R&D Incentives with Uncertain Probability of Success

Abstract

A firm’s decision to invest in R&D depends on a number of factors such as the availability of funds, extent of R&D spillovers, market structure and success probability. However, the probability of success depends, to a large extent, on factors endogenous to a firm. This means that the success probability can be known to the firm undertaking R&D investment, but not to the rivals; hence, there is incomplete information about probability of success in R&D. There are also uncertainties about the rivals’ R&D decision and R&D status. In a duopoly, we show that there is a non-monotone relation between R&D incentives and the level of information.

JEL Classifications: D43, D82, LI3, O3I

Keywords

R&D incentives duopoly incomplete information type distribution

Introduction

The role of incomplete information as a decisive factor behind R&D investment is a relatively less explored area in R&D literature. Some of the known works in this direction include Grishagin et al. (2001), Brocas (2004), Bacchiega and Garella (2008), Conti (2013), Kabiraj and Chattopadhyay (2015) and Chattopadhyay and Kabiraj (2015). Among these, Brocas (2004), Kabiraj and Chattopadhyay (2015) and Chattopadhyay and Kabiraj (2015) discuss the role of incomplete information in choosing the R&D organization, that is, whether to perform R&D collaboratively or single-handedly. Conti (2013) discusses the same issue in the presence of spillovers. Bacchiega and Garella (2008) talk about a firm’s choice between withholding as opposed to disclosing R&D information. Grishagin et al. (2001) consider a context where firms involve in a patent race, and do not know each other’s relative position in terms of R&D - a firm decides to perform R&D only if it initially starts in the same position as its rival; the set-up deals more with imperfect information rather than incomplete information.

Two recent papers by Chatterjee et al. (2018, 2019) have directly addressed the issue of how incomplete information may affect the R&D decision. Both these papers consider a Cournot duopoly where the firms decide whether to invest in R&D prior to competition in the product market. The firms always perform R&D non-cooperatively, and there is no uncertainty in R&D outcomes. In Chatterjee et al. (2019), incomplete information arises due to unobservability in the reduction of marginal cost of the rival in the absence of spillover. Chatterjee et al. (2018) have, on the other hand, analysed the problem where spillovers are present, but the extent to which a firm can benefit from spillover of its rival’s R&D outcomes is private information. Both these papers conclude that incomplete information may enhance, under some parametric situations, the incentive to invest in R&D activities compared to the case of complete information.

The present paper seeks to extend the aforementioned analysis of incomplete information to another direction. Now, we consider not only that the R&D outcome is uncertain, and hence probabilistic, but also that the firms have incomplete information about the probability of success of the rival. Therefore, the probability of success $p_{i} \in [0, 1]$ in R&D is private information to the concerned firm; that means, the probability of success of R&D for a particular firm constitutes its type, having a probability distribution function F(.) and density function f(.) with full support. We restrict to the scenario where innovation, if successful, is non-drastic, and therefore in the post-innovation situation, all firms will operate at a positive output level. Thus the purpose of the paper is to study how R&D incentives of firms get affected when the probability of success is private information, which has not been studied so far.

One interesting result of this paper is the non-monotonicity of R&D incentive with respect to availability of information about the R&D characteristics of the rival firm, resulting from a signalling effect.

The organisation of the paper is as follows: The section ‘Model SetUp’ specifies the model set-up; ‘Complete Information: A Benchmark Case’ briefly illustrates the complete information benchmark case; ‘Incomplete Information’ elaborates on the three alternative incomplete information structures, namely Level I, Level II and Level III; and the section ‘Comparison’ compares the different information structures and derives the important results. Finally, the section ‘Conclusion’ concludes the paper.

Model Set-Up

Consider two firms, A and B. They compete in quantities in the product market. The market price of the product is given by $P = max \{0, a - Q\}$ , where $a > 0$ is the demand parameter and $Q$ is the aggregate output produced in the market. The marginal cost of each firm is c. The firms can invest in a costreducing R&D. The cost of research, M>0, is the same for both firms. If a firm invests in R&D and is successful, its marginal cost will be $(c - D)$ ; $c > D > 0$ . The probability of success in $R & D$ for firm $i$ is $p_{i} \in [0, 1]$ . We assume that $p_{i}$ is private information and it is distributed according to distribution function $F (\cdot)$ and density function $f (\cdot)$ with full support. For technical reasons we assume $a > c + 3 D$ .¹

Consider the following notations: $K : = a - c, q (x) : = \frac{K + x}{3}, \prod (x) : = q^{2} (x)$ and $Θ (x) : = \int_{x}^{1} y \frac{d F (y)}{1 - F (x)} .$ .² It is easy to check that $q^{'} > 0, \prod^{'} = \frac{2}{3} q (x) > 0$ , $Θ^{'} (x) > 0$ when $x \in (0, 1)$ and ${lim}_{x \to 1} Θ (x) = 1.$ .³

We further denote ‘doing research’ by $R$ , ‘not doing research’ by $N$ , ’success’ in R&D by $S$ and ‘failure’ by $F$ . When firm $A$ chooses to invest in R&D and succeeds and firm $B$ does not invest in R&D, we denote profit (or expected profit) of firm $A$ by $Π_{A}^{[S N]} (E Π_{A}^{[S N]})$ and that of firm $B$ by $Π_{B}^{[S N]} (E Π_{B}^{[S N]})$ . Similar notations will be used for other cases.

