# Vertical Cooperation Stability in R&D Programs: A Game Theory Analysis of the Semiconductor Industry

Mathilde Aubry, Yihan Wang

► To cite this version:

Mathilde Aubry, Yihan Wang. Vertical Cooperation Stability in R&D Programs: A Game Theory Analysis of the Semiconductor Industry. Technological Forecasting and Social Change, 2024, 209, pp.123738. 10.1016/j.techfore.2024.123738. hal-04852219

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# Vertical cooperation stability in R&D programs: A game theory analysis of the semiconductor industry

Mathilde Aubry<sup>a, *</sup>, Yihan Wang<sup>b</sup>

## ARTICLE INFO

### Keywords:

Uncertainty

R&D program

Vertical cooperation

Semiconductor

Stability

Cross-cost-sharing

## ABSTRACT

This paper explores the formation mechanisms and stability of vertical cooperation in research and development (R&D) programs between upstream and downstream firms under technological and market uncertainty. Specifically, we concentrate on the dynamics of the semiconductor industry, which is characterized by accelerating technological improvements and an increasingly complex end market. Grounded in game theory, we propose a two-stage model and compare cooperative and noncooperative equilibria. Then, we illustrate the results by referring to a case study of successful vertical cooperation between Google and NXP Semiconductors. We highlight the contribution of vertical cooperation to the success of R&D programs, measured by the probabilities of technological improvement and market acceptance as well as the gross profits of both firms. Meanwhile, to ensure the stability of such an agreement, even in the case of initial collaboration, we advocate the introduction of a cross-cost-sharing mechanism. Finally, we conclude with the strategic implications of maintaining the stability of vertical cooperation in R&D programs in high-tech industries.

## 1. Introduction

Technological breakthroughs in information and communication technologies (ICT) and related industrial sectors propel innovation-driven global economic growth and create new value for various stakeholders in the market (Lisboa et al., 2011; Lu and Weng, 2018; Yalcinkaya et al., 2007). Meanwhile, the proliferating portfolio and the accelerating replacement rate of new technologies increase the uncertainty inherent in the innovation process (Milliken, 1987; Sicotte and Bourgault, 2008; Stockstrom and Herstatt, 2008). To overcome the impact of uncertainty in technological development and market exploration, innovation-oriented enterprises need to proactively adjust and reshape the governance mechanisms of their research and development (R&D) processes (Frank et al., 2019; Kohtamäki et al., 2019; Zott et al., 2011). Subsequently, these enterprises collaborate in R&D programs and build diverse relationships integrated into the open innovation ecosystem to encounter the challenges of technological and market uncertainty (Ahn, 2020; Chesbrough, 2003; Gupta et al., 2019; Lattacher et al., 2021; Sydow and Müller-Seitz, 2020).

In practice, these interorganizational collaborations encompass horizontal alliances among competitors from the same business sector

and vertical cooperation between upstream and downstream firms in the supply chain (Belderbos et al., 2018; Ishii, 2004; Pi and Li, 2022; Yu et al., 2021). Numerous prior studies have concentrated on the prevailing pattern of horizontal alliances in “coopetition” among competing stakeholders (Cozzolino and Rothaermel, 2018; Le Roy et al., 2022; Wilhelm, 2011). However, the high risk of key know-how dissemination and potential conflicts of interest among competitors in horizontal alliances compel firms to seek alternative forms of collaboration (Bahinipati and Deshmukh, 2012, Ge et al., 2021a, 2021b, Pi and Li, 2022, Yu et al., 2021). Vertical cooperation appears relevant, particularly for dealing with uncertainty (Bhaskaran and Krishnan, 2009; Letterie et al., 2008; Wang and Hu, 2020).

Multiple recent studies urge us to continue exploring the incentives, mechanisms, and effects of building vertical cooperation in knowledge-complex industries (Belderbos et al., 2018; Ge et al., 2021a, 2021b; Pi and Li, 2022; Wang and Hu, 2020; Yu et al., 2021). Furthermore, to avoid the loss of high initial investments and the fluctuating valuation of R&D programs, key stakeholders need to maintain the stability of vertical cooperation (Bloom, 2007; Czarnitzki and Toole, 2013; Farrukh et al., 2000; Oriani and Sobrero, 2008; Sicotte and Bourgault, 2008).

In this research, we explore the functioning mechanisms and stability

<footnote><sup>a</sup> Department of Economics, Territories, and Sustainable Development, Chair of “Digitalization and Innovation in Organizations and Territories”, EM Normandie Business School, Métis Lab, 9 Rue Claude Bloch, Caen 14052, France</footnote>

<footnote><sup>b</sup> Department of Strategy and Entrepreneurship, EM Normandie Business School, Métis Lab, 20 Quai Frissard, Le Havre 76600, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: maubry@em-normandie.fr (M. Aubry), yhwang@em-normandie.fr (Y. Wang).</footnote>
of vertical cooperation in R&D programs between upstream and downstream firms in the context of uncertainty. Specifically, we concentrate on the dynamics of the semiconductor industry, which is characterized by accelerating technological improvements and an increasingly complex end market. Indeed, in the digitalization of the global economy, semiconductor component producers manufacture the key products that control the processing and storing/retrieval functions of all ICT products and services (Chiaroni and Chiesa, 2007; Keil et al., 2019; Schneider et al., 2021). Nonetheless, as producers of high-tech intermediary products, semiconductor manufacturers encounter double uncertainty regarding the technological improvement of their own products, as well as the market acceptance of the downstream products (Aubry and Renou-Maissant, 2013; Chiaroni and Chiesa, 2007; Sydow et al., 2012). The recent worldwide semiconductor shortage further highlights the tension in the global supply chain of components affecting many sectors of the economy (Krolikowski and Naggert, 2021; Ramani et al., 2022; Vakil and Linton, 2021).

