Competition and strategic alliance in R&D investments: a real option game approach with multiple experiments

Abstract

In this paper, we analyse the effects that the number and outcomes of R&D experiments have on the strategic equilibria between two firms that can both compete and cooperate in a context of uncertainty. As is well known, R&D projects are characterised by the sequentiality of investments and by the outcomes obtained from the success or failure of their experiments. Furthermore, the positive results and the number of tests carried out in R&D increase the market value of the innovative product. The Real Option Approach evaluates the flexibility of R&D investments and the strategic scenarios. According to Nash equilibria, we show how the market value threshold, for which the investment policy is optimal, depends on the number of experiments and on the information revelation.

Appendix

Proof of Proposition 1

We analyse the strategic payoffs assuming as variable the market value V. We can observe that:

$$\begin{aligned} L_A(0,n)= & {} -R_A;\quad W_A(0,n)=0;\\ \frac{\partial L_A}{\partial V}= & {} N(d_1(P,T))\,e^{-\delta _v T} \sum _{k=0}^n \left( \begin{array}{c} n\\ k \end{array}\right) q^k(1-q)^{n-k}\, \\&\qquad \left[ \sum _{h=0}^n \left( \begin{array}{c} n\\ h \end{array}\right) {\left( p^{\varepsilon ,\theta }\right) }^h {\left( (1-p^{\varepsilon ,\theta }\right) }^{n-h}\, K_{t_{0_k}t_{1_h}}\right] \\ \frac{\partial W_A}{\partial V}= & {} N_2\left( d_1\left( \frac{P}{P^*_w},t_1\right) ,d_1(P,T);\rho \right) \,e^{-\delta _v T} \\&\quad \times \sum _{h=0}^{n}\left( \begin{array}{c} n\\ h \end{array}\right) p^{h}(1-p)^{n-h} \left[ \sum _{k=0}^n \left( \begin{array}{c} n\\ k \end{array}\right) q^k(1-q)^{n-k}\, K_{t_{1_k}t_{1_h}}\right] \end{aligned}$$

As $N(d_1(P,T))> N_2\left( d_1\left( \frac{P}{P^*_w},t_1\right) ,d_1(P,T);\rho \right)$, moreover, $p^{ \varepsilon ,\theta }>p$ and $K_{t_{0_k}t_{1_h}}>K_{t_{1_k}t_{1_h}}$, it results that $\displaystyle \frac{\partial L_A}{\partial V}>\frac{\partial W_A}{\partial V}>0$. This condition assures us a unique critical market value $V^*_{A,W}$. The same results occur for firm B. $\square$

Proof of Proposition 2

We observe that:

$$\begin{aligned} S_A(0,n)= & {} -R_A;\quad F_A(0,n)=0;\\ \frac{\partial S_A}{\partial V}= & {} N(d_1(P,T)) \,e^{-\delta _v T} \sum _{k=0}^n\left( \begin{array}{c} n\\ k \end{array} \right) q^k(1-q)^{n-k} \\&\qquad \left[ \sum _{h=0}^n\left( \begin{array}{c} n\\ h \end{array} \right) p^h(1-p)^{n-h}\, K^A_{t_{0_k}t_{0_h}} \right] \\ \frac{\partial F_A}{\partial V}= & {} N_2\left( d_1\left( \frac{P}{P^*},t_1\right) ,d_1(P,T);\rho \right) \,e^{-\delta _v T}\\&\quad \times \sum _{h=0}^{n}\left( \begin{array}{c} n\\ h \end{array}\right) p^{h}(1-p)^{n-h} \left[ \sum _{k=0}^n \left( \begin{array}{c} n\\ k \end{array}\right) {\left( q^{j,y}\right) }^k { \left( 1-q^{j,y} \right) }^{n-k} \, K^A_{t_{1_k}t_{0_h}} \right] \end{aligned}$$

with $\frac{\partial F_A}{\partial V}>0$ and $\frac{\partial S_A}{\partial V}>0$ and $\frac{\partial F_A}{\partial V} \gtrless \frac{\partial S_A}{\partial V}$. The same results occurs for firm B. $\square$