We consider three pieces of information that will affect the decision to invest in R&D of a firm:

The type of the rival, that is, whether the rival’s type is known to the firm.

The decision of the rival to invest, that is, whether it is observable that the rival has invested or not.

The status of the rival’s research, that is, whether the rival has come up with success or failure in R&D when it has invested in R&D.

Our objective is to find out how the decision to perform R&D depends on the type of a firm and the level of information available to it. We construct a two-stage game. In the first stage, all firms decide simultaneously whether to invest in research or not, and in the second stage they compete in the product market a la Cournot. To facilitate our analysis we first provide the result for the complete information model.

Complete Information: A Benchmark Case

We assume in this section that everything is common knowledge, including the types of the firms. Since we are considering duopoly, at equilibrium, several cases can happen: SS, SF, SN, FS, FF, FN, NS, NF and NN.⁴ The lemma given below further summarizes the payoffs of a firm under different equilibrium situations.

Lemma 1. The following holds for firm A:

\begin{array}{l} Π_{A}^{[N N]} = Π_{A}^{[N F]} = Π (0), & Π_{A}^{[F F]} = Π_{A}^{[F N]} = Π (0) - M, \\ Π_{A}^{[S S]} = Π (D) - M, & Π_{A}^{[S N]} = Π_{A}^{[S F]} = Π (2 D) - M, \\ Π_{A}^{[F S]} = Π (- D) - M, & Π_{A}^{[N S]} = Π^{[S S} (- D) . \end{array}

It holds similarly for firm B.

Note that, when a firm is going to invest in R&D, it does not know whether it will succeed or not.

When firm i does not invest in R&D but its rival does, then its expected profit is $p_{j} Π (- D) + (1 - p_{j}) Π (0)$ . When neither of them invests, its expected profit is $Π (0)$ .

Similarly, the expected profit of firm i when both firms invest is

p_{j} [p_{i} Π (D) + (1 - p_{i}) Π (- D)] + (1 - p_{j}) [p_{i} Π (2 D) + (1 - p_{i}) Π (0)] - M .

But if firm i alone invests, its expected profit is $p_{i} Π (2 D) + (1 - p_{i}) Π (0) - M$ .

Therefore when the rival is not doing $R & D$ , the expected ‘gain’ from doing $R & D$ of firm i is $p_{i} [Π (2 D) - Π (0)] - M$ , and when the rival is doing research, the expected "gain" of doing $R & D$ for firm i is $p_{i} [p_{j} (Π (D) - Π (- D)) + (1 - p_{j}) (Π (2 D) - Π (0))] - M$ .

When a firm does not invest in $R & D$ , its type is taken as zero. Define $U (p_{i}, p_{j})$ for all $i, j = \{A, B\}$ and $i \neq j$ , as

\begin{matrix} U (p_{i}, p_{j}) : = p_{i} [p_{j} Π (D) + (1 - p_{j}) Π (2 D)] \\ + (1 - p_{i}) [p_{j} Π (- D) + (1 - p_{j}) Π (0)] \end{matrix}

So the firms have the following payoff matrix for doing and not doing R&D (Table 1):

Table 1.

Payoff Matrix Under Complete Information.

Firm A/Firm B	R	NR
R	$U (p_{A}, p_{B}) - M, U (p_{B}, p_{A}) - M$	$U (p_{A}, 0) - M, U (0, p_{A})$
NR	$U (0, p_{B}), U (p_{B}, 0) - M$	$U (0, 0), U (0, 0)$

If we define gross strategic incentive (GSI) as the difference in gross payoffs between performing and not performing R&D when the rival firm is performing R&D, and define gross non-strategic incentive (GNSI) as the difference in gross payoffs between performing and not performing R&D when the rival firm is not performing R&D, then

$G S I (p_{i}, p_{j}) : = p_{i} [p_{j} [Π (D) - Π (- D)] + (1 - p_{j}) [Π (2 D) - Π (0)]]$ ,

$GNSI (p_{i}) : = p_{i} [Π (2 D) - Π (0)]$ .