To cope with rising costs and overcome the challenges associated with uncertainty in R&D programs, the upstream stakeholders, including semiconductor chip components designers, manufacturers, and service providers (e.g., Intel, Samsung Electronics, Toshiba, ST Microelectronics, NXP Semiconductors, etc.) are likely to collaborate with downstream firms (e.g., Apple, Philips, Toyota, Airbus, etc.) that embed semiconductor components in the final products of a wide range of related sectors (e.g., electronic appliances, mobile communications, automobiles, healthcare, professional services, aerospace) (Bahinipati and Deshmukh, 2012; Schubert et al., 2013; Sydow and Müller-Seitz, 2020; Sydow et al., 2012). To understand how vertical cooperation contributes to the success of R&D programs, as well as the conditions required to maintain the agreement's stability, we propose a two-stage game theory model and compare the cooperative and noncooperative equilibria between upstream and downstream firms (D'Aspremont and Jacquemin, 1988; Kamien et al., 1992).

This paper contributes both to the literature on interfirm R&D collaborations and to the literature on the semiconductor industry. We take the initiative to explore the mechanism of vertical cooperation in R&D programs considering technological and market uncertainty, as well as the instability of the agreement. Inspired by the semiconductor industry, this pioneering research applies a mixed method, combining game theory model with a practical case study. The model was built and validated through exchanges with industry professionals. They underlined its relevance and usefulness for setting up such agreements and defending them vis-à-vis national or international institutions (e.g., Important Projects of Common European Interest (IPCEI)<fnref n="1" /> of the European Union).

Among the first papers to analyze the vertical cooperation of R&D programs in the semiconductor industry, we make three major theoretical contributions. First, we demonstrate that cooperative behavior linking semiconductor supply chain partners encourages both to invest in R&D programs. We confirm previous academic results in situations of double uncertainty. Vertical cooperation contributes to the success of R&D programs, limiting the impact of uncertainty and increasing the gross profits of both firms. Second, we explain the rationale for agreement deviation as the source of instability in vertical cooperation. Third, to avoid the instability of vertical cooperation incurred because of agreement deviation, we introduce and analyze the cross-cost-sharing mechanism whereby both upstream and downstream firms invest in technological improvement and market acceptance programs as solutions.

The paper is structured into five parts. Following this introduction (Section 1), the literature review in Section 2 clarifies the positioning of the paper. The model setting and analysis are detailed in Section 3,

followed by a case study in Section 4 to illustrate the main results that have emerged. The strategic implications of the mechanism from both the theoretical and managerial perspectives are discussed in Section 5, which concludes the paper.

## 2. Literature review

### 2.1. Uncertainty and vertical cooperation in R&D programs

Defined as “an individual's perceived inability to predict something accurately,” uncertainty plays a central role in building interorganizational relationships and the success of innovation activities (Bloom, 2007; Czarnitzki and Toole, 2013; Milliken, 1987; Sicotte and Bourgault, 2008). Enterprises build up broad and diverse R&D collaborations to share the initial costs and limit the uncertainty in technological improvement and market acceptance they face in the innovation process (Bhaskaran and Krishnan, 2009; Combs, 1992; Dai et al., 2017; Gupta et al., 2019). More precisely, they establish cost-sharing mechanisms that involve different approaches regarding the sharing of the development cost (investment sharing) and the sharing of the development work (innovation sharing) (Bhaskaran and Krishnan, 2009; Ge et al., 2012a, 2012b; Mukherjee and Ramani, 2011; Wang et al., 2018). Furthermore, these collaborative relationships contribute to the market competitiveness of enterprises in terms of acquiring state-of-the-art knowledge and transforming their R&D outputs into market success (Chesbrough, 2003; Letterie et al., 2008; Sydow and Müller-Seitz, 2020; Wang et al., 2018).

In general, interorganizational collaborations are categorized as horizontal alliances among partners with similar specializations and vertical cooperation between arm's-length suppliers and buyers (Giroud and Scott-Kennel, 2009; Le Roy et al., 2022; Turkina et al., 2016). Considering the difficulties associated with the competitive relationships between companies in horizontal alliances, increasing research advocates for an in-depth understanding of the mechanisms and dynamics of alternative vertical cooperation in R&D activities (Belderbos et al., 2011; Conner and Prahalad, 1996; Crespo et al., 2014; Wallenburg and Schäffler, 2014). Researchers applied game theory to analyze vertical cooperation and its advantages (Atallah, 2002; Banerjee and Lin, 2003; Ge et al., 2014; Karbowski and Prokop, 2019). Through the joint decision-making mechanisms of the supply chain, partners of vertical cooperation can limit uncertainty and improve innovation performance (Bhaskaran and Krishnan, 2009; Letterie et al., 2008; Wang and Hu, 2020). By establishing vertical cooperation, firms can focus on their most advantageous business activities while reciprocally benefitting from their partners via knowledge sharing and capability upgrading (Buckley et al., 2009; Giroud and Scott-Kennel, 2009; Le Roy et al., 2022; Turkina and Van Assche, 2018). In light of this research progress, we advocate further exploration of vertical cooperation dynamics in the context of uncertainty and address their contribution to the success of R&D programs.

### 2.2. Stability of vertical cooperation in R&D programs

Although supply chain partners have strong incentive to establish vertical cooperation in R&D programs, heterogeneous expertise and economic interest can lure partners to break away from the agreement (Casas-Arce et al., 2019; Das and Rahman, 2010; Ge et al., 2021a, 2021b). Each stakeholder can decide to respect or deviate from the agreement to obtain a higher level of individual profit without taking into account the positive externalities for its partner or the society (Crespo et al., 2014; Iacobucci and Perugini, 2021; Tognazzo et al., 2016; Xu et al., 2015). If existing research on game theory has addressed and modeled the problems of instability in multi-party collaboration (Dai et al., 2017; Morasch, 1995; Takemoto and Arizono, 2020; Xu et al., 2015), in the case of vertical cooperation, the situation is specific. Different stakeholders have various expertise and bargaining power.

