Strategic Technology Switching Under Risk Aversion and Uncertainty

Sequential investment opportunities or the presence of a rival typically hasten investment under risk neutrality. By contrast, greater price uncertainty or risk aversion increase the incentive to postpone investment in the absence of competition. We analyse how price and technological uncertainty, reflected in the random arrival of innovations, interact with attitudes towards risk to impact both the optimal technology adoption strategy and the optimal investment policy within each strategy, under a proprietary and a non-proprietary duopoly. Results indicate that technological uncertainty increases the follower's investment incentive and delays the entry of the non-proprietary leader, yet it does not affect the proprietary leader's optimal investment policy. Additionally, we show that technological uncertainty decreases the relative loss in the value of the leader due to the follower's entry, while the corresponding impact of risk aversion is ambiguous. Interestingly, we also find that a higher first-mover advantage with respect to a new technology does not affect the leader's entry, and that technological uncertainty may turn a pre-emption game into a war of attrition, where the second-mover gets the higher payoff.


Introduction
Emerging technologies are subject to frequent upgrades that become available at random points in time and the firm that adopts them first can capture a greater market share (Lieberman & Montgomery, 1988;Zachary et al., 2015). Hence, firms investing in emerging technologies must take into account both strategic interactions and the sequential nature of such investments. Furthermore, emerging technologies typically entail technical risk that cannot be diversified, and, therefore, firms are likely to exhibit risk aversion. Indeed, the underlying commodities of such projects are typically not freely traded, thus preventing the construction of a replicating portfolio. Consequently, riskneutral valuation may not be possible as the assumption of hedging via spanning assets breaks down.
Although various models have been developed in order to analyse sequential investment under price and technological uncertainty, most of these either ignore strategic interactions (Grenadier & Weiss, 1997;Doraszelski, 2001;Chronopoulos & Siddiqui, 2015) or assume risk neutrality (Huisman & Kort, 2003, 2004Weeds, 2002). In this paper, we analyse how strategic interactions impact sequential investment decisions and how price and technological uncertainty interact with risk aversion to impact the optimal investment policy.
Incorporating such features in an analytical framework for sequential investment is crucial as these are pertinent to various industries, e.g., computer software, telecommunications, pharmaceutical, etc. For example, firms producing brand-name drugs enjoy high revenues so long as their patents are protected. In the early 1980s, a drug which soothes both pain and inflammation was a costly patented product. Today, Boots, a British chemist, sells generic tablets for just 2.5 pence per pill (Wall Street Journal, 2013b). In the area of telecommunications, Apple's iPhone sales declined prior to the introduction of iPhone 4s in 2012, while, at the same time, Samsung's Galaxy S 3 , the closest rival to Apple's market leading iPhone, took close to 18% of the market (Financial Times, 2012). The legal debate between Apple and Samsung reflects a highly competitive environment in which firms can potentially profit from adopting other firms' patented technologies. Of course, there are various other competitive advantages that a firm may have, that may not be related to the adoption of patented technologies, however, their analysis is beyond the scope of this paper.
For example, Samsung is more vertically integrated than Apple, and, thus, can bring products to the market more quickly (Wall Street Journal, 2013a).
We consider the case of duopolistic competition, where two identical firms invest sequentially in technological innovations facing price and technological uncertainty. Within this context, we analyse the case of proprietary and non-proprietary duopoly. The former may occur when a firm controls the innovation process, and, therefore, does not face the threat of pre-emption. By contrast, the latter may occur when the innovation process is exogenous to both firms, and, therefore, firms have to fight for the leader's position. Hence, we contribute to the existing literature by first developing a utility-based framework for sequential investment in order to analyse how price and technological uncertainty interact with risk aversion to impact investment under duopolistic competition. Second, we derive analytical expressions, where possible, for the optimal entry threshold of the leader and the follower. Thus, for each firm, we determine both the optimal technology adoption strategy, and, within each strategy, the optimal investment rule. Finally, we provide managerial insights for investment decisions based on analytical and numerical results.
We proceed by discussing some related work in Section 2 and introduce assumptions and notation in Section 3. We begin the analysis with the benchmark case of monopoly in Section 4. In Section 5, we assume that firms adopt each technology that becomes available (compulsive strategy) and analyse the case of proprietary and non-proprietary duopoly in Sections 5.1 and 5.2, respec-tively. In Section 5.3, we also consider how pre-emption may lead to a war of attrition. In Section 6, we assume that a firm may wait for a new technology to become available before deciding to either skip an old technology and invest directly in the new one (leapfrog strategy) or to adopt the old technology first and then the new one (laggard strategy). In Section 7, we provide numerical results for each case and illustrate how attitudes towards risk interact with price and technological uncertainty to impact the optimal technology adoption strategy and the associated investment rule. Section 8 concludes the paper and offers directions for further research.