We must have $G N S I (p_{i}) \geq G S I (p_{i}, p_{j})$ , because $Π (\cdot)$ is strictly increasing and strictly convex, hence, $[Π (2 D) - Π (0)] > [Π (D) - Π (- D)]$ .

Now define

V (x) : = \frac{M}{x (Π (D) - Π (- D)) + (1 - x) (Π (2 D) - Π (0))} .

Then $V^{'} (x) > 0$ .

Proposition 1. Under complete information,

If $p_{i} < V (0)$ , then firm i will never invest in research;

If $p_{i} > V (p_{j})$ , then firm i will always invest in research, and

If $V (p_{j}) > p_{i} > V (0)$ , then firm i will invest in research provided firm $j$ does not invest in research.

In the next section we develop the analysis under incomplete information.

Incomplete Information

In this section we discuss R&D incentives of firms under various levels of incomplete information. We consider the following three situations:

Level I incomplete information: Here a firm knows the rival’s R&D decision and its R&D status, but does not know the type of its rival.

Level II incomplete information: Here a firm knows its rival’s R&D decision, but does not know the type and the status of R&D of its rival.

Level III incomplete information: Here the rival’s R&D decision, R&D status and its type all are unknown to a firm.

Level I Incomplete Information: Type Unknown, R&D Decision and Status Known

In level I incomplete information, we assume incomplete information about the rival’s type. Thus a firm does not know the probability of success of the rival if the rival is doing R&D. But the firm knows, just before production takes place, whether the rival has invested in R&D or not, and whether the rival is successful or not if it has invested in R&D. However, at the time of investment in R&D, the firm does not know the type, decision and status (i.e. whether success or failure) of its rival firm. Let τ₁ be the threshold type of a firm such that if its type is greater than this threshold, it will invest in R&D. So, our objective will be to find this threshold value, given M.

The expected profit of the firm when it does not invest in R&D is as follows:

F (τ_{1}) Π (0) + (1 - F (τ_{1})) [Θ (τ_{1}) Π (- D) + (1 - Θ (τ_{1})) Π (0)]

and the expected profit of the firm when it does invest in R&D is as follows:

\begin{array}{r} p_{i} [F (τ_{1}) Π (2 D) + (1 - F (τ_{1})) [Θ (τ_{1}) Π (D) + (1 - Θ (τ_{1})) Π (2 D)]] \\ + (1 - p_{i}) [F (τ_{1}) Π (0) + (1 - F (τ_{1})) [Θ (τ_{1}) Π (- D) + (1 - Θ (τ_{1})) Π (0)]] - M . \end{array}

Define $T_{1} (x; τ_{1})$ as gross opportunity ‘gain’ of doing $R & D$ , when the type of the firm is $x$ . Here

\begin{array}{l} T_{1} (x; τ_{1}) : = x [F (τ_{1}) Π (2 D) + (1 - F (τ_{1})) [Θ (τ_{1}) Π (D) + (1 - Θ (τ_{1})) Π (2 D)]] \\ + (1 - x) [F (τ_{1}) Π (0) + (1 - F (τ_{1})) [Θ (τ_{1}) Π (- D) + (1 - Θ (τ_{1})) Π (0)]] \\ - [F (τ_{1}) Π (0) + (1 - F (τ_{1})) [Θ (τ_{1}) Π (- D) + (1 - Θ (τ_{1})) Π (0)]] \\ = x [[Π (2 D) - Π (0)] - (1 - F (τ_{1})) Θ (τ_{1}) [Π (2 D) - Π (0) - Π (D) + Π (- D)]] \end{array}

$T_{1} (x; τ_{1})$ is increasing in $x$ . Also a firm whose type is $τ_{1}$ must be indifferent between investing and not investing in R&D. Therefore, the threshold value $τ_{1}$ will be solved from the equation $T_{1} (τ_{1}; τ_{1}) = M$ . We have the following result.

Proposition 2. Given any $M > 0$ , suppose $M < T_{1} (1; 1)$ . Then there exists a unique $τ_{1}$ such that a firm will invest in research if and only if its type is greater than or equal to $τ_{1}$ , where $τ_{1}$ is solved from $T_{1} (τ_{1}; τ_{1}) = M$ . If $M > T_{1} (1; 1)$ , then no firm will invest in research.

Proof. We have $T_{1} (0; 0) = 0$ . With slight abuse of notation, $T_{1} (1; 1) =$ ${lim}_{τ_{1} \to 1} T_{1} (τ_{1}; τ_{1}) = Π (2 D) - Π (0) > 0$ . Further, $T_{1} (x; x)$ is increasing in $x \in$ $(0, 1)$ . To see this,

\begin{matrix} T_{1} (x; x) = x {[Π (2 D) - Π (0)] - (1 - F (x)) Θ (x) \\ \times [Π (2 D) - Π (0) - Π (D) + Π (- D)]} \end{matrix}

Hence,

\begin{array}{l} T_{1}^{'} (x; x) = [Π (2 D) - Π (0)] - (1 - F (x)) Θ (x) \\ \times [Π (2 D) - Π (0) - Π (D) + Π (- D)] \\ + x^{2} f (x) [Π (2 D) - Π (0) - Π (D) + Π (- D)] > 0 \end{array}

given $Π (x)$ strictly increasing and convex in $x$ so that $Π (2 D) - Π (0) > Π (D) - Π (- D)$ . Hence the result.

