A duopoly preemption game with two alternative stochastic investment choices
Abstract
This paper studies a duopoly investment model with uncertainty. There are two alternative irreversible investments. The first firm to invest gets a monopoly benefit for a specified period of time. The second firm to invest gets information based on what happens with the first investor, as well as cost reduction benefits. We describe the payoff functions for both the leader and follower firm. Then, we present a stochastic control game where the firms can choose when to invest, and hence influence whether they become the leader or the follower. In order to solve this problem, we combine techniques from optimal stopping and game theory. For a specific choice of parametres, we show that no pure symmetric subgame perfect Nash equilibrium exists. However, an asymmetric equilibrium is characterized. In this equilibrium, two disjoint intervals of market demand level give rise to preemptive investment behavior of the firms, while the firms otherwise are more reluctant to be the first mover.
Keywords: Duopoly, stochastic game, preemption, optimal stopping, irreversible investment.
JEL subject classification: L130, L260, C720, C730.
1 Introduction
This paper studies how a twofirm preemptive investment game is affected when the firms involved have two investment opportunities instead of one. We compare this to the classic duopoly investment model, see e.g. Fudenberg and Tirole [7].
The analysis is motivated by Décamps et al [4]. They show that adding another investment choice to a monopoly game gives rise to a dichotomous investment region. That is, the investment region is no longer a connected set, and it is this dichotomy that causes the preemption game to change structure.
We model a duopoly which operates in a market where there is a stochastic, exogenous demand, driven by a geometric Brownian motion (GBM). The firms can at any time carry out a costly investment, where they must choose between two different technologies. The investment is irreversible, and can only be done one time by each firm. The investment reward is a higher profit rate and hence a greater income. However, how much the technologies increase the firm’s profit is uncertain; each investment can either end up in a high profit state or a low profit state. As soon as the leader has invested, the profit state of his investment choice is revealed to both firms. Also, for the follower the investment cost of copying the same technology choice is reduced. Hence, being the second investor has benefits. On the other hand, the first mover will attain a monopoly benefit due to technological advantage until the second mover has invested. We solve this problem by using results by Décamps et al [4] combined with game theoretic techniques based on the analysis of ChevalierRoignant and Trigiorgis [3].
1.1 Literature overview
Duopoly games often have complicated structures where the firms both compete and cooperate. An important example is the case of cooperative research effort, which is studied in Aspremont and Jacquemin [1]. The paper models this cooperation to be active, in the sense that the firms at a R&D stage maximize their joint profit function, and then compete at a "market stage". Several other papers study this idea, but many exclude an active cooperative setting, and instead model the benefits of other firms’ investment by technology spillovers. Technology spillover is the concept that when one firm attains a new technology, it becomes easier for other firms to acquire the same technology, for instance because they can learn from the innovator firm’s research advances. An example of such a paper is Femminis and Martini [6]. In this paper, investment is a "oneshot" action which increases the firms’ profit rate. A certain time after the first firm has invested, the investment cost is reduced for the second firm. Thus, the first firm can be seen as an innovator, while the follower can be seen as an imitator.
Some papers that focus on second mover advantages, due to for instance information benefits, are Hoppe [8] and Y. J. Yap, S. Luckraz and S. K. Tey [17]. A paper with an indepth study of imitation versus innovation is Bessen and Maskin [2]. They consider a setting where the more firms that invest, the bigger the probability of innovating becomes. However, if one firm succeeds, the other firms can copy this. This gives incentives to not invest and instead simply imitate. The paper focuses on social optimality, and compares situations with and without patents. Also, they cover both a static and a sequential investment context. Another relevant paper that examines the choice of doing innovation versus imitation is Jin and Troege [13].
Huisman and Kort [9] analyzes a choice between two investments. However, the setting of this paper is different from ours, as one does not know when the second investment opportunity arrives. Other papers studying investment choice are Décamps et al. [4] and Nishihara and Ohyama [14]. The solution method we apply in this paper is based on the results in Décamps et al. [4]. However, in contrast to Decamps et al. [4], we study a game between two firms and we have an additional uncertainty in the profit rate obtained from the investment. Nishihara and Ohyama [14] studies a game, but without uncertain profit rate.