<footnote n="1">¹ https://competition-policy.ec.europa.eu/state-aid/legislation/mode
rnisation/ipcei_fr#several-approved-ipceis.</footnote>
Upstream firms are mostly oriented toward the technological improvement of the R&D process, while downstream firms tend to focus on market acceptance in channelling diversified products and services to meet the evolving needs of the final consumer market (Pi and Li, 2022; Singer and Donoso, 2008; Un and Asakawa, 2015). Consequently, they do not share the same interests to maintain the stability of the relationship. Downstream firms tend to acquire a technological advantage at the initiative stage of R&D collaboration, while upstream firms may benefit from preferential access to the end user (Bhaskaran and Krishnan, 2009; Dai et al., 2017; Ghosh and Shah, 2015; Yang et al., 2017).

Under a fair and transparent vertical cooperative agreement, each partner should be able to monitor the other's inputs, conduct, and performance to overcome the potential conflicts of interests. If this highly theoretical condition is not met, based on prior collaboration success, the vertical partners can build up long-term mutual commitment partnerships, restrain the opportunistic behavior and ensures the stability of the collaboration (Belderbos et al., 2018; Ge et al., 2021a, 2021b). In contrast, in newly formed vertical cooperation deals, due to insufficient monitoring and sanctioning mechanisms, the threat of agreement deviation exist in real practice (Banerjee and Lin, 2001; Dibiaggio, 2007; Morasch, 1995; Silipo, 2008; Takemoto and Arizono, 2020).

Regardless of the increasing research interest in vertical cooperation in R&D programs, very little research has proposed feasible solutions that avoid agreement deviation and instability from the very first cooperation (Dai et al., 2017; Ge et al., 2021a, 2021b; Mishra and Shah, 2009; Tripsas et al., 1995; Xu et al., 2015; Yu et al., 2021). Given this research gap, our research tackles the crucial issue on how to ensure the stability of the first collaboration.

## 2.3. Semiconductor industry and vertical cooperation R&D program

Semiconductors are high-tech intermediate goods. As such, enterprises in the industry must manage with heavy investment in R&D activities to maintain a high pace of technological improvement. They also have to deal with increasingly diverse and demanding user sectors (Bahinipati and Deshmukh, 2012; Letterie et al., 2008; Sydow and Müller-Seitz, 2020; Sydow et al., 2012).

Characterized by Moore's law<fnref n="2" /> (Moore, 1965), technological advances are leading to the increasing integration of material support functions on a single silicon chip. Semiconductor products are being replaced faster and faster. Patent applications accumulate rapidly, and the lifespans of semiconductor products are short (Chung and Sohn, 2020). Moreover, the “More than Moore” law (Graef, 2021; Zhang et al., 2005) addresses how this emerging trend is partially triggered by increasing social needs for high-level system integration, including non-digital functionalities. The interest is in multiplication but also in diversification of the functions in the final products. These so-called derivative technologies offer many opportunities to companies: they open up a line of less capitalistic research and create buoyant markets in numerous fields, such as transport, health, and safety. Along with the increasing number of end products incorporating semiconductor components, the R&D program development of the semiconductor industry extends its scope from upstream technological innovation to downstream market development (Chiaroni and Chiesa, 2007). Semiconductor industry players must therefore manage technological and market uncertainty (Mousavi et al., 2022; Ramani et al., 2022). They need to rethink the organization of their R&D activities and collaborate in R&D programs (Aubry and Renou-Maissant, 2014; Keil et al., 2019; Sydow and Müller-Seitz, 2020). More precisely, the increasing need to understand customer market dynamics highlights the importance of vertical cooperation in R&D programs (Atallah, 2002; Bahinipati and

Deshmukh, 2012; Cao and Zhang, 2011; Kilkki et al., 2018; Sydow et al., 2012). However, to a large extent, existing research on R&D collaborations in the semiconductor industry focuses on the prevailing pattern of horizontal alliances among competing stakeholders (Cozzolino and Rothaermel, 2018; Le Roy et al., 2022; Wilhelm, 2011). Consequently, numerous recent research studies point out the need to continue exploring the incentives, mechanisms, and effects of building vertical cooperation in the semiconductor industry, a representative knowledge-intensive industry, while taking technological and market uncertainty into account (Belderbos et al., 2018; Ge et al., 2021a, 2021b; Pi and Li, 2022; Yu et al., 2021).

In sum, Table 1 highlights the main themes of the research, its observed gaps, the main arguments, the positioning of the paper, and key references.

# 3. Formation of vertical cooperation: a game theory model

## 3.1. Model setting

In this paper, we analyze the functioning mechanisms and stability of vertical cooperation under uncertainty. Collaborative agreements between upstream and downstream firms contribute to the success of R&D programs, that is, “the degree of success attained by the firm at achieving its goals pertaining to product-market or technological innovation” (Goodale et al., 2011). The success of vertical cooperation in R&D programs can be measured by (1) technological improvement: the innovation performance of new knowledge generation and new product development; and (2) market acceptance: the economic gains and sales revenues in the product market (Bhaskaran and Krishnan, 2009; Ge et al., 2021a, 2021b; Yu et al., 2021). On this basis, we consider that successful vertical cooperation in terms of an R&D program must (1) reduce the uncertainty by enhancing the probabilities of technological improvement and market acceptance and (2) lead to an increase in the gross profits of both firms.

The model proposed in this article incorporates research outcomes of game theory, industrial economics, and innovation management. In the conventional vertical cooperation of semiconductor R&D programs, the partners have different focuses of interest in the innovation process (Bhaskaran and Krishnan, 2009; Chiaroni and Chiesa, 2007). The upstream firms concentrate on technological improvement in optimizing the productivity of technological inputs, in this case, enhancing the performance of intermediary semiconductor components. Thus, they indirectly improve the quality of the final products. In contrast, the downstream firms concentrate on improving the market acceptance of the final products with regard to the consumer market. They adapt the product to the evolving consumer market needs using improved technology and convince the consumers to adopt the new technological applications. Therefore, downstream firms encourage customers to purchase the final products containing semiconductor components. As a result, both partners benefit, directly or indirectly, from the technological improvement and the market acceptance of the R&D program.