Level II Incomplete Information: Type Unknown, Investment Decision Known but R&D Status Unknown

In level II incomplete information, we assume that a firm does not know the type of its rival. Also, if its rival has taken up R&D investment, the firm does not know whether the rival will succeed or fail. However, the firm can observe whether the rival does R&D or not. Let $τ_{2}$ be the threshold value such that if the type of a firm is greater than or equal to $τ_{2}$ , then it will invest in R&D. Our objective in this case is to find out such a $τ_{2}$ , given M.

The expected payoffs of the firms under level II incomplete information are given in the following lemma.

Lemma 2. The following results hold:

If none of the firms invests in R&D, the profit of firm i is $Π_{i}^{[N N]} = Π (0)$ .

If firm A does R&D but firm B does not, then the respective expected profits are $E Π_{A}^{[S N]} = Π (\frac{(3 + Θ (τ_{2})) D}{2}), E Π_{A}^{[F N]} = Π (\frac{Θ (τ_{2}) D}{2})$ and $E Π_{B}^{[R N]} =$ $Π (- Θ (τ_{2}) D)$ .

If both firms are investing in $R & D$ , then he expected profits of firm $A$ are given by $E Π_{A}^{[S R]} = Π (\frac{(3 - Θ (τ_{2})) D}{2})$ and $E Π_{A}^{[F R]} = Π (\frac{- Θ (τ_{2}) D}{2})$ .

Similar expected profit expressions can be derived for firm B routinely.

Derivation of the payoffs underlying Lemma 2(b) and 2(c) are given in Appendix 1.

Proceeding the same way as we did for level I incomplete information, we get the gross expected "gain" of doing research for the firm whose type is x, to be given by

\begin{matrix} T_{2} (x; τ_{2}) : = F (τ_{2}) [x Π (\frac{(3 + Θ (τ_{2})) D}{2}) + (1 - x) Π (\frac{Θ (τ_{2}) D}{2}) - Π (0)] \\ + (1 - F (τ_{2})) [x Π (\frac{(3 - Θ (τ_{2})) D}{2}) + (1 - x) Π (\frac{- Θ (τ_{2}) D}{2}) \\ - Π (- Θ (τ_{2}) D) \end{matrix}

So, $T_{2} (0; 0) = Π (- \frac{Θ (0) D}{2}) - Π (- Θ (0) D) > 0$ , and with slight abuse of notation, $T_{2} (1; 1) : = {lim}_{τ_{2} \to 1} T_{2} (τ_{2}, τ_{2}) = Π (2 D) - Π (0)$ .

Lemma 3. If $K > 3 D$ , then $T_{2} (x; x)$ is strictly increasing.

Proof of Lemma 3 is given in Appendix 2. We can now write the following proposition.

Proposition 3. Given the function $T_{2} (x; τ_{2})$ and $M$ , the following must hold:

If $M < T_{2} (0; 0)$ , then both the firms will invest in $R & D$ ,

If $M > T_{2} (1; 1)$ , then no firm will invest in $R & D$ , and

When $T_{2} (0; 0) < M < T_{2} (1; 1)$ , there exists a unique $τ_{2}$ such that a firm will invest in $R & D$ if and only if its type is greater than or equal to $τ_{2}$ where $τ_{2}$ is solved from the equality $T_{2} (τ_{2}; τ_{2}) = M$ ; and $τ_{2}$ is increasing in $M$ .

Since in the second stage the firms know the R&D decision of their respective rivals, this information can act as a signal. So, it is important now to check the incentive compatibility. We have claimed that a firm will invest in R&D if and only if the type of the firm is greater than or equal to $τ_{2}$ . Suppose firm A follows this strategy and believes that firm B also follows the same strategy. Further, firm B knows firms A ’s strategy and belief.