Related Work
Although traditional real options models often address the problem of optimal investment without considering strategic interactions Siegel, 1985 and1986;He & Pindyck, 1992; Malchow-Møller & Thorsen, 2005), the game-theoretic real options literature has increased considerably over the last years. Yet, models that analyse the impact of strategic interactions on investment decisions typically ignore either the sequential nature of investment opportunities and the different strategies they entail (Pawlina & Kort, 2006;Siddiqui & Takashima, 2012) or attitudes towards risk (Huisman & Kort, 2015). Examples of early work in the area of competition include Spatt & Sterbenz (1985), who analyse how the degree of rivalry impacts the learning process and the decision to invest. Specifically, they find that increasing the number of players hastens investment and that the investment decision resembles the standard NPV rule. Via a deterministic model of duopolistic competition, Fudenberg & Tirole (1985) show that a high first-mover advantage results in a pre-emption equilibrium with dispersed adoption timings, as it increases a firm's incentive to pre-empt investment by its rivals. Also, Smets (1993) first developed a continuous-time model of strategic real options allowing for product market competition, stochastic demand and irreversibility. Extending the framework of Fudenberg & Tirole (1985), Huisman & Kort (1999) find that uncertainty creates a positive option value of waiting that raises the required investment threshold. Specifically, they find that, in deterministic models, a high first-mover advantage leads to a pre-emption equilibrium, yet, in stochastic models, higher uncertainty may turn a pre-emption into a simultaneous investment equilibrium.
More recent examples of models that analyse investment decisions under rivalry and uncertainty include Lambrecht & Perraudin (2003), who incorporate incomplete information into an equilibrium model in which firms invest strategically. Also, Pennings (2004) develops a Stackelberg model for irreversible investment and examines quality choice and entry timing under demand uncertainty.
He finds that, under high demand uncertainty, a leader may choose to invest in a low quality product, while the follower's response is to differentiate herself by supplying a high quality product which has a greater investment cost. Consequently, the follower enters the market later and the leader enjoys an extended period of monopoly profits. A two-factor rivalry model is presented in Paxson & Pinto (2005), who find that an increase in the correlation between profits per unit and quantity of units produced raises their aggregate volatility, and, in turn, the investment trigger of both the leader and the follower. Takashima et al. (2008) assess the effect of competition on the investment decision of firms with asymmetric technologies under price uncertainty. They show how mothballing options facilitate investment, thereby offering a competitive advantage to a thermal power plant over a nuclear power plant. By contrast, lower variable and construction costs favour coal-and oil-thermal power plants.
In a setting including three firms, Bouis et al. (2009) find that if the entry of the third firm is delayed, then the second firm has an incentive to invest earlier so that it can enjoy the duopoly market structure for a longer time. This increases the incentive for the first firm to delay investment, as it faces a shorter period in which it can enjoy monopoly profits. In the same line of work, competitor makes the follower of the two more eager to invest in order to avoid being squeezed out of the market by the hidden competitor. Allowing for capacity sizing as well as entry and exit decisions under duopolistic competition, Lavrutich (2017) finds that the follower can set capacity strategically, so that the leader has an incentive to exit.
Although the aforementioned literature offers crucial insights on strategic investment under uncertainty, these are developed under the assumption of risk neutrality. Yet, the rapid growth of the R&D-based sector of the economy and the associated market incompleteness implies that insights reflecting a risk-neutral setting may not carry over to the risk-averse paradigm or a context of sequential investment opportunities. Examples of analytical models for investment under uncertainty that allow for risk aversion but ignore strategic interactions include Henderson & Hobson (2002), who introduce market incompleteness in the framework of Merton (1969) by allowing for a non-tradable asset and address the question of how to price and hedge this random payoff. Alvarez & Stenbacka (2004) develop a utility-based framework for optimal regime switching utilising a hyperbolic absolute risk aversion (HARA), and show that if the decision-maker is risk seeking, then increasing price uncertainty does not necessarily decelerate investment. A similar result is indicated in Henderson (2007), who shows that idiosyncratic risk raises the incentive to accelerates investment and lock in the investment payoff. In the same line of work, Hugonnier & Morellec (2013) use the framework of Karatzas & Shreve (1999) in order to determine the analytical expression for the expected utility of a perpetual stream of cash flows that follows a geometric Brownian motion (GBM), and find that greater risk aversion lowers the expected utility of a project and reduces the probability of investment. By contrast, Chronopoulos et al. While the literature on investment under risk aversion indicates that attitudes towards risk can have an ambiguous impact on the optimal investment policy, within the context of investment in emerging technologies, the decision-making process may be further complicated by technological uncertainty. Balcer & Lippman (1984) analyse the optimal timing of technology adoption taking into account the expected flow of technological progress, while a model for sequential investment in technological innovations is developed by Grenadier & Weiss (1997). The latter assume that a riskneutral firm may either adopt each technology that becomes available (compulsive), or wait for a new technology to arrive before adopting either the new (leapfrog) or the old technology (laggard), or purchase only an early innovation (buy and hold). They find that a firm may adopt an available technology despite the likely arrival of valuable innovations, while decisions on technology adoption are path dependent. Assuming that innovations follow a Poisson process, Farzin et al. (1998) investigate the impact of technological uncertainty on the optimal timing of technology adoption, yet ignore price uncertainty and attitudes towards risk. Doraszelski (2001) revisits the framework of Farzin et al. (1998) and shows that, compared to the net present value (NPV) approach, a firm will defer technology adoption when it takes the option value of waiting into account. Weeds The former involve a pre-emptive competition where firms invest sequentially and a symmetric outcome in which investment is more delayed than in the case of monopoly. The latter involves sequential investment, yet compared to the non-cooperative (pre-emptive leader-follower) game, the investment triggers are higher. Also, compared to the optimal cooperative investment pattern, investment is found to be more delayed when firms act non-cooperatively as each refrains from investing in the fear of starting a patent race. Miltersen & Schwartz (2004)  where the second mover gets the highest payoff. Alternatively, a follower may benefit from knowledge spillover as in Femminis & Gianmaria (2011). They study a duopoly setting where knowledge spillover reduces the investment cost, and find that even for low levels of spillover, the follower invests as soon as he attains the cost benefit.
More pertinent to our analysis is Siddiqui & Takashima (2012) and Chronopoulos & Lumbreras (2017). The former analyse the extent to which sequential decision making offsets the impact of competition under risk neutrality. They find that a duopoly firm's value relative to a monopolist's decreases (increases) with uncertainty as long as the loss in market share is high (low). Also, they show that this loss in value decreases if a firm adopts a sequential investment approach. The latter develop a utility-based framework that incorporates technological uncertainty via Markovregime switching. Although, they do not consider strategic interactions, they analyse different technology adoption strategies under risk aversion and technological uncertainty. Similar to Siddiqui & Takashima (2012), we consider a spillover-knowledge duopoly in which both firms invest sequentially in technological innovations and the follower can enter the market after the leader.
We develop a duopolistic competition framework and analyse how attitudes towards risk interact with price and technological uncertainty to affect the technology adoption strategy (compulsive, leapfrog, and laggard) of each firm. We assume that technological innovations arrive according to a Poisson process, while price uncertainty is modelled via a geometric Brownian motion (GBM).
Results indicate that technological uncertainty has a non-monotonic impact on the required investment threshold of the follower and the non-proprietary leader, yet it does not impact the proprietary leader's optimal investment policy. Furthermore, we find that the likely arrival of a new technology decreases the leader's relative loss in value due to the presence of a rival, and that increasing risk aversion raises the incentive to delay investment, yet it has an ambiguous impact on the relative loss in the value of the leader. Surprisingly, by comparing a compulsive with a leapfrog/laggard strategy under proprietary duopoly, we find that the latter strategy may dominate even under risk aversion, provided that the rate of innovation and the output price are sufficiently high. Finally, we find that a higher first-mover advantage with respect to a new technology does not affect the leader's entry, and that technological uncertainty may turn a pre-emption game into a war of attrition.

