input 1 and input 2) sequentially if and only if there is a time t, \(t\in [ {0,\infty } )\), where \(\phi _1 (t)\) reaches \(\phi _{1_L }^{*} (t)\) and \(\phi _{1_F }^{*} (t)\) the first time with \(\phi _{12} (t)<\phi _{12_L }^{*} (t)\) and \(\phi _{12} (t)<\phi _{12_F }^{*} (t)\). Proof: see “Appendix B”.
3.2.2 Illustrative results
The above results show that an increase in \(\sigma _{X} \) delays the investment for both firms and an increase in \(\rho _{X/I_1 } \) accelerates the investment in input 1 for both firms, with the follower slightly more sensitive than the leader to changes in these variables (Fig. 3).
The above results show that an increase in \(\mu _{I_1 } \) accelerates the investment for both firms and an increase in \(D_{1_L 1_F } \) delays the investment of the follower and has no effect on the investment threshold of the leader (Fig. 4).
The above results are like those of Fig. 5 and, therefore, similar comments apply.
The above results show that an increase in \(\xi \) accelerates the investment in the second input for both firms and an increase in \(D_{1_L 1_F } \) delays the investment of the leader in the second input (input 2) and has no effect on the investment threshold of the follower (Fig. 6).
3.2.3 PK market
PK market, terminalstate: follower
For the terminal state of sequentialinput investments with the leader active with the two inputs, the threshold expression of the follower is the same for both the PK and the SK markets. Therefore, the following condition holds:
Proposition 3
PK market, firststate: follower
In the firststate of “sequentialinput investments” the investment threshold expression of the leader for the PK and SK markets are the same. Therefore, the following proposition holds:
Proposition 4
3.2.4 Illustrative results
The results show that the investment threshold of the leader is lower than that of the follower (as expected) and that both increase with the revenue volatility. Comparing the thresholds of the lefthand side with those of the righthand side, we conclude that for both firms, the threshold to invest in input 1 if active with input 2 is lower than the threshold to invest in input 2 if active with input 1. This is because in our base parameters we use \(\mu _{I_1 } =\,0.05\) and \(\mu _{I_2 } =\,0.10\) (i.e. the cost of input 2 decreases more rapidly than the cost of input 1). This result supports the conclusion that when the cost of two complementary inputs decreases at different rates it might be optimal to invest sequentially, first, in the input with the cost decreasing less rapidly, a result which is in line with that of Smith (2005).
The above results show that an increase in \(D_{12_L 1_F } \) delays the investment of the follower in input 1 if inactive and has no effect on the investment threshold of the leader. The sensitivity of the investment thresholds of both firms to changes in \(\xi \) are similar to those described in the previous subsections for the SK market (Fig. 8) and therefore similar comments apply.
4 Results
In this section we provide further sensitivity analysis regarding the most relevant model parameters. As for the previous results, we use the following base parameters where for simplicity of notation \(\delta =\mu _{I_1 } \mu _{I_2 } \).
In our model, exante, firms hold three options to invest (in input 1 alone, input 2 alone and inputs 1 and 2 at the same time) whose values are driven by independent (stochastic) underlying variables. Therefore, in order to characterize the market conditions that (at a given time t) imply sequentiallyinput and/or simultaneouslyinput investments for each firm and market structures, we should analyse if the investment thresholds related to each of the above options are crossed. If firms are inactive and the threshold to invest in input 1 (or input 2) alone is crossed before the threshold to invest in the two inputs at the same time, they invest sequentially, starting with input 1 (or input 2). If firms are inactive and the threshold to invest in input 1 and input 2 at the same time is crossed before the threshold to invest in the input 1 (or input 2) alone, they invest in the two inputs at the same time.^{28} Therefore, comparing the threshold values stated in Tables 4 and 5 below with the value of the (respective) underlying variable (\(\phi _{12}(t)=1\), \(\phi _1(t)=2\) or \(\phi _2(t)=2)\) we can identify the market conditions (in terms of \(\xi \) and \(\delta )\) that determine whether the investments in the two inputs are sequential or simultaneous.