Let

N R_{I I} (x; τ_{2}) : = F (τ_{2}) Π (0) + (1 - F (τ_{2})) Π (- Θ (τ_{2}) D)

be the expected profit of a firm when it does not invest in R&D. Similarly, let

\begin{array}{r} R_{I I} (x; τ_{2}) : = & x [F (τ_{2}) Π (\frac{(3 + Θ (τ_{2})) D}{2})] + (1 - F (τ_{2})) Π (\frac{(3 - Θ (τ_{2})) D}{2}) \\ + (1 - x) [F (τ_{2}) Π (\frac{Θ (τ_{2}) D}{2}) + (1 - F (τ_{2})) Π (- \frac{Θ (τ_{2}) D}{2})] \end{array}

be the gross expected profit of a firm that does invest in R&D. Note that $R_{I I} (x; τ_{2})$ is an increasing function of $x$ . Also $T_{2} (x; τ_{2}) = R_{I I} (x; τ_{2}) -$ $N R_{I I} (x; τ_{2})$ . Then we can make the following two observations.

First, suppose firm B’s type is less than $τ_{2}$ but it decides to invest in $R & D$ , then the expected profit of firm $B$ under $R & D$ investment is $R_{I I} (p_{B}; τ_{2})$ . However, if it had not invested, then its expected profit would have been $N R_{I I} (p_{B}; τ_{2})$ . Given the definition of $τ_{2}$ , and that $T_{2} (x; τ_{2})$ is strictly increasing in $x$ , the following must hold for all $p_{B} < τ_{2}$ :

N R_{I I} (p_{B}; τ_{2}) > R_{I I} (p_{B}; τ_{2})

So, if firm B’s type is less than $τ_{2}$ , then given firm $A$ ’s strategy and belief, it will never invest in R&D. Similarly it can be shown that if firm B’s type is greater than $τ_{2}$ , then given firm $A$ ’s strategy and belief, it will always invest in research.

By optimal strategy under level II incomplete information we mean that the firm will invest in R&D if and only if its type is greater than or equal to $τ_{2}$ and it believes that the rival is following the same strategy. So, both the firms following this strategy is a perfect Bayesian Nash equilibrium.

Level III Incomplete Information: Type, R&D Decision and Status Unknown

In level II incomplete information, we assume that a firm does not know the type of its rival. Also, whether the rival does or does not invest in R&D is unobservable. Therefore, whether the rival has succeeded or failed in R&D is also unknown to the firm. As before, assume that a firm will invest in R&D if its type is greater than or equal to $τ_{3}$ . Thus our objective in this section is to find out such a $τ_{3}$ , given $M$ . Let $H (x) : = \frac{(1 - F (x) ΘΘ (x) D}{2}$ ; then $H^{'} (x) = - \frac{1}{2} x f (x)$ $D < 0$ .

The payoffs of a firm under various contingencies are given in Lemma 4 and are derived in Appendix 3.

Lemma 4. The following results hold:

If firm i invests in R&D and succeeds, then its expected profit is

E Π_{i}^{[S]} = Π (\frac{3 D}{2} - H (τ_{3})) - M,

If firm i invests in $R & D$ and fails, then its expected profit is

E Π_{i}^{[F]} = Π (- H (τ_{3})) - M,

and

If firm i does not invest in $R & D$ , then its expected profit is $E Π_{i}^{[N]} =$ $Π (- H (τ_{3}))$ .

Hence the gross expected ‘gain’ from investing in R&D for firm i is as follows:

T_{3} (x; τ_{3}) : = x [Π (\frac{3 D}{2} - H (τ_{3})) - Π (- H (τ_{3}))]

$T_{3} (x; τ_{3})$ is increasing in $x$ . Define a firm of type $τ_{3}$ that is indifferent between investing and not investing in $R & D$ . Then $τ_{3}$ is solved from $T_{3} (τ_{3}; τ_{3}) =$ $M$ . We have $T_{3} (0; 0) = 0$ , and (with slight abuse of notation) $T_{3} (1; 1) : =$ ${lim}_{τ_{3} \to 1} T_{3} (τ_{3}, τ_{3}) = Π (\frac{3 D}{2}) - Π (0)$ . Finally, $T_{3} (x; x)$ is strictly increasing in $(0, 1)$ .⁵ Then we have the following results, as given in Proposition 4.

Proposition 4. The following results hold:

If $T_{3} (0; 0) < M < T_{3} (1; 1)$ , there exists a unique $τ_{3}$ such that a firm will invest in research if and only if its type is greater than or equal to $τ_{3}$ when $τ_{3}$ can be obtained by solving the equality $T_{3} (τ_{3}; τ_{3}) = M$ .

If $M > T_{3} (1; 1)$ , then no firm will invest in $R & D$ .

Comparison

This section compares the various situations arising under the alternative information structures, complete as well as incomplete.

Comparison of Various Incomplete Information Levels

On comparing different cases of incomplete information we derive what we call the non-monotonicity of R&D incentive with respect to different levels of incomplete information. In particular, we show that as less and less information becomes available, a firm’s R&D incentive first increases, and then it starts falling.