Assumptions and Notation
Given a complete probability space (Ω, F, P), we introduce technological uncertainty by assuming that innovations follow a Poisson process {M t , t ≥ 0}, where t is continuous and denotes time. We assume that the output price {E t , t ≥ 0} is independent of the process {M t , t ≥ 0}, and evolves according to a GBM, as in (1), where µ is the annual growth rate, σ is the annual volatility, and dZ t is the increment of the standard Brownian motion. Also, ρ > µ denotes the subjective discount rate and r is the risk-free rate.
We assume that the firms' risk preferences are described by a specific utility function taken from the HARA class of utility functions, namely a power-function indicated in (2). Note that standard economic theory assumes that decision-makers are typically risk averse and that riskseeking behaviour is less plausible (Pratt, 1964). Nevertheless, we examine the implications of both risk-averse and risk-seeking behaviour to enable comparisons with both Hugonnier & Morellec (2013) and Chronopoulos & Lumbreras (2017).
We let a = p, n denote proprietary and non-proprietary duopoly, respectively, and b = m, , f denote the monopolist, the leader and the follower, where the leader is the first firm to enter the market in the case of competition. Also, we assume that each firm holds perpetual options to invest in two technologies, each with an infinite lifetime. There is no operating cost associated with each technology, while the investment cost is I i , i = 1, 2 (I 1 ≤ I 2 ) and the corresponding output is D i , where D i or D i indicates that there is either one (i) or two (i) firms in the market, respectively.
Hence, D i is decreasing in the number of active firms and increasing in i. Thus, depending on the number of firms in the industry, a firm's option to invest in technology i while operating technology , and the expected utility from operating technology i inclusive of embedded options is denoted by Φ ab i (·). Also, the time and output price at investment are denoted by τ ab i−1,i and E ab i−1,i , respectively, while the optimal investment threshold is denoted by ε ab i−1,i . For example, F n 0,1 (·) is the non-proprietary leader's option to invest in the first technology with a single embedded option to upgrade it by adopting the second one, τ n 0,1 is the time of investment and ε n 0,1 is the optimal investment threshold.
As in Chronopoulos & Siddiqui (2015), we assume that each technological version has greater output than the older one, yet it is more costly. In essence, this implies that there is a trade-off between the two technologies, i.e. the first technology is more lucrative for low output prices, while the second technology is preferred when output prices are high. This condition implies that there is an output price E = α, where the expected NPVs of the profits of the two technologies are equal, In terms of context, a firm may possess an investment opportunity to develop a production facility, and the investment decision is divided in two steps. In step one, the firm develops the energy production facility with an embedded option to increase its utilisation or retrofit it with new technology later. For example, oil production facilities have been converted to utilise gas reserves, but at a substantial cost in order to implement export facilities and retrofitting (Støre et al., 2018). we present the analysis here for ease of exposition and to allow for comparisons. Since U (·) is not separable, the key insight is to decompose all the cash flows of the project into disjoint time intervals. Hence, we assume that the monopolist has initially placed the amount of capital required for investment in a certificate of deposit and earns a risk-free rate, r. Thus, until time τ m 0,1 , the monopolist earns the instantaneous utility U (rI 1 ). At time τ m 0,1 , the monopolist swaps this risk-free cash flow in return for the instantaneous utility U (ED 1 ), as shown in Figure 1. Figure 1: Irreversible investment under monopoly