Grounded in these arguments, we set up the conceptual model as follows: First, we denote a producer of intermediary semiconductor components as an upstream firm (firm A), and a producer of final products incorporating semiconductor components (e.g., cellphones, automobiles, family electronic appliances, aircrafts) as a downstream firm (firm B). In this model, firm B uses *k* units of intermediary goods to produce a unit of a homogeneous final product, whereas *k* represents the number or the size of the chips that are necessary for a specific function. Therefore, the marginal cost of the product of firm B, *cm*<sub>B</sub>, is affected by *k* and the price of the intermediary good. The miniaturization of the chip size reduces the size of the final product, increases the frequency of operations per second, enhances the number of functions per unit, and reduces the energy consumption of the final product. In sum, a decrease in *k* indicates an improvement in the quality of the final product.

For the upstream firm A aiming at technological improvement, un-

<footnote n="2">² The original Moore's law (1965) argues that the number of transistors per chip doubles every 18 to 24 months thanks to miniaturization.</footnote>
<table><caption>**Table 1**
Research themes and positioning.</caption><thead><tr><th>Research themes</th><th>Research gaps</th><th>Main arguments</th><th>Positioning of the paper</th><th>Key references</th></tr></thead><tbody><tr><td>Uncertainty and vertical cooperation</td><td><ul><li>Limited research on vertical cooperation in R&D programs compared to horizontal alliances</li><li>The context of uncertainty in vertical cooperation in R&D programs is not sufficiently considered.</li></ul></td><td>The technological and market uncertainty, as well as the limitations of horizontal alliances, highlight the increasing importance of understanding the contribution of vertical cooperation to the success of R&D programs.</td><td>Pioneering research that develops a theoretical model based on game theory to analyze the advantages of vertical R&D cooperation in a context of technological and market uncertainty</td><td>Atallah (2002), Banerjee and Lin (2001), Belderbos et al. (2018), Bhaskaran and Krishnan (2009), Crespo et al. (2014), Karbowski and Prokop (2019), Yu et al. (2021)</td></tr><tr><td>Instability and vertical cooperation</td><td><ul><li>Research on the instability of vertical cooperation is not sufficient.</li><li>Insufficient proposition of a solution to maintain the stability of vertical cooperation and resolve the tendency of agreement deviation</li></ul></td><td><ul><li>The agreement deviation from vertical cooperation is economically rational but threatens the overall success of the R&D programs.</li><li>Cross-cost-sharing between vertical partners resolves the tendency of agreement deviation and contributes to the success of R&D programs.</li></ul></td><td><ul><li>In-depth analysis on the sources of instability of vertical cooperation</li><li>Proposition of original cross-cost-sharing solution to agreement deviation</li></ul></td><td>Bhaskaran and Krishnan (2009), Dai et al. (2017), Ge et al. (2021a, 2021b), Ghosh and Shah (2015), Mishra and Shah (2009), Tripsas et al. (1995), Xu et al. (2015), Yang et al. (2017), Yu et al. (2021).</td></tr><tr><td>Semiconductor industry and vertical cooperation R&D program</td><td><ul><li>Insufficient reflection on the characteristics of vertical cooperation in the semiconductor industry</li><li>Limited research on maintaining vertical cooperation stability and R&D program success in the semiconductor industry</li></ul></td><td><ul><li>Considering the importance and specificity of the semiconductor industry in the global economy, it is important to understand the mechanism of vertical cooperation and its contribution to the success of the R&D program.</li><li>To offset the negative impact of the agreement deviation of R&D programs, vertical partners can establish cross-cost-sharing mechanisms in technological and market development.</li></ul></td><td><ul><li>Novel proposition of two-stage game theory model that takes the complex intermediary good, fast-changing life cycle, and high initial R&D investment into account.</li><li>Integration of mixed methods of mathematical modelling with a case study of the industry to illustrate the results</li></ul></td><td>Atallah (2002), Bahinipati and Deshmukh (2012), Cao and Zhang (2011), Chiaroni and Chiesa (2007), Chung and Sohn (2020), D'Aspremont and Jacquemin (1988), Kilkki et al. (2018), Sydow et al. (2012)</td></tr></tbody></table>

certainty affects the value of k. We note that $k = k_1$ if there is no technological improvement (either because the program is not implemented, or it is a failure) and $k = k_2$ if there is a technological improvement (because the program is a success).

For the downstream firm B aiming at market acceptance, the uncertainty affects the level of demand. To illustrate this second source of uncertainty, let us take, for the sake of simplicity, the case of a double monopoly (Aubry, 2012), where firm A is the only upstream firm in the market and firm B is the only downstream firm in the market.

We suppose here, in the classic way, that the inverse demand function of the final goods is linear:

$$p_B(q_B) = a - bq_B \quad (1)$$

where $p_B$ is the price of the final goods, and $q_B$ is the quantity of goods required. Uncertainty affects here the value of a. In this inverse demand function, the constant term a may take two different values: $a = a_1$ if the final product is refused by consumers (no market acceptance) and $a = a_2$ with $a_2 > a_1$ if the final product is accepted by consumers (market acceptance).

With $\Pi_B^{(1)}$ the gross profit of the downstream firm when there is no technological improvement, $\Pi_B^{(2,1)}$ the gross profit of the downstream firm when there is a technological improvement, but the final product is rejected by the market (no market acceptance) and $\Pi_B^{(2,2)}$ the gross profit of the downstream firm when there is a technological improvement and market acceptance, the gross profits of firm B can be presented in the following way:

$$\Pi_B^{(1)} = (p_B^{(1)} - cm_B) q_B^{(1)} \quad (2)$$

$$\Pi_B^{(2,1)} = (p_B^{(2,1)} - cm_B) q_B^{(2,1)} \quad (3)$$

$$\Pi_B^{(2,2)} = (p_B^{(2,2)} - cm_B) q_B^{(2,2)} \quad (4)$$

where $cm_B$, the marginal cost of the downstream firm, can be expressed as follows $cm_B = k p_A$.