This table provides complementary results for the combined effect of changes in \(\xi =\gamma _{12} (\gamma _1 +\gamma _2 )\) and \(\delta =(\mu _1 \mu _2 )\) on the threshold to invest in the two inputs at the same time, for the leader (top panel) and the follower (bottom panel)

This table provides further results on the effect the difference between the cost growth rates of the two inputs on the threshold to invest in input 1 alone, input 2 alone and input 1 and input 2 at the same time
\(\mu _{I_1 }\)  \(\mu _{I_2 } \)  \(\delta =(\mu _1 \mu _2 )\)  Sequentialinput investments  Simultaneousinput investments  

\(\phi _{1_F }^{*\mathrm{SK}} \)  \(\phi _{1_F }^{*\mathrm{PK}} \)  \( \phi _{1_\mathrm{L} }^{*\mathrm{PK} \& \mathrm{SK}} \)  \(\phi _{2_F }^{*\mathrm{SK}} \)  \(\phi _{2_F }^{*\mathrm{PK}} \)  \( \phi _{2_\mathrm{L} }^{*\mathrm{PK} \& \mathrm{SK}} \)  \( \phi _{12_\mathrm{L} }^{*\mathrm{PK} \& \mathrm{SK}} \)  \( \phi _{12_\mathrm{F} }^{*\mathrm{PK} \& \mathrm{SK}} \)  
0.00  \(\,{ 0.05}\)  0.05  2.12  2.44  1.06  3.00  3.75  1.50  0.41  0.82 
0.00  \(\,{ 0.10}\)  0.10  2.12  2.44  1.06  3.94  4.93  1.97  0.48  0.97 
0.00  \(\,{ 0.15}\)  0.15  2.12  2.44  1.06  4.91  6.149  2.46  0.56  1.11 
0.00  \(\,{ 0.20}\)  0.20  2.12  2.44  1.06  5.89  7.36  2.95  0.64  1.28 
0.00  \(\,{ 0.25}\)  0.25  2.12  2.44  1.06  6.88  8.60  3.44  0.72  1.44 
0.00  \(\,{ 0.30}\)  0.30  2.12  2.44  1.06  7.87  9.83  3.93  0.80  1.60 
Wait  Wait  Invest  Wait  Wait  \({ Invest}/\hbox {wait}\)  Invest  \({ Invest}/\hbox {wait}\) 
The above results show that while the leader’s thresholds were crossed for most scenarios, the follower’s thresholds were crossed for few scenarios only, which suggests that simultaneousinput investments are less likely for the follower. We can also see that \(\xi \) and \(\delta \) have opposite effects on the investment thresholds of both firms. An increase in \(\xi \) accelerates the investment and an increase in \(\delta \) delays the investment.
These contrasting effects make the conventional wisdom, which says that “when a production process requires two extremely complementary inputs, a firm should upgrade (or replace) them simultaneously...”, less likely to hold for the follower when input drift differences are large, a result which was highlighted by Smith (2005) for a monopoly market. Notice that the marginal changes used for \(\xi \) and \(\delta \) are of the same size (\(\Delta \xi =\Delta \delta =0.05)\) and yet the threshold values of the diagonals of both matrices decrease slightly as \(\xi \) and \(\delta \) increase (see threshold values underlined), which means that the effect of changes in the former parameter values slightly dominates that of the latter.
Table 5 shows a sensitivity analysis for all the investment thresholds, using \(\mu _{I_1} =0\), and changing \(\mu _{I_2 }\), from \(\,0.05\) to 0.30, ceteris paribus, in order to illustrate more clearly our findings (Fig. 9).
4.1 Further analysis
In most real option models such as those of Decamps et al. (2006), Nishihara (2012) and Siddiqui and Takashima (2012), the value of the exante real options are driven by the same stochastic variable. However, in our model, the value of the exante real options (to invest in input 1 alone, or input 2 alone or input 1 and input 2 at the same time) are driven by three independent stochastic variables (\(\phi _1 (t)\), \(\emptyset _2 (t)\) and \(\phi _{12} (t)\), respectively). While this modelling setting is more realistic, it complicates significantly the characterization of the market conditions that justify sequentialinput or simultaneousinput investments for the two firms because the investment thresholds (related to \(\phi _1 (t)\), \(\emptyset _2 (t)\) and \(\phi _{12} (t)\)) are not directly comparable. In this section we summarize some further relevant insights on the effect of \(\mu _{I_1 } \), \(\mu _{I_2 } \) and \(\delta \) on firms’ investment behavior (Fig. 11).
4.1.1 Input cost growth rates
We start our analysis by the scenarios where the set of (\(\mu _{I_1 } \),\(\mu _{I_2 } \)) points are in the southeast or northwest quadrants, and conclude, respectively: if \(\mu _{I_1 } \downarrow \), with \(\mu _{I_1 } \in ( {\,\infty ,0} ]\), and \(\mu _{I_2 } \uparrow \), with \(\mu _{I_2 } \in [ {0,+\,\infty } )\), \(\phi _{1_L }^{*} \uparrow \), \(\phi _{1_F }^{*} \uparrow \), \(\phi _{2_L }^{*} \downarrow \) and \(\phi _{2_F }^{*} \downarrow \), and sequentialinput investments (starting with input 2) are more likely for both firms. If \(\mu _{I_1 } \uparrow \), with \(\mu _{I_1 } \in [ {0,+\,\infty } )\), and \(\mu _{I_2 } \downarrow \), with \(\mu _{I_2 } \in ( {\,\infty ,0} ]\), \(\phi _{1_L }^{*} \downarrow \), \(\phi _{1_F }^{*} \downarrow \), \(\phi _{2_L }^{*} \uparrow \) and \(\phi _{2_F }^{*} \uparrow \), and sequentialinput investments (starting with input 1) are more likely for both firms. When the set of (\(\mu _{I_1 } \),\(\mu _{I_2 } \)) points are in the northeast or southwest quadrants, we conclude, respectively: if \(\mu _{I_1 } \downarrow \), with \(\mu _{I_1 } \in [ {0,+\,\infty } )\), and \(\mu _{I_2 } \uparrow \), with \(\mu _{I_2 } \in [ {0,+\,\infty } )\), \(\phi _{1_L }^{*} \uparrow \), \(\phi _{1_F }^{*} \uparrow \), \(\phi _{2_L }^{*} \downarrow \) and \(\phi _{2_F }^{*} \downarrow \), and sequentialinput investments (starting with input 2) are more likely for both firms. If \(\mu _{I_1 } \uparrow \), with \(\mu _{I_1 } \in ( {\,\infty ,0} ]\), and \(\mu _{I_2 } \downarrow \), with \(\mu _{I_2 } \in ( {\,\infty ,0} ]\), \(\phi _{1_L }^{*} \downarrow \), \(\phi _{1_F }^{*} \downarrow \), \(\phi _{2_L }^{*} \uparrow \) and \(\phi _{2_F }^{*} \uparrow \), and sequentialinput investments (starting with input 1) are more likely for both firms.