Benchmark Case: Monopoly
The time-zero expected discounted utility of all the cash flows of the project is described in (3), denotes the expectation operator that is conditional on the initial output price, E.
By decomposing the first integral, we can rewrite (3) as in (4).
Notice that the first term in (4) is deterministic, as it does not depend on the investment threshold.
Therefore, the optimisation objective is reflected in the second term and can be written as in (5) using the law of iterated expectations and the strong Markov property of the GBM. The latter states that the values of the process {E t , t ≥ 0} after time τ m 0,1 are independent of the values of the process before time τ m 0,1 and depend only on the value of the process at time τ m 0,1 . Note that the stochastic discount factor is E E [e −ρτ ] = E Eτ β 1 (Dixit & Pindyck, 1994), β 1 > 0, β 2 < 0 are the roots of the quadratic 1 2 σ 2 β(β − 1) + µβ − ρ = 0, and S is the set of stopping times generated by the filtration of the process {E t , t ≥ 0}.
Using Theorem 9.18 of Karatzas & Shreve (1999), the maximised expected value of the option to invest can be expressed as in (6) where Φ m 1 (·) is the expected utility of the active project and is described in (7).
Solving the unconstrained optimisation problem (6), we obtain the optimal investment threshold that is indicated in (8). Note that, although the investment threshold is commonly expressed in terms of β 1 , it is more expedient to use β 2 in our case, due to the relationship β 1 β 2 = −2ρ / σ 2 .
Also, the second-order sufficiency condition (SOSC) requires the objective function to be concave at ε m 0,1 , which is shown in Chronopoulos & Lumbreras (2017). Note that the analysis of sequential technology adoption for the monopolist is identical to the follower's (see Section 5.1), except for replacing D i by D i . Therefore, we omit this for ease of exposition.
From existing literature (Dixit & Pindyck, 1994 and Hugonnier & Morellec, 2013), we know that, in the benchmark case, increasing price uncertainty and risk aversion delay investment by raising its opportunity cost and decreasing the expected utility of the project, respectively. However, the benchmark model does not allow for sequential investment opportunities that may be subject to technological uncertainty or strategic interactions. Uncertainty over the arrival of innovations raises the incentive to adopt and existing technology (Chronopoulos & Siddiqui, 2015), while the presence of a rival may also induce earlier investment due to the fear of pre-emption. Consequently, these features introduce opposing forces that are not accounted for in the benchmark model and will be addressed in the following sections.

Proprietary Duopoly Follower
We extend the benchmark case of Section 4 by assuming that there are two firms in the market competing in the adoption of technological innovations. First, we consider the optimal investment policy of the follower. As illustrated in Figure 2, the follower is initially in state (0, 1) and holds the option to invest in the first technology, and, thus, move to state 1. Once an innovation takes place, the follower moves to state (1,2), where she has the option to invest in the second technology and move to state 2. We denote a transition due to an innovation (investment) by a dashed (solid) line.
Also, note that the follower will always adopt each technology after the leader. Hence, to alleviate notation, we will indicate the presence of two firms via 1 and 2 only when it is necessary to avoid confusion, i.e. when it is not implied by the superscript. For example, ε f 0,1 reduces to ε f 0,1 since the follower will adopt the first technology after the leader. Similar to the benchmark case, we assume that the amount of capital required for the adoption of each technology is exchanged at investment for the risky cash flows of the project. For example, at time τ f 0,1 the follower exchanges the capital required for investing in the first technology for the risky cash flows of the project. Analogously to (4) and (5), this results in the instantaneous utility U (ED 1 ) − U (rI 1 ), which now accrues from τ f 0,1 until τ f 1,2 , as indicated in Figure 3. Similarly, at τ f 1,2 the follower exchanges the capital required for investing in the second technology for the risky cash flows it generates.
waiting region Figure 3: Sequential investment under a compulsive strategy The follower's objective is to maximise the time-zero discounted expected utility of all the cash flows of the project, which is described in (9). The first (second) integral in (9) indicates the expected utility of the cash flows from operating the first (second) technology.
By decomposing the first integral, we can rewrite (9) as in (10).
Next, we determine the follower's value function in each state using backward induction. Therefore, we first assume that the follower has already adopted and operates the first technology. The expected utility of the project's cash flows is indicated in (11), where the first term is the expected utility from operating the first technology and the second term is the maximised expected value of the embedded option to adopt the second one.
Like in (4), the first term in (11) does not depend on the investment threshold, and, therefore, the optimisation objective is reflected in the second term. The latter, is expressed in (12) as the maximised discounted expected utility from adopting the second technology.
Solving this unconstrained optimisation problem, we obtain the expression of the optimal investment threshold that is indicated in (13) (all proofs can be found in the appendix).
Equivalently, we can express the follower's value function in state (1, 2) as in (14). The first two terms on the top part reflect the expected utility of the cash flows from operating the first technology, while the third term represents the option to adopt the second one. The bottom part represents the expected utility of the profits from operating the second technology, Φ f Next, we step back to state 1, where the follower is operating the first technology and holds an embedded option to invest in the second one, that has yet to become available. The dynamics of the follower's value function are described in (15), where the first term on the right-hand side represents the instantaneous utility of the profits from operating the first technology and the second term is the expected utility of the project in the continuation region. As the second term indicates, with probability λdt the second technology will arrive and the follower will receive the value function, , whereas, with probability 1 − λdt, no innovation will occur and the follower will continue to hold the value function, Φ f 1 (E).
By expanding the right-hand side of (15) using Itô's lemma, we can rewrite (15) as in (16), where Λ = Υ λΥ+1 and δ 1 > 0, δ 2 < 0 are the roots of the quadratic 1 2 σ 2 δ(δ − 1) + µδ − (ρ + λ) = 0. Also, A f 1 > 0 and B f 1 < 0 are determined analytically by applying value-matching and smooth-pasting conditions to the two branches of (16). The first two terms on the top part represent the expected utility of the revenues and cost, respectively. The third term is the option to invest in the second technology, adjusted via the last term since the second technology is not available yet. The first three terms on the bottom part, represent the expected utility of operating the second technology, while the fourth term represents the likelihood of the price dropping into the waiting region prior to the arrival of an innovation.
Finally, the follower's value function in state (0, 1) is indicated in (17). By applying valuematching and smooth-pasting conditions to the two branches of (17), we can solve for the optimal investment threshold, ε f 0,1 , and the endogenous constant, A f 0,1 , numerically.
Note that by setting γ = 1, we can retrieve the same value functions and investment thresholds as Chronopoulos & Siddiqui (2015), who analyse sequential investment under risk neutrality.

