Elsevier

Fuzzy Sets and Systems

Volume 434, 30 April 2022, Pages 117-134
Fuzzy Sets and Systems

Real R&D options under fuzzy uncertainty in market share and revealed information

https://doi.org/10.1016/j.fss.2021.07.011Get rights and content

Abstract

In this paper, we study Nash equilibria for real R&D options in a context of non-cooperative oligopoly under fuzzy uncertainty in market shares and revealed information. In particular, we consider a duopoly fuzzy game in which each firm has two available strategies, that is to invest or to defer its research investment. The deferring option gives the benefit of the information revelation about market reaction but, on the other hand, the disadvantage of a lower market share. Since these two effects are not easily to quantify, we describe the revealed information and the market shares as fuzzy numbers. As shown by numerical examples, the proposed fuzzy framework allows to reach different firms' strategic behaviours and new Nash equilibria with respect to the crisp case.

Introduction

Research and Development (R&D) investments are considered an important driving force for the growth of the modern economy. R&D is a broad category describing the entity of basic and applied research and development activities in order to increase knowledge and its use to develop new products, processes, or services. R&D investment presents some uncertainties that may come from the limitation of the R&D capabilities, external market volatility, or complexity of the project. An R&D investment can affect the changes of firms' relative competitive positions; for instance, the learning experience cost effect may be an important factor influencing the timing of R&D investment in a competitive context.

R&D investments evaluation is rather complex since these opportunities are characterized by several factors of uncertainty and, generally, involve multiple phases with or without overlapping. If the investment is planned in a phased manner, with the beginning of a subsequent phase being dependent on the success achieved in the previous phase, it is known as sequential investment. Since each stage provides information for the next phase, creating an opportunity (option) for subsequent investment, such projects can be valued using the technique of real option methodology, that allows to take into account the managerial flexibility implicit in R&D investment, such as the opportunity to postpone for better market conditions (see Shenhar and Dvir [34]).

According to Margrabe [27], an R&D project investment can be explained as owning an investment opportunity that allows to exchange the investment cost for the project value at maturity time. Thus, in order to evaluate these opportunities embedded in projects or processes, real exchange option models have been employed in literature, as witnessed by Carr [8], Lee and Paxson [24], Armada et al. [1], Cortelezzi and Villani [9], just to name a few.

In this paper, we deal with R&D investment opportunities in a context of oligopoly with few companies having comparable market and competitive positions, in order to study firms' strategic behaviours and to determine Nash market equilibria. In recent years, R&D investments in competitive markets have been addressed as games among firms and a large number of papers in the real option literature incorporating game theoretic concepts have been published. The reason for this tendency is that such approach is often suitable in terms of real applications since many industries are characterized by both uncertainty and strategic interactions. In this framework, a standard real options game can be considered as an oligopoly market where firms' payoffs are derived combining game theory concepts with the real options methodology, since firms implicitly take into account what they think will be the other firms' reactions to their own actions. The first paper in real options game literature to consider interactions between firms was Smets [35]. Other papers in the same research line include Grenadier [17], Trigeorgis [38], Smit and Trigeorgis [36], [37], Huisman [18], Murto and Keppo [29], Weeds [41], Lambrecht and Perraudin [23], Huisman and Kort [19], [20], Paxson and Pinto [31], Pawlina and Kort [30] and Kong and Kwon [22] and Azevedo and Paxson [2].

In particular, we focus on a duopoly R&D investment game between two firms with comparable market and competitive positions, that can decide if to realize their research investment at initial time or to postpone their decision waiting for better market conditions, under uncertainty about the success or the failure of their research plan. Each of the two firms has two available strategies, to invest or to defer, which can lead to three different scenarios: i) both firms invest; ii) one firm invests and the other one defers; iii) both firms defer. To each of these scenarios correspond different firms' payoffs. In the scenario ii), for convenience in terminology, we denote as Leader the firm that invests earlier and as Follower the one that defers its investment. We assume that Leader receives a first mover advantage consisting in a higher market share in case of research investment success. Conversely, the Follower obtains an information revelation from the research investment carried out by Leader. This information revelation can be assumed in a legal way, for instance observing the result obtained by an oil drilling or the results of a vaccine trial, or illegally through industrial espionage. Indeed, a company could decide to invest early in technology research, assuming a leader position, and spend significantly on R&D obtaining only modest overall earnings; differently, a close competitor waiting for the innovations and picking up the good ideas without having to pay for the errors and mistakes, can realize better overall returns.