$p_A$ is the price of the intermediary good, and k is the amount of intermediary goods necessary to produce one unit of the final products. We assume that the downstream firm has no other operating costs beside this one.

To be more precise,

$$\Pi_B^{(1)} = (p_B^{(1)} - k_1 p_A^{(1)}) q_B^{(1)} \quad (5)$$

$$\Pi_B^{(2,1)} = (p_B^{(2,1)} - k_2 p_A^{(2,1)}) q_B^{(2,1)} \quad (6)$$

$$\Pi_B^{(2,2)} = (p_B^{(2,2)} - k_2 p_A^{(2,2)}) q_B^{(2,2)} \quad (7)$$

Similarly, with $\Pi_A^{(1)}$ the gross profit of the upstream firm when there is no technological improvement, $\Pi_A^{(2,1)}$ the gross profit of the upstream firm when there is a technological improvement, but the final product is rejected by the market (no market acceptance) and $\Pi_A^{(2,2)}$ the gross profit of the upstream firm when there is a technological improvement and market acceptance, the gross profits of firm A can be presented in the following way:

$$\Pi_A^{(1)} = (p_A^{(1)} - cm_A) q_A^{(1)} \quad (8)$$

$$\Pi_A^{(2,1)} = (p_A^{(2,1)} - cm_A) q_A^{(2,1)} \quad (9)$$

$$\Pi_A^{(2,2)} = (p_A^{(2,2)} - cm_A) q_A^{(2,2)} \quad (10)$$

where $q_A^{(1)} = k_1 q_B^{(1)}$ $q_A^{(2,1)} = k_2 q_B^{(2,1)}$ $q_A^{(2,2)} = k_2 q_B^{(2,2)}$.

Where the marginal cost of the upstream firm is noted $cm_A$ and is supposed to be constant.

In the rest of the paper, detailed calculations will be given, notably in the appendix, for the case of a double monopoly, presented above, which serves as an illustration. However, the global reasoning to come could be applied to any markets.

We assume that the results of the R&D program can be characterized
as probabilities of success, which depend on the expenditures of each firm. $\rho$ and $\eta$ are, respectively, the probability of success of the technological improvement if firm A spends $e_1$ and the probability of market acceptance success if firm B spends $e_2$. In the semiconductor industry, increasing expenditure on technological improvement and market acceptance is associated with the success of an R&D program.

Therefore, we propose the inverse functions $e_1 = e_1(\rho)$ and $e_2 = e_2(\eta)$ upon which we make the following assumption.

**Assumption 1.** The probability of success increases at a lower rate than the expenditures.

By convention, the expenditure functions are twice differentiable, increasing with the probability of success, and convex: $e'_1(\rho) \ge 0$, $e'_2(\eta) \ge 0$ and $e''_1(\rho) \ge 0$, $e''_2(\eta) \ge 0$.

Moreover, $e_1(e_2)$ tends to infinity when $\rho(\eta)$ tends to 1.

Finally, $e'_1(0) = 0$ and $e'_1(1) = \infty$; $e'_2(0) = 0$ and $e'_2(1) = \infty$.

Our conceptual model is shown in Fig. 1. The probability of achieving technological improvement is noted as $\rho$ and the probability of market acceptance, under the condition of technological improvement, as $\eta$.

Detailed explanations of the notations are presented in Appendix 1.

Based on a two-stage model grounded in game theory, our aim is to compare the cooperative and non-cooperative equilibria of upstream and downstream firms.

In the first stage, based on their different expertise in the R&D process, the upstream firm and the downstream firm simultaneously choose the level of R&D expenditure to obtain technological improvement and market acceptance. The equilibrium achieved in this first stage may be either a non-cooperative solution (N): the firms do not coordinate their decisions and decide individually on their level of investment; or a cooperative agreement (J): the firms coordinate their decisions and reach an agreement about the level of their investment.

In the second stage, the vertical partners trade their products in the market and maximize their individual profits to determine prices and quantities. We assume here that the upstream firm establishes the price of intermediary goods. The model is solved using the standard backward induction procedure.

## 3.2. Comparisons of equilibria

In our model, we note the gross profits of the upstream firm A and the downstream firm B as $\Pi_A$ and $\Pi_B$ respectively.

If a cooperation agreement is not reached or fails (situation called N), the expected net profits for firms A noted $E(\pi_A^N)$ and B noted $E(\pi_B^N)$ are, according to gross profits, as follows:

$$E(\pi_A^N) = (1 - \rho)\Pi_A^{(1)} + \rho(1 - \eta)\Pi_A^{(2,1)} + \rho\eta\Pi_A^{(2,2)} - e_A(\rho) \quad (11)$$

<figure><img src="image_3.png" /><figcaption>Fig. 1. The conceptual model of vertical cooperation in R&D programs.</figcaption></figure>

$$E(\pi_B^N) = (1 - \rho)\Pi_B^{(1)} + \rho(1 - \eta)\Pi_B^{(2,1)} + \rho\eta\Pi_B^{(2,2)} - e_B(\eta) \quad (12)$$

Partners choose the optimal level of expenditures for the R&D program, maximizing their expected net profits, to $\rho$ for firm A and $\eta$ for firm B.

The first parts of the Eqs. (11) and (12), $((1 - \rho)\Pi_A^{(1)}$ and $(1 - \rho)\Pi_B^{(1)}$ correspond to the situation where firm A does not achieve a technological improvement ($k = k_1$), the second parts of the Eqs. (11) and (12), $(\rho(1 - \eta)\Pi_A^{(2,1)}$ and $\rho\eta\Pi_B^{(2,1)}$ correspond to the situation where firm A achieves a technological improvement but the market rejects the new product ($k = k_2$, $a = a_1$), and the third parts of the Eqs. (11) and (12), $(\rho\eta\Pi_A^{(2,2)}$ and $\rho\eta\Pi_B^{(2,2)}$ correspond to the situation where firm A achieves a technological improvement and the market accepts the final product ($k = k_2$, $a = a_2$). The expenditures in the R&D programs ($e_A(\rho)$ and $e_B(\eta)$) appear each time.