Notice that if the set of (\(\mu _{I_1 } ,\mu _{I_2 } \)) points are in the northwest or southeast quadrants, sequentialinput investments are more likely than if these are in the southwest or northeast quadrants. This is because, in the former cases, \(\mu _{I_1 } \) and \(\mu _{I_2 } \) have different signs which means that the cost of one input is decreasing whereas the cost of the other input is increasing and, therefore, sequentialinput investments, starting with the input whose growth rate is positive, are more likely. Finally, if the set of (\(\mu _{I_1 } ,\mu _{I_2 }\)) points are on the 45 degree dotted line, \(\delta =0\), simultaneousinput investments are more likely regardless of the relative values of \(\mu _{I_1 } \) and \(\mu _{I_2 } \). This is because, in these cases, the cost growth rates of the two inputs are the same (either positive or negative).^{29}
Proposition 5
In nonpreemption duopoly SK and PK markets, ceteris paribus, an increase in the difference between the cost growth rates of the two inputs (\(\delta \)), makes both firms more likely to invest in the two inputs sequentially. Proof: See “Appendix B”.
Corollary 5.1
For an inactive leader (follower), ceteris paribus, if there is a \(+\Delta \mu _{I_1 } \) and a \(\Delta \mu _{I_2 } \) so \(\mu _{I_{12} } \) is kept unchanged, \(\phi _{1_L }^{*} \downarrow \) (\(\phi _{1_F }^{*} \downarrow \)), \(\phi _{2_L }^{*} \uparrow \) (\(\phi _{2_F }^{*} \uparrow \) ) and \(\phi _{12_L }^{*} \) (\(\phi _{12_F }^{*} \) ) is kept unchanged—and sequentialinput investments, starting with input 1, are more likely for both firms. Proof: See “Appendix B”.
Corollary 5.2
For an inactive leader (follower), ceteris paribus, if there is a \(\Delta \mu _{I_1 } \) and a \(+\Delta \mu _{I_2 } \) so \(\mu _{I_{12} } \) is kept unchanged, \(\phi _{1_L }^{*} \uparrow \) (\(\phi _{1_F }^{*} \uparrow \)), \(\phi _{2_L }^{*} \downarrow \) (\(\phi _{2_F }^{*} \downarrow \)) and \(\phi _{12_L }^{*} \) (\(\phi _{12_F }^{*} \)) is kept unchanged—and sequentialinput investments, starting with input 2, are more likely for both firms. Proof: See “Appendix B”.
Corollary 5.3
For an inactive leader (follower), ceteris paribus, if there is a \(+\Delta \mu _{I_1 } \) and a \(\Delta \mu _{I_2 } \) so as there is a \(+\Delta \mu _{I_{12} } \), \(\phi _{1_L }^{*} \downarrow \) (\(\phi _{1_F }^{*} \downarrow \)), \(\phi _{2_L }^{*} \uparrow \) (\(\phi _{2_F }^*\uparrow \)) and \(\phi _{12_L}^{*} \downarrow \) (\(\phi _{12_F }^{*} \downarrow \)), and both sequentialinput investments starting with input 1 and simultaneousinput investments are possible, with the predominant investment behavior dependent on the (exante) relative values of \(\mu _{I_1 } \) and \(\mu _{I_{12} } \) and (exante) how far away \(\phi _1 (t)\) and \(\phi _{12} (t)\) are from \(\phi _{1_L }^{*} (t)\) and \(\phi _{12_L }^{*} (t)\), respectively. Proof: See “Appendix B”.
Notice that, although in general the quadrant where the set of (\(\mu _{I_1 } ,\mu _{I_2 } \)) points is located determine to some extent firms behavior (sequentialinput or simultaneousinput investments), in all quadrants the thresholds of the leader and the follower have different sensitivities to changes in the input cost growth rates, as shown in Sect. 3.
4.1.2 Input complementarity
As shown in Sect. 3, the complementarity between the two inputs plays a very important role in firms’ behavior. Below we discuss with further detail some of the results of Sect. 3.