The novelty of our model is to handle in a fuzzy way the uncertainty related to the information revelation, concerning both the success/failure of the R&D investment and the learning effect and, furthermore, the uncertainty about the firms' market shares. An adequate estimation of these economic variables is very useful since they influence the cost of research phase, the market position and, consequently, the firms' payoffs. Nevertheless, their correct estimation turns out to be not easy and often they have to be evaluated by experts in presence of scarce data. In order to menage with imprecise knowledge about market shares and revealed information, we propose to address the problem to study R&D firms' strategic behaviours under random and fuzzy uncertainty. Moving on this general context of uncertainty, we will show that new economically interesting Nash equilibria occur.

In the framework of fuzzy game theory, several papers have addressed the problem to analyse strategic behaviours and determine Nash equilibria. For instance, Butnariu [5] introduces in the analysis the player's fuzzy preference relations about the crisp pure strategies. Buckley [4] investigates a two-party non-cooperative games involving both uncertainty and multiples goals. Campos [6] models a two-person zero-sum game with fuzzy payoffs by transforming the game into an optimization problem using Yager's fuzzy number ranking index. Maeda [25] studies parametric bimatrix games with fuzzy payoffs described by symmetric triangular fuzzy numbers. Vijay et al. [40] employ a suitable defuzzification function for studying a two-person zero-sum game with fuzzy payoffs. Cunlin and Qiang [10] investigate a two-person zero-sum game in fuzzy environment introducing the concept of crisp bimatrix game with parameters. Our study contributes to this research line analysing the R&D investment as a two-person fuzzy game with non zero-sum payoffs, using real option methodology under ambiguity in information revelation and market shares.

The paper is organized as follows. Section 2 contains a brief presentation of the exchange option valuation models and basic concepts about fuzzy numbers. In Section 3, the R&D investment model is analyzed in a fuzzy environment and, in Section 4, explicit formulas for fuzzy payoffs are presented. In Section 5, the fuzzy game is solved computing Nash equilibria and some numerical applications are offered. Finally, Section 6 concludes.

Section snippets

Exchange options and fuzzy preliminaries

In this section we present the basic concepts concerning European exchange options and fuzzy numbers.

R&D investment model

We consider a duopoly R&D investment game between two firms, A and B, with comparable market and competitive positions, that can decide if to realize their research investment at initial time t0=0 or to postpone their decision at time t1 waiting for better market conditions. The timing of the investment decision has remarkable effects on the firms' payoffs since early entry into the market offers the possibility to intercept a greater market share but, on the other hand, deferring the

Explicit formulas for fuzzy payoffs

In this section we will provide explicit formulas for fuzzy payoffs obtained in Section 3 assuming that the values of Ri, D and V are known at time t0, while their future values are described by the geometric Brownian motions1

Fuzzy R&D game and numerical applications

In previous sections we have presented our model for R&D investment with fuzzy uncertainty in market share and revealed information parameters. In particular, we have established explicit expressions of fuzzy payoffs for different strategic behaviours. We now investigate the existence of pure Nash equilibria. The main result we obtain is that, by considering different levels of uncertainty, different Nash equilibria appear and, furthermore, new strategic outcomes are obtained in comparison with

Concluding remarks

In this study, we have considered an R&D investment game between two firms that have the opportunity to realize their research investment at initial time or to postpone their decision waiting for better market conditions, under uncertainty about the success or the failure of their research plan. The value of this deferring option is not easy to compute since it depends on unobservable variables, such as revealed information and market shares. For this reason, we have modelled the R&D investment

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (41)

  • K. Huisman et al.

    Strategic technology adoption taking into account future technological improvements: a real options approach

    Eur. J. Oper. Res.

    (2004)
  • J. Kong et al.

    Real options in strategic investment games between two asymmetric firms

    Eur. J. Oper. Res.

    (2007)
  • B. Lambrecht et al.

    Real options and preemption under incomplete information

    J. Econ. Dyn. Control

    (2003)
  • T. Maeda

    On characterization of equilibrium strategy of two person zero-sum game with fuzzy payoffs

    Fuzzy Sets Syst.

    (2003)
  • S. Majd et al.

    Time to build, option value and investment decisions

    J. Financ. Econ.

    (1987)
  • D. Paxson et al.

    Rivalry under price and quantity uncertainty

    Rev. Financ. Econ.

    (2005)
  • A. Shenhar et al.

    Toward a typological theory of project management

    Res. Policy

    (1996)
  • V. Vijay et al.

    Matrix game with fuzzy goals and fuzzy payoffs

    Omega

    (2005)
  • A. Azevedo et al.

    Real Options Game Models: A Review

    (2010)
  • P. Carr

    The valuation of sequential exchange opportunities

    J. Finance

    (1988)
  • Cited by (5)

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