We use $\delta_i^{(1)}$ to represent the difference in gross profit between the situation in which there is a technological improvement, but it is rejected by the market and the situation in which there is no technological improvement, and $\delta_i^{(2)}$ to represent the difference in gross profit between the situation in which there is a technological improvement, and it is accepted by the market, and the situation in which there is a technological improvement, but it is rejected by the market.

Thus we introduce
$$ \delta_i^{(1)} = \Pi_i^{(2,1)} - \Pi_i^{(1)} \\ \delta_i^{(2)} = \Pi_i^{(2,2)} - \Pi_i^{(2,1)} \quad (i = A, B) $$

Thereafter, we reformulate the expected net profits, if a cooperation agreement is not reached or fails (situation called N), for firms A noted $E(\pi_A^N)$ and B noted $E(\pi_B^N)$, according to gross profits, as follows:

$$E(\pi_A^N) = \Pi_A^{(1)} + \rho\delta_A^{(1)} + \rho\eta\delta_A^{(2)} - e_A(\rho) \quad (13)$$

$$E(\pi_B^N) = \Pi_B^{(1)} + \rho\delta_B^{(1)} + \rho\eta\delta_B^{(2)} - e_B(\eta) \quad (14)$$

**Assumption 2.1.** $\delta_A^{(2)} > 0$ and $\delta_B^{(2)} > 0$.

Firms A and B agree to invest in the R&D program only if Assumption 2.1 is respected, i.e., a total success of the R&D program allows an increase of their gross profit.

**Assumption 2.2.** $\delta_A^{(1)} > 0$ and $\delta_B^{(1)} > 0$.

The partners refuse to invest if they suppose that a technological improvement rejected by the market could affect their profits. Moreover, it is possible to demonstrate that these assumptions are true in the case of a double monopoly. Indeed, in Appendix 2, the relationships between the gross profits in each situation is detailed, as well as the quantities produced by each firm and the prices of the product. We demonstrate that $q_B^{(1)} < q_B^{(2,1)} < q_B^{(2,2)}$ and $q_A^{(2,1)} < q_A^{(2,2)}$. However, we cannot reach a conclusion about the relationship between $q_A^{(2,2)}$ or $q_A^{(2,1)}$ and $q_A^{(1)}$. In the same way, $p_A^{(1)} < p_A^{(2,1)} < p_A^{(2,2)}$, $p_B^{(1)} < p_B^{(2,1)}$ and $p_B^{(2,2)} > p_B^{(2,1)}$ but the relationship between $p_B^{(1)}$ and $p_B^{(2,2)}$ depends on the relative effects of the two programs (for the technological improvement $k_2$ or the market acceptance $a_2$). Finally, $\Pi_B^{(2,2)} > \Pi_B^{(2,1)} > \Pi_B^{(1)}$ and $\Pi_A^{(2,2)} > \Pi_A^{(2,1)} > \Pi_A^{(1)}$.

If both firms agree to cooperate and coordinate their investment (situation called J), their expected net joint profit, noted $E(\pi_{A+B}^J)$, is, according to gross profits, as follows:

$$E(\pi_{A+B}^J) = (1 - \rho)(\Pi_A^{(1)} + \Pi_B^{(1)}) + (1 - \eta)\rho(\Pi_A^{(2,1)} + \Pi_B^{(2,1)}) \\ + \rho\eta(\Pi_A^{(2,2)} + \Pi_B^{(2,2)}) - e_1(\rho) - e_2(\eta) \quad (15)$$

Partners choose the optimal level of expenditures for the R&D
program, maximizing the expected net joint profits, to $\rho$ for firm A and $\eta$ for firm B.

The relationships between $\rho$ and $\eta$ are presented in Figs. 2.1 and 2.2 on the equilibrium in the non-cooperative and cooperative scenarios.

Based on Eqs. (A1)–(A2), we demonstrate, in Appendix 3, that the success probabilities of each firm in relation to the success probabilities of the other can be represented by curves A (for firm A) and B (for firm B) in the non-cooperative scenario. Fig. 2.1 represents the non-cooperative equilibrium of R&D program success corresponding to point N.

Based on Eqs. (A5)–(A6), we demonstrate, in Appendix 3, that the success probabilities of each firm in relation to the success probabilities of the other can be represented by curves A*(for firm A) and B* (for firm B) in the cooperative scenario.

Fig. 2.2 represents the cooperative equilibrium of R&D program success corresponding to point J.

The success probability curves of both firms shift toward A* and B*, under the cooperative scenario, at a higher level than curves A and B, under the non-cooperative scenario, (see Fig. 2.2). Subsequently, the equilibrium level of profit shifts from point N to point J.

Comparing the probability of reaching technological improvement ($\rho$) and the probability of market acceptance as ($\eta$) at the cooperative equilibrium ($\rho^J, \eta^J$) with those at the non-cooperative equilibrium ($\rho^N, \eta^N$), we find that $\rho^J > \rho^N$ and $\eta^J > \eta^N$. Therefore, we propose the following:

**Proposition 1.1.** Vertical cooperation formation enhances the probability of the success of R&D programs in terms of technological improvement and market acceptance.

Additionally, we compare the expected gross profits of both firms, noted $E(\Pi_A^N)$ and $E(\Pi_B^N)$, in the scenario of non-cooperation:

$$E(\Pi_A^N) + E(\Pi_B^N) = (\Pi_A^{(1)} + \Pi_B^{(1)}) + \rho^N (\delta_A^{(1)} + \delta_B^{(1)}) + \rho^N \eta^N (\delta_A^{(2)} + \delta_B^{(2)}) \quad (16)$$

and the expected gross joint profit, noted $E(\Pi_{A+B}^J)$, in the scenario of cooperation:

$$E(\Pi_{A+B}^J) = (\Pi_A^{(1)} + \Pi_B^{(1)}) + \rho^J (\delta_A^{(1)} + \delta_B^{(1)}) + \rho^J \eta^J (\delta_A^{(2)} + \delta_B^{(2)}) \quad (17)$$

Since $\rho^J > \rho^N$ and $\eta^J > \eta^N$ and $\delta_A^{(1)}, \delta_B^{(1)}, \delta_A^{(2)}, \delta_B^{(2)}$ are all >0, $E(\Pi_A^N) + E(\Pi_B^N) < E(\Pi_{A+B}^J)$.