Let us first compare Level I and Level III incomplete information. It can be shown (see Appendix 4) that:

T_{1} (x; x) > T_{3} (x; x) \forall x \in (0, 1)

Further, we have $T_{1} (0; 0) = T_{3} (0; 0)$ , but $T_{1} (1; 1) > T_{3} (1; 1)$ . Given that both $T_{1}$ and $T_{3}$ are increasing in $x$ , this means we must have $τ_{1} < τ_{3}$ . This suggests that the R&D incentive of a firm under Level I incomplete information is larger than that under Level III incomplete information. The reason is that compared to Level I incomplete information, there are many characteristics of the rival unknown to the firm.

Now we compare Level I and Level II incomplete information. However, it may be recalled that under Level II incomplete information, there is signalling effect through which a firm may get larger information about its rival. Now given that $T_{1} (0; 0) < T_{2} (0; 0), T_{1} (1; 1) = T_{2} (1; 1)$ , and both $T_{1} (\cdot; \cdot)$ and $T_{2} (\cdot; \cdot)$ are increasing functions of $x$ , then if the signalling effect is strong enough, $T_{2} (\cdot; \cdot)$ will always be above $T_{1} (\cdot; \cdot), \forall x \in (0, 1)$ . This is equivalent to restricting the $F (\cdot)$ function.

Assume that $F (\cdot)$ satisfies the following condition for all $x \in (0, 1)$ :

(Θ (x) - x) \frac{K}{D} + Θ (x) [(2.5 - F (x)) x + F (x) Θ (x)] > 1.75 x + 0.75 Θ {(x)}^{2}

(*)

If the given condition holds, we can say that the signalling effect is strong, and that under this condition we must have $T_{2} (x; x) > T_{1} (x; x) \forall x \in (0, 1)$ . The implication of this result is that,

τ_{2} < τ_{1},

that is R&D incentive of a firm is larger under Level II incomplete information than that under Level I incomplete information.

One may easily check whether the condition (*) necessarily holds for a uniform distribution function (a linear distribution function) and many other distribution functions.⁶ Therefore, combining the aforementioned analysis we can write the following result:

Proposition 5. Suppose the condition (*) holds. Then comparing all the threshold values we must have $τ_{2} < τ_{1} < τ_{3}$ , that is, if the signalling effect is strong enough, $R & D$ incentive of a firm is larger under Level II incomplete information.

Example 1. Let us assume $a = 10, c = 2, M = 2, D = 1$ , and $p_{i}$ is distributed uniformly for all $i \in \{A, B\}$ . Then in this case $τ_{1} = 0.5211, τ_{2} = 0.4689$ and $τ_{3} = 0.696$ . So here, clearly we have, $τ_{2} < τ_{1} < τ_{3}$ .

Note that the restriction we impose on the distribution function is a sufficient condition for getting the non-monotone relationship; the condition is not necessary. So when the condition does not hold, suppose $T_{2}$ intersects $T_{1}$ at an intermediate probability, say $τ_{c}$ ⁷ then given both $T_{2}$ and $T_{1}$ strictly increasing and $0 = T_{1} (0) < T_{2} (0) < T_{2} (τ_{c}) = T_{1} (τ_{c})$ , there exists $M \in (0, T_{2} (τ_{c}))$ for which the desired non-monotonic relationship holds. This possibility is shown in Figure 1.

Complete vs. Incomplete Information

In this subsection, we show that there are situations when incomplete information may enhance the incentive for R&D investment compared to complete information case.⁸

Proposition 5(A). There always exists $M > 0$ and types of firms $A$ and $B$ , namely $p_{A}$ and $p_{B}$ , such that both the firms will invest in $R & D$ under Level II incomplete information, but will never invest in $R & D$ under complete information.

Proposition 5(B). If both the firms are investing under Level I incomplete information, then at least one firm will definitely invest under complete information.

Proposition 5(C). If both the firms invest under Level III incomplete information, then they will definitely invest under complete information.

We can explain the results underlying Proposition 5(.). As we move from complete to incomplete information, depending on the extent of uncertainty, there are three levels of incomplete information, where maximum uncertainty arises under Level III incomplete information. So under such a situation, if for some $(p_{a}, p_{b})$ , both the firms do R&D, then under complete information they will certainly do. In case of Level I incomplete information, uncertainty is lowest among these three cases of incomplete information. So for some $(p_{a}, p_{b})$ , when at least one firm does invest in R&D under complete information, under Level I incomplete information both the firms might do R&D because one firm does not know the other’s probability of success with certainty. On the other hand, consideration of R&D investment under Level II incomplete information takes care of the signalling effect in the sense that if one firm invests in R&D, it may signal that its probability of success is possibly high. Therefore, in the situation when no firm does R&D under complete information, both firms may invest because of the signalling effect. Thus our results show that there are situations when incomplete information may lead to higher incentives for R&D investment compared to complete information.