The papers by Huisman et al. [10] and [11] both analyze stochastic games related to real options models. The paper [11] is quite general, but the firms only have one investment choice. We extend this by adding another technology. They study a preemption example, and we extend this example in a way which results in a more complicated structure of behavior of the firms (contrast Figure 1 in Huisman et al. [11] to our Figure 2). The paper Huisman et al. [10] differs from ours in that they don’t have a stochastic demand, but rather a Poisson process for when new information is revealed to the firms.
1.2 Structure of the paper
The paper is organized as follows. In Section 2, we give a mathematical formulation of the model. Then, Section 3 describes the different payoff functions the firms may attain (depending on the investment outcome). This includes the case of a simultaneous Cournot investment, being the market follower and being the market leader. Section 4 presents a stochastic game in this framework. The game is defined in Section 4.1, based on the approach of Huisman [11]. Then a specific scenario where double preemption intervals occur is analyzed in Section 4.2. We show that no subgame perfect symmetric Nash equilibrium exists in this case, however, we characterize an asymmetric equilibrium. Finally, we conclude in Section 5.
2 The model
Consider a framework with two firms, both facing two different irreversible investment opportunities. We call these investment 1 and investment 2. Both firms may invest at any time in either of the investment opportunities, but not in both. When the first firm has invested, thus becoming the leader, nature chooses one of two outcomes: a high profit outcome or a low profit outcome. This outcome is determined according to some probability measure. Hence, we consider a finite probability space (the subscript stands for “finite”), where , is the corresponding algebra and is a probability measure. Here, is the scenario where investing in alternative 1 leads to a high profit, denoted , while investment 2 leads to low profit, . Similar interpretations hold for and . These probabilities are the same for both firms.
Remark 2.1
The motivation for this problem is to study a model for an investment game involving uncertainty, since this is more realistic than a deterministic model. Only considering two possible outcomes is a simplification which is made to make the computations more manageable. However, similar kinds of arguments can be made for some number of potential outcomes, though this will become chaotic as grows.
The second firm to invest, thus becoming the follower, can choose what to do after the leader has invested. The follower observes the outcome of the leader’s investment. Hence, the follower has the advantage of having more information than the leader.
The profit rates of the two firms vary according to the state of the game. In accordance to what is often done in the literature, we normalize such that both agents get no profit before any investment has occurred. When the first firm has invested, it immediately gets either profit rate (high profit, investment 1), (low profit, investment ), or , depending on whether the firm chooses investment 1 or 2, and on which scenario is realized. Until the second firm invests, the leader also enjoys a monopoly benefit . The follower continues to have profit rate until it invests. If it chooses to copy the technology of the first investor, by for example choosing investment 1, the follower gets the same level of profit as the first investor (either or depending on the outcome of the draw). However, if the second investor chooses not to follow the first investor (i.e. by choosing investment 2), it will get a random profit rate, either or , according to the probability measure given that only or can happen.
To simplify, we assume that the problem is symmetric, i.e. that , and that the probabilities of high or low outcomes, respectively, are the same for both investments. This means that we can remove the technology choice for the first firm, and just look at nature’s choice of high or low profit.
To model the exogenous and uncertain demand for the duopoly’s product, we introduce another probability space , and consider a Brownian motion in this space. Let be the filtration generated by the Brownian motion. The process modeling the uncertainty in demand is a geometric Brownian motion which we denote by , such that
Here, . Also, let be the interest rate. We assume .
For the first firm to invest, we let the cost of any investment be . If the second investor chooses to follow the leader, the investment cost is reduced to , , while the price for the other investment choice is still . The intuition is that it is more expensive to be innovative than to imitate technology that already exists. After both firms have invested, no more choices are made and the profit rates of the investors stay constant.
3 The payoff functions
We want to describe a game where the firms can choose when to invest, so they can influence whether they become the leader or the follower. In order to do this, we must compute the expected payoff functions attained for the different roles. More specifically, we will compute the following functions for all times , where we assume that the stochastic demand process has value at time :