<figure><img src="image_4.png" /><figcaption>Fig. 2.1. Non-cooperative equilibrium.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2.2. Cooperative equilibrium.</figcaption></figure>

**Proposition 1.2.** Vertical cooperation formation contributes to the gross profitability of both partners.

For further clarification and to illustrate, we show in Appendix 2 that, in the special case of a double monopoly, the quantities of final goods and the prices of intermediary goods are higher with vertical cooperation. The quantities of intermediary goods exchanged, and the prices of final goods only increase, in a situation of vertical cooperation, when the effects of R&D programs are significant on market acceptance and relatively weaker on technological improvement.

### 3.3. Opportunism and cross-cost-sharing

Although it is generally profitable for both upstream and downstream firms to collaborate in R&D programs, vertical cooperation faces the challenges of agreement deviation (Morasch, 1995; Mukherjee and Ramani, 2011; Takemoto and Arizono, 2020). Without an efficient and transparent monitoring mechanism, each partner may be incentivized to conduct opportunistic behaviors, such as reducing its investment input or free riding on the successful outcomes of others (Ge et al., 2021a, 2021b; Gilbert and Cvsa, 2003).

As a result, the cooperative equilibrium (J) faces the risk of agreement deviation, which can shift the profit level back to the non-cooperative equilibrium (N) (see Appendix 4 for a detailed demonstration of Proposition 2). Therefore, we propose the following:

**Proposition 2.** There is a risk of agreement deviation in vertical cooperation in R&D programs.

To overcome the risk of agreement deviation in vertical cooperation in R&D programs, we propose the cross-cost-sharing mechanism based on implementation theory (Hurwicz and Reiter, 2006; Maskin and Sjöström, 2002). In the original paradigm of implementation theory, partners build up cost-sharing mechanisms, whereby, at the equilibrium level, both of them respect the agreed input-output level, and neither partner has an incentive to deviate (Bhaskaran and Krishnan, 2009; Combs, 1992; Dai et al., 2017; Ghosh and Shah, 2015; Yang et al., 2017). Considering the specificity of the semiconductor industry, we further propose a cross-cost-sharing solution, whereby both upstream and downstream firms engage in the technological improvement and market acceptance programs. Thereafter, we determine the levels of input allocation in both stages and examine the equilibrium output levels
accordingly.

In the cross-cost-sharing model, let $\alpha$ be firm A's share of expenditures on technological improvement and $\beta$ be firm B's share of expenditures on market acceptance. Then, the values of $\alpha$ and $\beta$ are determined such that the new non-cooperative equilibrium is the same as the cooperative one.

In a situation of no cooperation with cross-cost sharing (situation called NJ), the expected net profits for firm A noted $E(\pi_A^{NJ})$ and firm B noted $E(\pi_B^{NJ})$ are, according to gross profits, as follows:

$$E(\pi_A^{NJ}) = \Pi_A(\rho, \eta) - \alpha e_1(\rho) - (1-\beta)e_2(\eta) \quad (18)$$

$$E(\pi_B^{NJ}) = \Pi_B(\rho, \eta) - (1-\alpha)e_1(\rho) - \beta e_2(\eta) \quad (19)$$

As already shown in Appendix 3 (Eqs. (A7) and (A8)), the conditions for a cooperative equilibrium can appear in the following way (this equilibrium is the same as the previous one):

$$e'_1(\rho^J) = (\delta_A^{(1)} + \delta_B^{(1)}) + \eta^J (\delta_A^{(2)} + \delta_B^{(2)}) \quad (20)$$

$$e'_2(\eta^J) = \rho^J (\delta_A^{(2)} + \delta_B^{(2)}) \quad (21)$$

And, as shown in Appendix 5 (Eqs. (A11) and (A12)), the conditions for a cooperative equilibrium with cross-cost sharing can appear in the following way:

$$e'_1(\rho^{NJ}) = \frac{\delta_A^{(1)} + \eta^{NJ} \delta_A^{(2)}}{\alpha} \quad (22)$$

$$e'_2(\eta^{NJ}) = \frac{\rho^{NJ} \delta_B^{(2)}}{\beta} \quad (23)$$

To make partners respect their agreement, neither of them should have an interest in deviating. It is sufficient that $\rho^{NJ} = \rho^J$ and $\eta^{NJ} = \eta^J$ and, thus, that $e'_1(\rho^{NJ}) = e'_1(\rho^J)$ and $e'_2(\eta^{NJ}) = e'_2(\eta^J)$.

Consequently, to avoid agreement deviation, the cross-cost sharing must respect this condition:

$$\alpha = \frac{\delta_A^{(1)} + \eta \delta_A^{(2)}}{(\delta_A^{(1)} + \delta_B^{(1)}) + \eta (\delta_A^{(2)} + \delta_B^{(2)})} \quad (24)$$

$$\beta = \frac{\delta_B^{(2)}}{(\delta_A^{(2)} + \delta_B^{(2)})} \quad (25)$$

**Proposition 3.** A cross-cost-sharing mechanism maintains the stability of vertical cooperation in R&D programs.

To simplify, we can consider $\lambda_1 = \frac{\delta_A^{(1)}}{\delta_A^{(1)} + \delta_B^{(1)}}, \lambda_2 = \frac{\delta_A^{(2)}}{\delta_A^{(2)} + \delta_B^{(2)}},$ and $\varphi = \frac{\delta_A^{(1)} + \delta_B^{(1)}}{\delta_A^{(2)} + \delta_B^{(2)}}$.