Conclusion

In the present paper we have discussed R&D incentives of a firm in a duopoly under various information structures hitherto not properly dealt with in the literature. We have assumed that a firm’s probability of success in R&D is determined by factors that are endogenous to firms and hence is private information that constitutes its type. We have considered the benchmark case of complete information along with three incomplete information structures depending on the levels of uncertainty, or incomplete information. We have first shown that incentive to invest in R&D is the highest in the case of Level II incomplete information, where the rival’s type is unknown and the investment decision is known, but the status of R&D is unknown, and is lowest under Level III incomplete information, where the rival’s type, investment decision and status of R&D, are all unknown to a firm. Therefore, there is a non-monotone relation between the R&D incentive and the level of incomplete information. We also find that incentives for R&D may be higher under incomplete information compared to complete information under certain situations. R&D incentive under Level II is highest because of the presence of a signalling effect in this case. Despite a firm’s type being unknown, by means of its investment decision, it can give a signal to its rival that it is a high-type firm, irrespective of its true type. So the rival may believe that the firm’s type is high, that is, above the threshold value. Hence, in the presence of a signalling effect, a firm’s incentive for R&D is higher. But as further information available to a firm declines, the firm is discouraged to invest under increasing uncertainty.

Finally, we have proved our results under a broad class of probability distribution functions of types. We have provided an example supporting our results.

Appendices

Appendix I Proof of Lemma 2

Proof. (a) This is obvious.

(b) The expected profit function of firm A is

\begin{matrix} p_{A} [((K + D) - q_{A}^{[S N]} - q_{B}^{[R N]}) q_{A}^{[S N]}] \\ + (1 - p_{A}) [(K - q_{A}^{[F N]} - q_{B}^{[R N]}) q_{A}^{[F N]}] - M . \end{matrix}

Similarly the expected profit of firm B is

(K - q_{B}^{[R N]} - [Θ (τ_{2}) q_{A}^{[S N]} + (1 - Θ (τ_{2})) q_{A}^{[F N]}]) q_{B}^{[R N]} .

Solving the reaction function we get

\begin{array}{l} q_{A}^{[S N]} = q (\frac{(3 + Θ (τ_{2})) D}{2}), q_{A}^{[F N]} = q (\frac{Θ (τ_{2}) D}{2}) \\ and q_{B}^{[R N]} = q (- Θ (τ_{2}) D) . \end{array}

Hence the payoffs.

\begin{array}{r} p_{A} [(K + D) - q_{A}^{[S R]} - [Θ (τ_{2}) q_{B}^{[R S]} + (1 - Θ (τ_{2})) q_{B}^{[R F]}]] q_{A}^{[S R]} \\ + (1 - p_{A}) [K - q_{A}^{[F R]} - [Θ (τ_{2}) q_{B}^{[R S]} + (1 - Θ (τ_{2})) q_{B}^{[R F]}]] q_{A}^{[F R]} - M . \end{array}

The expected profit of firm B is given by

\begin{array}{r} p_{B} [(K + D) - q_{B}^{[R S]} - [Θ (τ_{2}) q_{A}^{[S R]} + (1 - Θ (τ_{2})) q_{A}^{[F R]}]] q_{B}^{[R S]} \\ + (1 - p_{B}) [K - q_{B}^{[R F]} - [Θ (τ_{2}) q_{A}^{[S R]} + (1 - Θ (τ_{2})) q_{B}^{[F R]}]] q_{B}^{[R F]} - M . \end{array}

By solving the reaction functions we get

q_{A}^{[S R]} = q_{B}^{[R S]} = q (\frac{(3 - Θ (τ_{2})) D}{2}) and q_{A}^{[F R]} = q_{B}^{[R F]} = q (\frac{- Θ (τ_{2}) D}{2}) .

Appendix 2 Proof of Lemma 3

Proof. To prove Lemma 3, first consider the expression of $T_{2} (x; x)$ . From the expression in the first brackets in $T_{2} (x; x)$ ,

\begin{array}{l} \frac{d}{d x} [x Π (\frac{(3 - Θ (x)) D}{2}) + (1 - x) Π (\frac{- Θ (x) D}{2}) - Π (- Θ (x) D)] \\ = [Π (\frac{(3 - Θ (x)) D}{2}) - Π (\frac{- Θ (x) D}{2})] \\ + \frac{2}{3} [2 q (- Θ (x) D) - x q (\frac{(3 - Θ (x)) D}{2}) - (1 - x) q (\frac{- Θ (x) D}{2})] \frac{Θ ‘ (x) D}{2} \end{array}

The first part of the RHS is clearly positive. For the second part, note that if $K > 3 D$ then $2 q (- Θ (x) D) > q (\frac{(3 - Θ (x)) D}{2}) > q (\frac{- Θ (x) D}{2})$ holds.