Proposition 6
For the SK and PK markets, ceteris paribus: (i) an increase in \(\xi \) accelerates the investments of the leader and the follower in the two inputs at the same time and the investments of both firms in input 2 if active with input 1; (ii) for investments in the two inputs at the same time, the sensitivity of the threshold of the follower to changes in \(\xi \) is greater than that of the leader. Proof: See “Appendix B”.
Corollary 6.1
As \(\gamma _{12} \rightarrow 0\): (i) the follower tends to delay forever the investment in input 2 if he is active with input 1; (ii) both firms tend to delay forever the investment in the two inputs at the same time. Proof: See “Appendix B”.
Corollary 6.2
As \(\gamma _{12} \rightarrow 1\): (i) the leader behaves as if she was in a monopoly regarding the investment in the two inputs at the same time; (ii) when the two firms are active with the two inputs and their market shares are symmetric (\(D_{12_L 12_F } =D_{12_F 12_L } =0.5\)), the threshold of the follower to invest in the two inputs at the same time is twice that of the leader. Proof: See “Appendix B”.
Corollary 6.3
(i) As \(\gamma _1 (\gamma _2 )\rightarrow 0\), the leader tends to delay forever her investment in input 1(2); (ii) as \(\gamma _1 (\gamma _2 )\rightarrow 1\) the leader tends to behave as if she was in a monopoly regarding her firststage investment in input 1(2); (iii) an increase in \(\gamma _1 (\gamma _2)\) accelerates the follower’s firststage investment in input 1(2) and delays the follower’s secondstage investment in input 2 (1). Proof: See “Appendix B”.
Proposition 7
(i) For the SK and PK markets where the leader is active with input 1 (input 2), an increase in \(\gamma _{12} \) accelerates the investment in input 2 (input 1); (ii) when the leader is active with input 1 (input 2), the threshold to invest in input 2 (input 1) is more sensitive to changes in \(\gamma _{12}\) if the leader is in a SK market. Proof: See “Appendix B”.
Proposition 8
(i) For the SK and PK markets where the leader is active with input 1 (input 2), an increase in \(\gamma _1 (\gamma _2 )\) accelerates the secondstage investment in input 2 (input 1); (ii) when the leader is active with input 1 (input 2) the threshold to invest in input 2 (input 1) is more sensitive to changes in \(\gamma _1 (\gamma _2 )\) if the leader is in a SK market. Proof: See “Appendix B”.
Proposition 9
(i) For the SK and PK markets where the leader is inactive, an increase in \(\gamma _1 (\gamma _2 )\) accelerates her firststage investment in input 1 (input 2); (ii) if the leader is inactive, her threshold to invest in input 1 (input 2) is more sensitive to changes in \(\gamma _1 (\gamma _2 )\) if in a PK market. Proof: See “Appendix B”.
This table provides information on the investment thresholds of both firms for all the investment scenarios
SK market  PK market  Note  

Sequentialinput investment  Firststage, starting with input 1  Leader:  \(\phi _{1_L }^{*}\)  1.50  1.50  The leader should invest in input 1 alone for both markets, since \(\phi _1(t)\ge \phi _{1_L }^{*} \)—Eq. (54) 
Proposition 4 holds  
Follower:  \(\phi _{1_F }^{*}\)  3.00  3.75  The follower should delay the investment in input 1 alone for both markets, since \(\phi _1 (t)<\phi _{1_F }^{*} \)—Eq. (42) for SK and Eq. (60) for PK  
The follower invests earlier in the first input (input 1) if in a SK market  
Firststage, starting with input 2  Leader:  \(\phi _{2_L }^{*} \)  1,97  1,97  The leader should invest in input 2 alone for both markets, since \(\phi _2(t)\ge \phi _{2_L }^{*} \)—Eq. (54)  
Proposition 4 holds  
Follower:  \(\phi _{2_F }^{*} \)  3.94  4.93  The follower should delay the investment in input 2 alone for both markets, since \(\phi _2(t)<\phi _{2_F }^{*} \)  
The follower invests earlier in the first input (input 2) if in a SK market—Eq. (42) for SK and Eq. (60) for PK  
Secondstage, with input 2  Leader:  \(\phi _{1+2_L }^{*} \)  1.46  1.10  The leader should invest in the second input (input 2) for both markets, since \(\phi _2(t)\ge \phi _{1+2_L }^{*} \)—Eq. (35) for SK and Eq. (67) for PK  
The leader invests earlier in the second input (input 2) if in a PK market  
Follower:  \(\phi _{1+2_F }^{*} \)  1.67  1.67  The follower should invest in the second input (input 2) for both markets, since \(\phi _2(t)\ge \phi _{1+2_L }^{*} \)—Eq. (35)  
Proposition 3 holds  
Secondstage, with input 1  Leader:  \(\phi _{2+1_L }^{*} \)  1.11  0.83  The leader should invest in the second input (input 1) for both markets, since \(\phi _1(t)\ge \phi _{2+1_L }^{*} \)—Eq. (49) for SK and Eq. (67) for PK  
The leader invests earlier in the second input (input 1) if in a PK market  
Follower:  \(\phi _{2+1_F }^{*} \)  1.27  1.27  The follower should invest in the second input (input 1) for both markets, since \(\phi _1(t)\ge \phi _{2+1_F }^{*} \)—Eq. (35)  
Proposition 3 holds 
The results above show that, for both firms and the SK and PK markets, in sequentialinput investments, the thresholds to invest in input 2 if active with input 1 (\(\phi _{1+2_L }^{*} \) and \(\phi _{1+2_F }^{*} \)) are higher than the thresholds to invest in input 1 if active with input 2 (\(\phi _{2+1_L }^{*} \) and \(\phi _{2+1_F }^{*} \)). This occurs because we assume in our base parameters that the cost of input 1 decreases more slowly than the cost of input 2 (\(\mu _{I_1 } =\,0.05\) and \(\mu _{I_1 } =\,0.10\)). Furthermore, in sequentialinput investments, the timing of the firststage investment of the leader is the same for both markets, but the leader invests earlier in the secondstage if in a PK market and the follower invests earlier in the firststage if in a SK market. The above results also support Propositions 3 and 4.