Leader
Next, we consider the optimal investment policy of the proprietary leader. Notice that once the where knowledge spillover allows the follower to enter immediately after the leader. Once the follower adopts the first technology, both firms share the market in state 1. The same process is then repeated with respect to the second technology.
Assuming that the follower chooses the optimal investment policy, the value function of the proprietary leader in state 2 is described in (18). The first two terms on the right-hand side reflect the monopoly profits from operating the second technology and the third term is the expected reduction in the proprietary leader's profits due to the follower's entry. The endogenous constant A p 2 is obtained by value-matching (18) with the bottom part of (14), i.e., the follower's value function Φ f 2 (E), at ε f 1,2 , and is indicated in (A-5).
Next, the value function of the proprietary leader in state 1, 2 is described in (19). The first two terms on the top part reflect the expected utility of the cash flows from operating the first technology, and the third term is the embedded option to invest in the second one.
The endogenous constant, A p 1,2 , and the optimal investment threshold, ε p 1,2 , can be obtained analytically via value-matching and smooth-pasting conditions and are indicated in (20).
Corollary 1 indicates the necessary condition for a trade-off to exist between the two technologies, and is a consequence of the assumption about the second technology being more efficiency yet also more costly in Section 3. Note that, by setting γ = 1, we can retrieve the condition under risk neutrality, as in Chronopoulos & Siddiqui (2015),

Corollary 1. A trade-off between the two technologies exists iff
Using Corollary 1, we can show that the proprietary leader will not invest in the second technology before the follower adopts the first one, as indicated in Proposition 1. Intuitively, the second technology is considerably more costly and will not be adopted when the output price is below the follower's required investment threshold for the first technology.
Interestingly, unlike Chronopoulos et al. (2014), the leader's required investment threshold in the second technology is lower than that of the monopolist, as shown in Proposition 2. Intuitively, the entry of the follower reduces the monopoly profits of the leader with respect to the first technology.
In turn, this raises the value of the leader's option to invest in the second technology and lowers the required adoption threshold, thereby extending the corresponding period of monopoly profits.
In state 1, the leader shares the market with the follower waiting for the arrival of the second technology. The dynamics of the value function of the leader are described in (21), where the first term on the right-hand side reflects the instantaneous profit from operating the first technology.
The second term is the discounted expected value in the continuation region, where the proprietary leader gets either F p 1,2 (E) or Φ p 1 (E), depending on whether an innovation occurs or not.
The proprietary leader's value function in state 1 is indicated in (22), where A p 1 and C p 1 are determined by value matching and smooth pasting the two branches, while B p 1 is obtained by value matching (22) with the top branch of (19) at ε f 1,2 . The first two (three) terms in the top (bottom) part of (22) reflect the expected utility of the profits under a low (high) output price. The third term on the top part is the option to invest in the second technology adjusted via the fourth term due to technological uncertainty. The fourth term on the bottom part reflects the reduction in the expected utility of the leader's profits due to the follower's entry adjusted for technological uncertainty via the fifth term. The last term reflects the likelihood of the price dropping in the waiting region.
The value function of the proprietary leader in state 1 is indicated in (23), where A p 1 < 0 is obtained by value matching (23) with the top branch in (22) at ε f 0,1 and is described in (A-6). The first two terms in (23) reflect the expected utility from operating the first technology, and, the third term, is the expected reduction in the proprietary leader's profits due to the follower's entry.
In state (0, 1), the proprietary leader holds the option to invest in the first technology with an embedded option to invest in the second one, that has yet to become available. The expression of F p 0,1 (E) is described in (24), where the top part is the value of the option to invest and the bottom part is the expected utility of the active project inclusive of the embedded option to invest in the second technology.
The expression of ε p 0,1 and A p 0,1 is indicated in (25). Notice that, as shown in Proposition 3, the leader's decision to adopt the first technology is independent of technological uncertainty.
Proposition 3. The proprietary leader's required investment threshold for the first technology is independent of λ.