: The payoff of both firms when they invest simultaneously at time , and the firms do not take the roles of leader and follower. Note that the stands for “Cournot” (see also ChevalierRoignant and Trigeorgis [3]). In this case, we assume that neither of the firms get any information advantage or reduced investment cost.

: The payoff of the second investor given that the leader stops at time t and that the follower is acting optimally.

: The payoff of the first investor who stops at time t and hence becomes the leader, given that the other firm will act optimally as the follower.
We divide the computation of these functions into several subsections.
In the following, denotes the expectation with respect to the product measure , also let and denote the expectation w.r.t. the measures and respectively.
3.1 The Cournot case:
Let be the random variable taking the values and with probabilities determined by . Let denote the expectation given that .
In the case where both firms invest exactly at time and the Cournot outcome is attained, their payoff functions are identical, and given by
(1) 
Here, the first and the third equalities follow from Tonelli’s theorem, the second equality from a change of variables and the final equality from that is a geometric Brownian motion.
3.2 The follower’s payoff:
We begin by finding the payoff function of the follower, hence we assume that one of the firms has already invested at time . Based on this we can later determine the payoff of the leader who’s optimal behavior depends on the behavior of the follower.
For the follower, two scenarios can occur: The leader got high profit or low profit . The follower knows which profit level the leader got, and can choose to copy or innovate based on this.
3.2.1 The case where the leader got low profit:
If the leader got the low profit , the payoff function of the follower at the leader’s investment time is:
(2) 
where is the stopping time from the investment time of the leader until the investment time of the follower (that is, is the investment time of the follower).
The first expression in the maximum corresponds to copying the leader, while the second expression corresponds to innovating. The first expression can be rewritten:
where the second equality follows from double expectation, the third equality from the strong Markov property for the Itô diffusion (see also Øksendal [15] exercise 7.11 for justification of the integral term) and the final equality follows from that is a geometric Brownian motion.
A similar computation shows that the second part of the maximum in problem (2) can be written
Hence, problem (2) is equivalent to
(3) 
where , , and . Note that and .
Problem (3) is of the same form as the problems in Decamps et al [4] and Nishihara and Ohyama [14]. As in these papers, we have to consider several cases. The first case is , however since , this never occurs.
Let . The next case to consider is . Then,
(4) 
where and as well as the investment thresholds are determined using value matching and smooth pasting conditions as in Dixit and Pindyck [5] (i.e. requiring continuity and continuous differentiability of the value function ). In particular, we find that
(5) 
Similarly, we can also find and , however we omit writing these out since the expressions are long, and not necessary to know for our purposes. However, we will use the fact that .
The values of this split function correspond to the following actions of the follower (starting at the top): 1) Wait until the demand reaches , then copy the leader. 2) Copy right away. 3) Wait, but make no decision to copy or innovate. This decision depends on whether demand hits (copy) or (innovate) first. 4) Innovate right away.
The optimal stopping time is
The reason for the waiting behavior in the interval is due to the structure of the follower’s payoff function
as shown in equation (3). Since and there is a point where , and at this point we have
where denotes the right and left derivatives respectively. Thus, at the profit gain of a marginal increase in market demand is larger than the profit loss of a marginal decrease in market demand. As a result, there is an open nonempty interval containing where the follower is interested in waiting to see how the market demand evolves. For more on this issue, see the paper by Decamps et al [4].
The final case to consider is . In this case,
(6) 
The optimal stopping time is . In this situation, the follower waits until the demand reaches the level and then innovates.
3.2.2 The case where the leader got high profit:
In the case that the leader got the high profit , the payoff function of the follower at time has a similar form as the above situation:
(7) 
However, since both and are true, it follows that the first term in the maximum expression always is higher than the second. Thus, the payoff function reduces to
(8) 
Here,
is the critical value such that the follower waits for all and copies immediately for (see e.g. Femminis and Martini [6] for solution procedure), where is as before.
Note that , i.e. the demandthreshold for the followerfirm investing when they are guaranteed a high profit is lower than the first investment threshold in the low profit case.
3.3 The leader’s payoff:
Now that we have determined the optimal behavior and payoff function for the follower, we can derive the leader’s optimal value function . This payoff function is the expectation (with respect to the probability measure corresponding to nature’s choice of high or low profit) of the payoff functions in the case where the leader gets high or low profit respectively. We let and be the value functions in the low and high profit cases respectively. Then,
(9) 
Since we have assumed that the rest of the problem is symmetric, we assume that
We derive the two value functions and in the following subsections.
3.3.1 The low profit case:
The leader’s payoff function depends on the behavior of the follower. Hence, we have to take the structure of the function into consideration, see Section 3.2.1. As in Section 3.2.1, we consider two cases.
First, we consider the case where (where the functions are defined as in Section 3.2.1). The payoff function is the payoff of the leading firm when it invests at time , given that the follower then behaves optimally.
By computations similar to those of Section 3.2.1, we can compute (as we have assumed, due to symmetry, that the leader has no investment choice, and the investment time is per definition of ). For example, in the case where , we have
where , . Note that in the first equality, the first integral corresponds to the monopoly benefit, which only lasts until the follower invests (at time ). Note also that the second equality follows by adding and subtracting the monopoly benefit, the third equality follows from a similar argument (using the strong Markov property for Itô diffusions) as in Section 3.2.1 and the fourth equality follows from that is a geometric Brownian motion, so we know the expectation of the first hitting time at the investment level (for the follower) (see also Jeanblanc [12], section 8.1.3). The final equality follows from some basic algebra.
Thus, we have
(10) 
where,
(11) 
Here, using that is a geometric Brownian motion (see Sigman [16])
where , and . By some basic algebra, we see that
The split in the value function of the leader, see equation (10), appears because the leader does not know whether the demand process will hit the innovation level of the follower, , or the copying level of the follower, , first. The splits have similar interpretations as those in Section 3.2.1.
The final case is where . From Section 3.2.1, we know that in this case the followerfirm waits until the demand reaches the level and then innovates. By similar calculations as in the previous case, we find that
(12) 
3.3.2 The high profit case:
Again, the leader’s payoff function depends on the behavior of the follower. We have
The term equals zero if the follower stops immediately after the leader, and equals
for (see Section 3.3.1).
Thus, we have that
(13) 
3.4 Combining the expressions
To summarize, we have that the leader’s ex ante expected payoff is
Similarly, the follower’s ex ante expected payoff is
By analyzing and (as functions of the demand ), we find that is a linear function satisfying . The leader payoff function also satisfies . However, is strictly concave on , then equal to for and . What happens to in the interval depends on the values of the parameters . The follower’s payoff satisfies . It is strictly convex in the interval . Also, it is greater than, but parallel to (and therefore as well) for . What happens in the interval depends on the parameters of the problem. For , is linear with a steeper slope than and it is also larger than . An example of how the three payoff functions may look is shown in Figure 2 below.
4 The stochastic game
In this section, we present a problem where the firms can choose when to invest, hence influencing whether they become the leader or the follower. Our aim is to illustrate that the timing of investment can become peculiar in a duopoly where there are benefits both of being leader (monopoly benefit) and follower (information benefit and reduced investment cost). The game we present is based on games discussed in Huisman et al. [11] and ChevalierRoignant and Trigiorgis [3].
4.1 Defining the game
We consider a duopoly, with the two firms denoted firm 1 and firm 2. For a subgame starting at time , a simple strategy for firm () is defined to be a function such that