4.1.3 Competition factor
We show that when competition is taken into account, the conventional wisdom described above does not necessarily hold, particularly for the follower. This investment behavior of the follower is more likely when the competition asymmetry (in terms of market shares) between the two firms is high (Fig. 4, righthand side). Thus, we conclude that competition has a negligible effect on the leader’s behavior and affects significantly the timing of the follower’s investment in the two inputs at the same time.
Regarding the firststage of the sequential inputinvestments, we find that the leader’s threshold is not affected by its market share, whereas the follower’s threshold increases with the leader’s market share for both the SK and PK markets (see, respectively, Fig. 6, righthand side, and Fig. 11 at the bottom). With respect to the secondstage of sequential inputinvestments, there is a more diverse set of investment behavior for both firms if in a SK market. The leader delays the investment in the second input as \(D_{1_L 1_F } \) increases, whereas the follower’s threshold is not affected by changes in \(D_{1_L 1_F } \)(see Fig. 8, righthand side). In addition, both firms invest earlier in the secondinput if \(D_{12_L 1_F }\) increases (see Fig. 9, lefthand side), the follower delays the investment in the second input if \(D_{12_L 12_F } \) increases, and the leader’s threshold is not affected by changes in \(D_{12_L 12_F}\) (see Fig. 9, righthand side). Furthermore, we also conclude that:
Proposition 10
For the SK and PK markets where the leader is active with the two inputs, ceteris paribus: (i) if the follower is inactive, an increase in \(D_{12_F 12_L } \), accelerates the investment in the two inputs at the same time; (ii) if the follower is active with input 1(2), an increase in \(D_{1_F 12_L } (D_{2_F 12_L } )\), accelerates the secondstage investment in input 2(1); (iii) if the follower is active with input 1(2), \(D_{12_F 12_L } \), accelerates the secondstage investment in input 2(1). Proof: See “Appendix B”.
5 Conclusions
We study the combined effect of uncertainty and competition on the timing optimization of investments in complementary inputs, for a leader–follower nonpreemption duopoly market where both revenue and input costs are uncertain, and either spilloverknowledge is allowed or proprietaryknowledge holds. We develop a multifactor real option game model where, exante, the two firms hold the option to invest in input 1 alone, input 2 alone, or input 1 and 2 at the same time.
Our results show that, for both firms, the threshold to invest in the two inputs at the same time is significantly affected by the degree of complementarity between the two inputs (Fig. 3, righthand side) and the input cost growth rates (Fig. 4, lefthand side). Also, the leader’s market share when both firms are active with the two inputs has no effect on the timing of the investment of the leader and affects significantly the timing of the investment of the follower (the higher the leader’s market share, the later the follower invests), see Fig. 4, righthand side. Thus, we conclude that when we mix revenue and input costs uncertainty with duopoly competition, the two firms can have very distinct behavior regarding the investment in the two inputs at the same time.
Notice that our model setting is for a nonpreemption game, thus the leader invests at the same time as a monopolist, and gets 100% of the market while active alone, regardless of which input she adopts. Therefore, competition (i.e., how the expost market share is divided between the two firms when both are active) does not affect the leader’s threshold to invest in the two inputs at the same time, but it influences significantly the timing of the follower’s investment. The more asymmetric is the expost market share between the two firms, favoring the leader, the later the follower invests, which implies that asynchronous inputinvestments are more likely.
Smith (2005) shows that input cost uncertainty makes a monopolistic firm less likely to follow the conventional wisdom regarding the investment in two complementary inputs. From our results we conclude that the leader is more prone to behave according to what is suggested by the conventional wisdom, synchronous inputinvestments are more likely, and the follower is more prone to behave according to what is advised for the monopolistic firm of Smith (2005), asynchronous inputinvestments are more likely (see Fig. 13 in the “Appendix C”). However, our findings also show that revenue market share competition reinforces an asynchronous inputinvestment behaviour for the follower, who is more likely than the monopolistic firm of Smith (2005) to invest in the two inputs sequentially.
The above behaviour is somewhat surprising, because in a leader–follower investment game as soon as the leader invests the follower is like a monopolist. Nevertheless, because the follower is exposed to competition, and the more asymmetric is the expost market share between the two firms, favoring the leader, the later the follower invests, which makes synchronous inputinvestments less likely.