Non-Proprietary Duopoly
With two firms in the market fighting for the leader's position, each one of them faces the risk of pre-emption. Note that, under a compulsive strategy, the follower will invest in each technology after the leader has already adopted it. Consequently, the value function of the follower in each state is the same as in Section 5.1. However, to determine the non-proprietary leader's optimal investment policy, starting with the second technology, we must consider the strategic interactions between the leader and the follower. Note that the leader's value function in state 2 is described in (18), i.e., Φ n 2 (E) ≡ Φ p 2 (E). We let ε n 1,2 denote the point of intersection between the value function of the leader and the follower. If E < ε n 1,2 , then a firm is better off being the follower, since F f 1,2 (E) > Φ n 2 (E). By contrast, if E > ε n 1,2 , then a firm is better off being a leader, since F f 1,2 (E) < Φ n 2 (E). Consequently, the point of indifference between being a leader and a follower is ε n 1,2 and is determined numerically by solving (26).
Note that there are two possible scenarios: i. ε f 0,1 > ε n 1,2 and ii. ε f 0,1 < ε n 1,2 . In the former scenario, the follower invests in the first technology after the leader can pre-empt the second one. This implies that the leader does not face the risk of pre-emption, since the follower is assumed here to adopt a compulsive strategy, and, therefore, will not skip the first technology. In the latter scenario, the follower adopts the first technology before the leader can pre-empt the second one. This implies that the leader faces the threat of pre-emption. Consequently, the leader's optimal investment threshold in the second technology is max ε f 0,1 , ε n 1,2 , as shown in Proposition 4. Intuitively, although the leader can pre-empt the second technology at ε n 1,2 , she may choose to delay adoption until the follower's entry at ε f 0,1 , provided that ε f 0,1 > ε n 1,2 . Doing so, the leader captures the same value function, albeit at a higher threshold, closer to the utility-maximising one. Proposition 4. The optimal investment threshold of the non-proprietary leader for the second technology is max ε f 0,1 , ε n 1,2 , where ε n 1,2 satisfies the condition F f 1,2 (E) = Φ n 2 (E).
Next, we step back prior to the arrival of the second technology and assume that the firm that pre-empts the first technology is better placed to also adopt the second technology, if the follower has not entered the market. The non-proprietary leader's value function is indicated in (27). The first two terms reflect the expected utility of the monopoly profits from operating the first technology and the third term is the reduction in expected utility due to the entry of the follower. Note that the latter depends on whether ε f 0,1 > ε n 1,2 or ε f 0,1 < ε n 1,2 .
Note that the last term on the right-hand side of (27) depends on whether the follower invests in the first technology before ε f 0,1 < ε n 1,2 or after ε f 0,1 > ε n 1,2 the leader can pre-empt the second one. In each case, the amount of reduction in the value of the leader due to the entry of the follower is different. If ε f 0,1 < ε n 1,2 , then the leader does not have an advantage regarding the second technology, since the technology will be adopted at the indifference point between being a leader and a follower. Consequently, A n 1 is determined by value matching (27) with (16). However, if ε f 0,1 ≥ ε n 1,2 , then upon the follower's entry the leader will receive the reduced value from operating the first technology with the expected value from pre-empting the second one. In this case, A n 1 is obtained by value-matching (27) with the value function whose dynamics are described in (28).
The first term indicates the instantaneous utility of leader's reduced profits due to follower's entry and the second term is the expected value of the profits in the continuation region. Note that with probability λdt the second technology will become available and the leader will get to pre-empt it, whereas with probability 1 − λdt the leader will continue to operate the first technology. Expanding (28) using Itô's lemma and solving the resulting ordinary differential equation yields (A-7).
Following the same reasoning as in (26), the leader's pre-emption threshold in the first technology, ε n 0,1 , is determined by solving (29).

War of Attrition
Due to the competitive advantage created by ignoring the first technology, and, therefore not incurring the associated cost, a firm may choose to invest in the second one directly. Here, we consider how pre-emption of the first technology by a rival motivates a firm to adopt the second technology directly and ignore a compulsive strategy. Note that the difference in investment strategies prevents a comparison between the two firms in a way similar to that of Section 5.2. Like Takashima et al.
(2008), we take the perspective of each firm separately and analyse their value functions assuming initially that it is possible for each firm to assume both roles, i.e., leader and follower. Then, we conclude which role is feasible for each firm. Since we have already determined the pre-emption threshold for the second technology under a compulsive strategy in (26), we only need to determine the pre-emption threshold when the first technology is ignored.
We denote as follower the firm that gets pre-empted in the adoption of the first technology, and, therefore, may have a greater incentive to adopt the second one directly. The follower's value function when investing in the second technology directly is described in (30). The top part is the value of the option to invest and the bottom part is the expected utility of the active project.
The expression of A f 0,2 and ε f 0,2 is obtained through value-matching and smooth-pasting conditions and is indicated in (31).
The corresponding leader's value function is denoted by Φ n 2 (·) and is described in (32). Note that A n 2 is determined by value matching (32) with the bottom part of (30) at ε f 0,2 . The pre-emptive leader's threshold, ε n 0,2 , satisfies the condition Consequently, skipping the first technology is a feasible strategy provided that ε n 0,2 < ε n 1,2 . Intuitively, the follower in the first technology can invest in the second technology first provided the pre-emption threshold of the compulsive leader is greater than the threshold of directly adopting the second technology. The feasibility of skipping the first technology and adopting the second one directly can be analysed by comparing the relative value of the two strategies, i.e., Φ n 2 (E)/F f 0,1 (E).