Similarly for the combined expression within the second and third brackets in $T_{2} (x; x)$ ,

\begin{array}{l} \frac{d}{d x} [Π (\frac{(3 + Θ (x)) D}{2}) - Π (\frac{(3 - Θ (x)) D}{2}) - Π (\frac{Θ (x) D}{2}) \\ + Π (\frac{- Θ (x) D}{2})] = \frac{2}{3} [q (\frac{(3 + Θ (x)) D}{2}) + q (\frac{(3 - Θ (x)) D}{2}) \\ - q (\frac{Θ (x) D}{2}) - (\frac{- Θ (x) D}{2})] \frac{Θ ‘ (x) D}{2} > 0. \end{array}

And that within the fourth brackets in $T_{2} (x; x)$

\begin{array}{l} \frac{d}{d x} [Π (\frac{Θ (x) D}{2}) - Π (\frac{- Θ (x) D}{2}) - Π (0) + Π (- Θ (x) D)] \\ = \frac{2}{3} [q (\frac{Θ (x) D}{2}) + q (\frac{- Θ (x) D}{2}) - 2 q (- Θ (x) D)] \frac{Θ ‘ (x) D}{2} > 0. \end{array}

Therefore all the four parts of the RHS of the $T_{2} (x; x)$ are strictly increasing. So, $T_{2} (x; x)$ is strictly increasing.

Appendix 3 Proof of Lemma 4

Proof. Note that the expected profit of firm A if it invests in research and succeeds is

\begin{array}{l} (1 - F (τ_{3})) [(K + D) - q_{A}^{[S]} - [Θ (τ_{3}) q_{B}^{[S]} + (1 - Θ (τ_{3})) q_{B}^{[F]}]] q_{A}^{[S]} \\ + F (τ_{3}) [(K + D) - q_{A}^{[S]} - q_{B}^{[N]}] q_{A}^{[S]} - M . \end{array}

Similarly, the expected profit of firm $A$ if it invests in research and fails is

\begin{array}{l} (1 - F (τ_{3})) [K - q_{A}^{[F]} - [Θ (τ_{3}) q_{B}^{[S]} + (1 - Θ (τ_{3})) q_{B}^{[F]}]] q_{A}^{[F]} \\ + F (τ_{3}) [K - q_{A}^{[F]} - q_{B}^{[N]}] q_{A}^{[F]} - M \end{array}

Finally, the expected profit of firm $A$ , if it does not invest in research, is

\begin{array}{l} (1 - F (τ_{3})) [K - q_{A}^{[N]} - [Θ (τ_{3}) q_{B}^{[S]} + (1 - Θ (τ_{3})) q_{B}^{[F]}]] q_{A}^{[N]} \\ + F (τ_{3}) [K - q_{A}^{[N]} - q_{B}^{[N]}] q_{A}^{[N]} . \end{array}

Looking at these expected profit functions it is clear that at equilibrium we have $q_{A}^{[F]} = q_{A}^{[N]}$ ; call it $q_{A}^{[N]}$ . Solving the reaction functions we get

q_{A}^{[S]} = q_{B}^{[S]} = q (\frac{3 D}{2} - H (τ_{2})) and q_{A}^{[N]} = q_{B}^{[N]} = q (- H (τ_{3})) .

The rest of the proof is trivial.

Appendix 4 Comparing Threshold Values Under Level I and Level III Incomplete Information

Proof. Note that $Π (2 D) - Π (0) > Π (D) - Π (- D) > Π (\frac{3 D}{2} - H (x)) - Π (- H (x))$ for all $x \in (0, 1)$ , as $K > 3 D$ and the $Π (\cdot)$ is strictly increasing and strictly convex to the origin. Let $G (x) : = Π (\frac{3 D}{2} - H (x)) - Π (- H (x)) .$ .

Therefore,

\begin{array}{l} F (x) [Π (2 D) - Π (0)] + (1 - F (x)) [Θ (x) (Π (D) - Π (- D)) \\ + (1 - Θ (x)) (Π (2 D) - Π (0))] > F (x) G (x) + (1 - F (x)) [Θ (x) G (x) \\ + (1 - Θ (x)) G (x)] = G (x) . \end{array}

Footnotes

Acknowledgement

The authors are greatly indebted to an anonymous referee of this journal for meaningful and productive comments. Authors would also like to thank Prabal Roy Choudhury and Krishnendu Ghosh Dastidar for their helpful comments and suggestions on an earlier draft. However, any errors are responsibility of the authors alone. This paper is a shortened version of our working paper ‘R&D incentives with uncertain probability of success’ https://mpra.ub.unimuenchen.de/98474/

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

Notes

ORCID iDs

Srobonti Chattopadhyay

Tarun Kabiraj

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