We also show that asymmetric changes in the cost growth rates of the two inputs accelerate the investment in the input whose cost growth rate change is positive, deters the investment in the input whose cost growth rate is negative, and may not affect the timing of the investment in the two inputs at the same time (Figs. 4 and 6, lefthand side, and Table 5). Also, for both firms, we find that an increase in the degree of complementarity between the two inputs, enhances the investment in the two inputs at the same time, if the firm is inactive, and the investment in the second input if the firm is active with one input (see, respectively, Fig. 3, righthand side, Fig. 8, lefthand side, and Fig. 11, righthand side). Finally, we conclude that, for simultaneousinput investments, the sensitivity of the follower’s investment threshold to changes in the degree of complementarity is slightly greater than that of the leader (Fig. 3, righthand side), dependent on the input growth rate differentials (Table 4).
By taking advantage of the natural homogeneity of degree one of our investment problem (regarding PDE 11) and the use of other standard real option modelling assumptions, we arrive at analytical solutions for both the firms’ value and the investment thresholds, for several realistic investment scenarios, which make a very complex (threedimensional) multifactor real option game model analysis tractable.
This research can be extended in several ways. For instance, it would be interesting to consider markets where preemption is allowed, or there is a secondmover advantage, or the degree of complementarity between the investment inputs and the expost market shares is stochastic.
Footnotes
 1.
Jovanovic and Stolyarov (2000, p. 13).
 2.
 3.
See Carree et al. (2011).
 4.
See http://www.pvmagazine.com/, 20 September 2013.
 5.
 6.
In R&D contexts, firms who do not have a dominant market position and intend to grow rapidly tend to manage their R&D efforts so as to launch new products which are compatible with those of their rivals who have dominant market positions. Firms who have dominant market positions tend to guide their R&D efforts in order to launch new products that are, as much as possible, not compatible with those of rivals. A practical illustration of the later strategy is, for instance, the nineyear battle between the European Union (EU) commission and Microsoft which culminated in October 2007 with a fine of €497 million due to a supposed misconduct in developing software that does not allow opensource software developers access to interoperability information for workgroup servers used by businesses and other large organizations (see Etro (2007), p. 221, and Financial Times, October 23, 2007, p. 1).
 7.
In our model an idle firm can be inactive or active but operating without the most recent production input(s). For instance, a firm operating with an old rail train with old tracks is idle in not yet adopting highspeed trains and new tracks, if available.
 8.
For simplicity of the notation, henceforth we drop the “t”.
 9.
Suppose that a train operator gets: a 10% reduction in operating costs per passenger if investing in a new train; 10% reduction in operating costs per passage if investing in a new track; and 30% reduction in operating cost per passenger if investing in both a new train and a new track. There is complementarity between the two investments and, within a given output range, savings increase with the sales.
 10.
We assume that the investment cost of the two inputs follows an independent stochastic process (i.e., it is not necessarily equal to the sum of the cost of the two inputs). This is a realistic assumption for some investments since suppliers may offer more favorable prices for higher investment commitments, and there may be different cost savings if firms invest in the two inputs at the same time.
 11.
The rationale for this assumption is that the leader gets higher cost savings due to the effect of complementarity between the two inputs and is able to use the cost savings advantage to earn a higher market share. For the sake of simplicity, we assume that the two inputs are symmetric in terms of cost savings.
 12.
See proof in Sect. 1 of “Appendix A”.
 13.
This analytical simplification leads to the following inputrelated ratios: \(\phi _1 =X/I_1 \), \(\phi _2 =X/I_2 \) and \(\phi _{12} =X/I_{12} \).
 14.
 15.
In (15) the superscripts “F” and “SK” stand for “follower” and “spilloverknowledge”, respectively.
 16.
To save space, in the next subsections we omit this step.
 17.
To save space, in the next subsections we do not show the expressions for the value functions.
 18.
Notice that this term equals the leader’s loss discounted back from the (random) time at which the follower invests in inputs 1 and 2. The term \((\phi _{12} /\phi _{12_F }^{*\mathrm{SK}} )^{\eta _1 }\) is interpreted as a stochastic discount factor which is equal to the present value of $1 received when the variable \(\phi _{12} \) hits \(\phi _{12_F }^{*\mathrm{SK}} \) (see Pawlina and Kort 2006, p. 8).
 19.
To save space, in the next subsections we do not show the expressions for the value functions.
 20.
Notice that the second term of the first row cancels the third term of the second row.
 21.
For a monopoly market, the derivation steps in order to get the investment thresholds to adopt input 1 and input 2 at the same time are the same as those we provide in this section. The only difference is that, in the valuematching and smoothpasting conditions, the competition factor (i.e., the market share) is absent—the threshold is given by \(\emptyset _{12}^{*,SK} =\frac{\eta _1 }{\eta _1 1}\frac{r\mu _{X} }{\gamma _{12}}\).
 22.
Notice that, in sequentialinput investments, the difference between the SK and PK markets is that, in the latter, the leader completes the two investment stages before the follower is allowed to proceed, whereas in the former, the follower is allowed to invest in the first stage (input 1) immediately after the firststage investment of the leader.