Leapfrog and Laggard Strategy
Assuming that the leader has proprietary rights on each technology, she may decide to ignore a technology temporarily in order to wait for a new one to arrive before deciding which one to invest in. If the leader ignores the first technology, then only the second one will be commercialised, and, therefore, the follower's value function is indicated in (30). Given the follower's optimal response, the proprietary leader can choose whether to adopt a leapfrog or a laggard strategy as illustrated in Figure 5. Instead of moving from (0, 1) to 1, the leader moves to state (0, 1 ∨ 2), and then, either invests in the first technology, holding the option to switch to the second one, i.e., state (1, 2), or (∨) invests directly in the second technology, thereby moving to state 2. Notice that the value function of the proprietary leader in states 1, 2 , 2, and 2 following a laggard strategy is the same as in Section 5.1, while her value function in state 2 following a leapfrog strategy is indicated in (32). Hence, we proceed directly to state (1,2), where the leader operates the first technology and earns monopoly profits until the follower enters at ε f 0,1 . The leader's value function is described in (33), where the third term on the right-hand side reflects the expected reduction in the leader's profits due to the follower's entry. The endogenous constant, A p 1,2 , is obtained by value matching (33) with the bottom part of (19) at ε f 0,1 .
Due to the presence of the second technology, there exist two waiting regions: i. E ≤ ε p 0,1 and ii. ε p 0,1 ≤ E ≤ ε p 0,2 (Décamps et al., 2006). Hence, the value function in state (0, 1 ∨ 2) is described in (34), where B p 0,1∨2 , C p 0,1∨2 , ε p 0,1 , and ε p 0,2 are obtained numerically via value-matching and smooth-pasting conditions between the bottom three branches, and Φ n 2 (E) is indicated in (32). Notice that, if E < ε p 0,1 , then the firm will wait until E = ε p 0,1 and then adopt the first technology. By contrast, if ε p 0,1 ≤ E ≤ ε p 0,2 , then the firm will either invest directly in the second technology if E ↑ ε p 0,2 , or it will invest in the first one and hold the option to switch to the second if E ↓ ε p 0,1 .
Finally, in state (0, 1) either the second technology will become available with probability λdt and the proprietary leader will receive the value function F p 0,1∨2 (E), or no innovation will take place with probability 1 − λdt and the leader will continue to hold the value function F p 0,1 (E).
The expression of the value function in state (0, 1) is indicated in (36), where D p 1 , G p 1 , H p 1 , J p 1 , K p 1 , L p 1 , and M p 1 are determined numerically via the value-matching and smooth-pasting conditions between the branches of (36). Notice that (36) has five branches, since the value function Φ n 2 (E) changes to the bottom branch of (30) when the follower enters the market.

Numerical Results
Proprietary duopoly with compulsive firms    The right panel of Figure 8 illustrates the impact of λ and γ on the required investment threshold of the non-proprietary leader for σ = 0.18, 0.20. Notice that, although the impact of γ and σ is the same as in Figure 7, greater λ induces later adoption for the leader. Intuitively, this happens because earlier entry of the follower, as illustrated in the right panel of Figure 7, reduces the period of monopoly profits for the non-proprietary leader, thereby decreasing the attractiveness of the first technology. Also, as the left panel illustrates, increasing the first-mover advantage raises the investment incentive and lowers the required entry threshold of the non-proprietary leader.  both panels illustrate, the leader's required investment threshold in the first technology is not affected by the first-mover advantage in the second one. More specifically, since the follower's entry threshold is not affected by changes in D 1 , the period of monopoly profits for the leader in the first technology is unchanged and so is the leader's optimal adoption threshold. By contrast, a greater first-mover advantage in the first technology accelerates investment, while the threat of pre-emption increases the investment incentive, as illustrated in the right panel.
In order to calculate the leader's relative loss in value, we use the follower's analysis from Section 5.1 to find the monopolist's option value under sequential investment. The impact of γ and σ on the relative loss in the value of the proprietary and non-proprietary leader is indicated in the leftand the right-hand side expression of (37), respectively, and is illustrated in Figure 10.
In line with Siddiqui & Takashima (2012) and Chronopoulos et al. (2014), the left panel in Figure   10 indicates that the relative loss in the value of the proprietary leader increases (decreases) with greater price uncertainty when the first-mover advantage is high (low). Intuitively, this happens because, under low discrepancy in market share, the increase in the proprietary leader's value of investment opportunity due to the follower's late investment is greater than the expected loss due to the entry of the follower. However, when the discrepancy is high the period of time with monopoly profits in the second technology is more pronounced causing the relative loss to increase. Also, as the right panel illustrates, greater price uncertainty and a lower first-mover advantage decreases the relative loss in value for the non-proprietary leader.
The impact of γ and λ on the relative loss in value for the proprietary (left panel) and nonproprietary leader (right panel) is illustrated in Figure 11. As both panels illustrate, a higher  innovation rate lowers the relative loss in the value of the leader by raising the expected utility of the embedded option to adopt a more efficient technology. Interestingly, risk aversion has an ambiguous effect on the relative loss in the value of the proprietary leader (left panel). More specifically, under a low (high) rate of technological innovation, greater risk aversion decreases (increases) the relative loss in the value of the leader. This happens because greater risk aversion postpones the entry of the follower, thereby allowing the leader to enjoy monopoly profits for a longer time. However, when λ is high, the second technology is more likely to become available, which in turn, gives the leader greater incentive to invest than the monopolist, as shown in Proposition 2. Consequently, the impact of greater risk aversion is mitigated by higher technological uncertainty.