is RCLL and right differentiable, and is measurable.

If for , then the right derivative of is positive.
Furthermore, let
and .
The intuition of can be interpreted in two steps. First, firm choosing means that for the given scenario , firm is not at all interested in becoming the leader firm at time . If this holds for both firms in a neighborhood of , then no investment will occur at this time. Second, firm choosing means that firm definitely wants investment to occur at time for the given scenario, and is willing to be the leader in order for this to happen. However, the higher the level of , the higher is the probability that firm actually becomes the leader given that also .
A closed loop strategy for firm is a collection of simple strategies
with the property that

for all such that we have whenever .
The point of the above property is to ensure that the strategies of two different subgames coincide at all times that is a shared first hitting time of a new state for both games. This ensures that strategies are consistent for the different subgames.
The expected discounted value of firm of the subgame starting at time given the simple strategies is
(16) 
Here, and
Finally, if , then
while
(17) 
if and
(18) 
if . The functions and are as in Section 3.
For a given a tuple of simple strategies
is a Nash equilibrium for the subgame starting at time if for
for all simple strategies .
A pair of closed loop strategies is a subgame perfect equilibrium if for all is a Nash equilibrium.
4.2 The case with a high monopoly benefit
We consider the case where , so the function is as in equation (4). Furthermore, we let the monopoly benefit be sufficiently large so that the following conditions hold

there exists such that ,

there exists such that .
This situation will always happen for sufficiently large, since we see from equation (10) that becomes arbitrarily large for and as increases.
Thus, we are in the situation depicted in Figure 2, where the follower and the leader value functions, and , intersect twice. Let and denote the two intervals where the leader’s payoff exceeds the follower’s . Also, recall that and are the investment thresholds of the follower in the low and high profit cases respectively, see Sections 3.2.1 and 3.2.2.
Let be the infimum of , and let similar definitions hold for and . Note that is the smallest level of demand such that is greater than or equal to . We let denote the stopping time for when the process is greater than or equal given we are at time , with similar definitions for and .
Under the assumptions above, we can prove the following result.
Theorem 4.1
There is no pure symmetric subgame perfect Nash equilibrium in this setting.
However, we have the following asymmetric equilibrium. For all , let