 23.
To save space we do not show these illustrative results.
 24.
For a monopoly market, the derivation steps in order to get the investment thresholds to adopt input 2 alone if active with input 1, and input 1 alone if active with inputs 2, are the same as those we provide in this section. The only difference is that in the valuematching and smoothpasting conditions the competition factor (i.e., the market share) is absent—the thresholds are given by: \(\emptyset _{1+2}^{*SK} =\frac{\psi _1 }{\psi _1 1}\frac{r\mu _{X} }{\gamma _{12} \gamma _1 }\) and \(\emptyset _{2+1}^{*,SK} =\frac{\psi _1 }{\psi _1 1}\frac{r\mu _{X} }{\gamma _{12} \gamma _2 },\) respectively.
 25.
Notice that \(\phi _{1+2}^*\) is the threshold which if reached justifies the follower investing in input 2 if active with input 1. Therefore, in the VM condition we replace the \(\phi _2 \) of Eq. (30) by \(\phi _{1+2_F }^{*} \).
 26.
Notice that if firms start with input 2, their threshold expression would be the same, only the notation “1” and “2” changes. To save space we show the derivations for the case where firms start with input 1 only, although illustrative results and sensitivity analyses are provided in Sect. 4 for both cases.
 27.
Notice that one of the differences between the PK and the SK markets is that in the latter the follower optimizes the investment in input 1 with the leader active with input 1 only.
 28.
Notice that if firms are inactive and the thresholds to invest simultaneously in input 1 and input 2 and to invest in input 1 (or input 2) alone are crossed at the same time, firms will invest in the two inputs at the same time.
 29.
Notice that \(\delta =\mu _{I_1 } \mu _{I_2 } \) and, for instance, if: (i) \(\mu _{I_1 } =0.05\) and \(\mu _{I_2 } =0.05\) or \(\mu _{I_1 } =\,0.05\) and \(\mu _{I_2 } =\,0.05\), \(\delta =0\); (ii) if \(\mu _{I_1 } =\,0.05\) and \(\mu _{I_2 } =0.10\), \(\delta =0.05\)—southwest quadrant; (iii) if \(\mu _{I_1 } =\,0.05\) and \(\mu _{I_2 } =0.10\), \(\delta =0.15\)—southwest quadrant; (iv) if \(\mu _{I_1 } =0.05\) and \(\mu _{I_2 } =0.10\), \(\delta =\,0.05\)—northwest quadrant; (v) if \(\mu _{I_1 } =0.05\) and \(\mu _{I_2 } =0.10\), \(\delta =0.15\)—northwest quadrant.
 30.
Empirical proof can be provided under request.
 31.
For simplicity of notation we drop the upper script on \(F_{12} \) and \(f_{12} (\phi _2 )\).
Notes
Acknowledgements
We thank Roger Adkins, Michael Flanagan, Michi Nishihara, Paulo Pereira, Tiago Pinheiro, Helena Pinto, Artur Rodrigues, Miguel Sousa, Mark Tippett, Andrianos Tsekrekos, Bruno Versaevel and the participants at the Real Options Conference 2015, held in Athens and Monemvasia, and at the seminars at Porto Business School, 2014, and Faculty of Economics, University of Porto, 2016, and the two anonymous referees for helpful comments on earlier versions. Alcino Azevedo gratefully acknowledges support from the Fundação Para a Ciência e a Tecnologia.
References
 Azevedo, A., & Paxson, D. (2014). Developing real option game models. European Journal of Operational Research, 237, 909–920.CrossRefGoogle Scholar
 BastianPinto, C., Brandão, L., & Lemos Alves, M. (2010). Valuing the switching flexibility of the ethanolgas fuel car. Annals of Operations Research, 176, 333–348.CrossRefGoogle Scholar
 Carree, M., Lokshin, B., & Belderbos, R. (2011). A note on testing for complementarity and substitutability in the case of multiple practices. Journal of Productivity Analysis, 35, 263–269.CrossRefGoogle Scholar
 ChevalierRoignant, B., Flath, C., Huchzermeier, A., & Trigeorgis, L. (2011). Strategic investment under uncertainty: A synthesis. European Journal of Operational Research, 215, 639–650.CrossRefGoogle Scholar
 Chronopoulos, M., & Siddiqui, A. (2015). When is it better to wait for a new version? Optimal replacement of an emerging technology under uncertainty. Annals of Operations Research, 235, 177–201.CrossRefGoogle Scholar
 Colombo, M., & Mosconi, R. (1995). Complementarity and cumulative learning effects in the early diffusion of multiple technologies. Journal of Industrial Economics, 43, 13–48.CrossRefGoogle Scholar
 Conrad, J. L. (1995). Drive that branch: Samuel Slater, the power loom, and the writing of America’s textile history. Technology and Culture, 36, 1–28.CrossRefGoogle Scholar
 Decamps, J.P., Mariotti, T., & Villeneuve, S. (2006). Irreversible investment in alternative projects. Economic Theory, 28, 425–448.CrossRefGoogle Scholar
 Dixit, A., & Pindyck, R. (1994). Investments under uncertainty. Princeton, NJ: Princeton University Press.Google Scholar