War of attrition.
The top panel in Figure 12 indicates that, for λ = 0.1, γ = 0.9 and σ = 0.2, the pre-emption threshold for the second technology when the first one has already been adopted is 14.58. Yet, the bottom panel illustrates that direct pre-emption of the second technology, without adopting the first one, requires a threshold of 8.31. Consequently, the competitive advantage from ignoring the first technology can facilitate the direct pre-emption of the second one. However, although skipping the first technology in order to pre-empt the second may be feasible, it may still be optimal to proceed with a compulsive strategy. Figure 13 illustrates the relative value of i. pre-empting the second technology directly and ii. adopting compulsive strategy under a low (left panel) and a high (right panel) output price. The relative value of these two strategies is described in (38). In this comparison, we ignore technological uncertainty by assuming that both technologies are available.
Note that if the output price is low, then it is always better to be a compulsive follower as the left panel of illustrates. However, under a high output price (right panel), increasing price uncertainty makes it optimal to skip the first technology in order to pre-empt the second one. Interestingly, lower risk aversion also increases the relative value of pre-empting the second technology directly.
In fact, even under risk neutrality (γ = 1), it is optimal to ignore the first technology and pre-empt the second one directly, provided that price uncertainty is adequately high.
Proprietary duopoly under a leapfrog/laggard strategy.
The left panel in Figure 14 illustrates Output Price, E 8.31 the proprietary leader. More specifically, the left panel illustrates the top branch in (39), which compares the first branch of (36) with (23) when the output price is low, i.e., E < ε p 0,1 . Similarly, the right panel illustrates the expression in the bottom-branch, which compares the bottom part of (36) with the top part of (22) when the output price is high, i.e., E = ε p 0,2 .
As the right panel illustrates, the compulsive strategy dominates when the output price is low. This happens because a firm must wait longer to invest in the more capital intensive technology and the associated payoff does not offset the foregone revenues from ignoring the existing one. In fact, greater risk aversion promotes the adoption of a compulsive strategy and makes the leapfrog/laggard strategy relative less attractive. Interestingly, unlike Chronopoulos & Siddiqui (2015), the same result holds even at a high output price, as long as the discrepancy in market share is large.
However, as the right panel illustrates, a leapfrog strategy may dominate, under a high output price and a low discrepancy in market share.

Conclusions
We analyse how attitudes towards risk interact with price and technological uncertainty to impact sequential investment decisions of firms within the context of duopolistic competition. The analysis is motivated by four main features of the modern economic environment: i. increasing competition due to the deregulation of many sectors of the economy, such as energy and telecommunications; ii. market incompleteness and associated attitudes towards idiosyncratic risk; iii. the sequential nature of investment decisions in emerging technologies, e.g., energy, and the R&D-based sector of the economy; and iv. technological uncertainty. We incorporate these features into a utilitybased framework for investment under uncertainty by assuming that two identical firms compete in the sequential adoption of technological innovations. More specifically, we assume that the firms compete in the adoption of two technologies, of which the first one is available, while the arrival of the second, more efficient one is subject to technological uncertainty.

Leader
In a state 2, the value function of the leader described in (18) will value match with the bottom part of the leader's value function (14), because for E ≥ ε f 1,2 the two firms will share the market. Hence, A p 2 is described in (A-5).
By contrast, in state 1, A p 1 is obtained by value matching (23) with the top branch in (22) at ε f 0,1 . Hence, the endogenous constant A p 1 is indicated in (A-6).
If the follower in the first technology chooses to ignore it in order to adopt the second one directly, then her value function is obtained by expanding (28)   The optimal investment rule is found by applying the first-order necessary conditions to (A- 14) with respect to E p 0,1 and is outlined in (A-15), where the marginal benefit (MB) of delaying the investment is equal to the marginal cost (MC). The first term on the left-hand side reflects the extra benefit from allowing the project to start at a higher price threshold and the second term is the increase in MB form postponing the investment cost. Similarly, the first term on the right-hand side represents the opportunity cost of forgone cash flows. The third term on the left-hand side represents the MB of postponing the loss in value due to the follower's entry, while the second term on the right-hand side is the MC from waiting, and, thus, incurring a greater loss in value when the follower enters. These opposing forces cancel each other, because the follower will enter before the second technology arrives. Consequently, the leader's investment threshold in the first technology does not impact her possible monopoly profits in the second technology, and, thus, the leader adopts a myopic investment strategy.

Proposition 4
The optimal investment threshold of the non-proprietary leader for the first technology is max ε f 0,1 , ε n 1,2 , where ε n 1,2 satisfies the condition F f 1,2 (E) = Φ n 2 (E). Proof: Ideally, the leader would invest at the threshold that maximises her expected utility, i.e., at ε p 1,2 . However, the threat of pre-emption lowers the adoption threshold to ε n 1,2 . The price threshold at which the firm is indifferent between being the leader or the follower is defined implicitly through the equality F f 1,2 (E) = Φ n 2 (E). Given that the follower adopts a compulsive strategy, there are two possible scenarios: i. ε f 0,1 > ε n 1,2 ii. ε f 0,1 < ε n