 Etro, F. (2007). Competition, innovation, and antitrust: A theory of market leaders and its policy implication. Berlin: Springer.Google Scholar
 Farzan, F., Mahani, K., Gharieh, K., & Jafari, M. (2015). Microgrid investment under uncertainty: A real option approach using closed form contingent analysis. Annals of Operations Research, 235, 259–276.CrossRefGoogle Scholar
 Femminis, G., & Martini, G. (2011). Irreversible investment and R&D spillovers in a dynamic duopoly. Journal of Economic Dynamics & Control, 35, 1061–1090.CrossRefGoogle Scholar
 Franklin, S. (2015). Investment decisions in mobile telecommunications networks applying real options. Annals of Operations Research, 226, 201–220.CrossRefGoogle Scholar
 Grenadier, S. (1996). The strategic exercise of options: Development cascades and overbuilding in real estate market. Journal of Finance, 51, 1653–1679.CrossRefGoogle Scholar
 Griffiths, T., Hunt, P., & O’Brien, P. (1992). Inventive activity in the British textile industry. Journal of Economic History, 52, 881–906.CrossRefGoogle Scholar
 Hsu, Y.W., & Lambrecht, B. (2007). Preemptive patenting under uncertainty and asymmetric information. Annals of Operations Research, 15, 5–28.CrossRefGoogle Scholar
 Huisman, K. (2001). Technology investment: A game theoretical options approach. Boston: Kluwer.CrossRefGoogle Scholar
 Huisman, K., & Kort, P. (2003). Strategic investment in technological innovations. European Journal of Operational Research, 144, 209–223.CrossRefGoogle Scholar
 Huisman, K., & Kort, P. (2004). Strategic technology adoption taking into account future technological improvements: A real options approach. European Journal of Operational Research, 159, 705–728.CrossRefGoogle Scholar
 Jovanovic, B., & Stolyarov, D. (2000). Optimal adoption of complementary technologies. American Economic Review, 90, 15–29.CrossRefGoogle Scholar
 Koussis, N., Martzoukos, S., & Trigeorgis, L. (2007). Real R&D options with timetolearn and learningbydoing. Annals of Operations Research, 151, 29–55.CrossRefGoogle Scholar
 Leung, C., & Kwok, Y. (2012). Patentinvestment games under asymmetric information. European Journal of Operational Research, 223, 441–451.CrossRefGoogle Scholar
 Martzoukos, S. (2000). Real options with random controls and value of learning. Annals of Operations Research, 99, 305–323.CrossRefGoogle Scholar
 Milgrom, P., & Roberts, J. (1990). Economics of modern manufacturing: Technology, strategy, and organization. American Economic Review, 80, 511–528.Google Scholar
 Milgrom, P., & Roberts, J. (1994). Complementarities and systems: Understanding Japanese economic organization. Stanford: Stanford University. Working Paper.Google Scholar
 Milgrom, P., & Roberts, J. (1995). Complementarities and fit strategy, structure, and organizational change in manufacturing. Journal of Accounting and Economics, 19, 179–208.CrossRefGoogle Scholar
 Moretto, M. (2008). Competition and irreversible investments under uncertainty. Information Economics and Policy, 20, 75–88.CrossRefGoogle Scholar
 Nishihara, M. (2012). Real options with synergies: Static versus dynamic policies. Journal of the Operational Research Society, 63, 107–121.CrossRefGoogle Scholar
 Pawlina, G., & Kort, P. (2006). Real options in an asymmetric duopoly: Who benefits from your competitive advantage? Journal of Economics and Management Strategy, 15, 1–35.CrossRefGoogle Scholar
 Paxson, D., & Pinto, H. (2005). Rivalry under price and quantity uncertainty. Review of Financial Economics, 14, 209–224.CrossRefGoogle Scholar
 Pereira, P., & Rodrigues, A. (2014). Investment decisions in finitelived monopolies. Journal of Economic Dynamics & Control, 46, 219–236.CrossRefGoogle Scholar
 Siddiqui, A., & Takashima, R. (2012). Capacity switching options under rivalry and uncertainty. European Journal of Operational Research, 222, 583–595.CrossRefGoogle Scholar
 Smets, F. (1993). Essays on foreign direct investment. Ph.D. thesis, Yale University.Google Scholar
 Smith, M. (2005). Uncertainty and the adoption of complementary technologies. Industrial and Corporate Change, 14, 639–650.CrossRefGoogle Scholar
 Sydsaeter, K., & Hammond, P. (2006). Mathematics for economic analysis. Englewood Cliffs, NJ: PrenticeHall.Google Scholar
 Thijssen, J. (2010). Preemption in a real option game with a first mover advantage and playerspecific uncertainty. Journal of Economic Theory, 145, 2448–2462.CrossRefGoogle Scholar
 Weeds, H. (2002). Strategic delay in a real options model of R&D competition. Review of Economic Studies, 69, 729–747.CrossRefGoogle Scholar
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