Since $\delta_A^{(2)} > 0$ and $\delta_B^{(2)} > 0, 0 \le \lambda_2 \le 1$, and $\varphi > 0$, we find that $\beta = 1 - \lambda_2$ and $\alpha = \frac{\eta \lambda_2 + \varphi \lambda_1}{\eta + \varphi} \alpha \in [\min(\lambda_1, \lambda_2), \max(\lambda_1, \lambda_2)]$.

Hereby, we observe that upstream firms can invest in market acceptance and, vice versa, downstream firms can invest in technological improvement. The degree of cross-segment investment depends on the gross profit ratio between firms A and B or vice versa. We observe that if firm A benefits as much as firm B from the R&D program ($\delta_A^{(1)} = \delta_B^{(1)}, \lambda_1 = \frac{1}{2}$), then firm A will bear half of the expenditures ($\alpha = 1 - \lambda_1 = \frac{1}{2}$). If firm A benefits more than firm B from the R&D program ($\delta_A^{(1)} > \delta_B^{(1)}, \lambda_1 < \frac{1}{2}$), then firm A will pay more than half of the expenses ($\alpha > \frac{1}{2}$). Finally, if firm A benefits less than firm B from the R&D program ($\delta_A^{(1)} < \delta_B^{(1)}, \lambda_1 > \frac{1}{2}$), it will pay less than half of the expenses ($\alpha < \frac{1}{2}$).

Similarly, if firm B benefits as much as firm A from the R&D program ($\delta_A^{(2)} = \delta_B^{(2)}, \lambda_2 = \frac{1}{2}$), then it will bear half of the expenses ($\beta = 1 - \lambda_2 = \frac{1}{2}$). If firm B benefits more than firm A from the R&D program ($\delta_A^{(2)} < \delta_B^{(2)}, \lambda_2 < \frac{1}{2}$), then it will pay more than half of the expenses ($\beta > \frac{1}{2}$). Finally, if firm B benefits less than firm A from the R&D program ($\delta_A^{(2)} > \delta_B^{(2)}, \lambda_2 > \frac{1}{2}$), it will pay less than half of the expenses ($\beta < \frac{1}{2}$).

**Proposition 4.** The more one firm benefits from the R&D program compared to the other firm, the larger the share of the expenditures it must assume to stabilize the agreement.

To better understand how the expenditures should be shared, we can consider the special case where the profit ratio of firms A and B is the same in the three situations:

$$\frac{\Pi_B^{(1)}}{\Pi_A^{(1)}} = \frac{\Pi_B^{(2,1)}}{\Pi_A^{(2,1)}} = \frac{\Pi_B^{(2,2)}}{\Pi_A^{(2,2)}} = \frac{\delta_B^{(2)}}{\delta_A^{(2)}} = \frac{\delta_B^{(1)}}{\delta_A^{(1)}} = \mu \quad (26)$$

This ratio remains constant if the law of demand is linear or, at least, the price elasticity of this demand is constant.

If this is the case, $\lambda_1 = \lambda_2 = \frac{1}{1+\mu} = \alpha, \beta = \frac{\mu}{1+\mu}$, and $\alpha + \beta = 1$. This means that each company will invest as much in technological improvement as in market acceptance.

We observe that the higher $\mu$ is, the bigger $\beta$ and the lower $\alpha$ are. The greater the profit of firm B is compared to that of firm A, the more it must invest in the R&D program to assure the stability of the agreement. Appendix 6 exhibits the application of this in the case of a double monopoly (Aubry, 2012).

## 4. Cross-cost-sharing in practice: vertical cooperation between NXP and Google

To confirm the practical implications of the model and illustrate the cross-cost-sharing mechanism, we present a real case study of the successful vertical cooperation between the Dutch semiconductor designer and manufacturer NXP Semiconductors as the upstream firm and the American smart phone developer Google as the downstream firm. Indeed, the internet giant has developed a smart cellphone, the NEXUS STM, under the Android Gingerbread operating system.

At the end of 2010, NXP Semiconductors and Google agreed to collaborate in the R&D program. The aim was then to integrate near-field communication (NFC) technology, which allows contactless mobile services, into the Android Gingerbread operating system. To this end, a technological brick needed to be developed around NFC technology (technological improvement), and efforts were required to attract end users to the new services offered by the phone, such as contactless payment (market acceptance).

If this innovation project that incorporates cutting-edge new technology in new product development is highly promising, there is also a high degree of uncertainty since neither technological improvement nor market acceptance of the services can be guaranteed. The vertical cooperation, incorporating the technological expertise of NXP in semiconductor component development with Google's market knowhow in new smartphone development, appears to be a helpful strategy for reducing uncertainty and convincing companies to undertake this project. However, each partner may have the incentive to break away from the deal, ensuring the existing gains at the cost of the unforeseen success of the R&D program. On the one hand, if NXP invests in the technological brick, and Google does nothing to secure a large customer base, both sides will bear the final loss of unsuccessful commercialization of the phone. On the other hand, if Google promotes a smartphone that incorporates NFC technology to its customers, but ultimately the chip cannot be integrated due to insufficient effort on the part of NXP, both sides will bear the failure of the R&D program.

The upstream firm and the downstream firm have organized themselves differently, which has helped to avoid agreement deviation. They
set up a cross-cost-sharing mechanism informally. From a technological improvement point of view, Google has placed a high value on NXP technology by working on NFC integration and validation within the Android Gingerbread operating system. Thus, the downstream firm takes part in the technological development of the components, leading its own teams to work on NFC technology and its compatibility with the Android system. In return, regarding the market acceptance perspective, to attract end users, NXP teams based at the Effiscience Campus in Colombelles (Caen-Calvados) have provided an open software library in Google's Android environment, allowing the development of applications by any developer in the world (open source). This opens the door to the introduction of new mobile services: payment, online tickets, and information. The multiplication of possible applications is a way of ensuring the sales of the phone and, by extension, of the integrated components. By crossing their efforts and investments, the firms not only ensured the stability of the agreement but also the success of the R&D program.

The commercial success of this cooperation has been demonstrated by the fact that the Android platform dominates the market after becoming the leading operating system for smartphones in the United States in the first quarter of 2011, with 37 % of the market share. In terms of revenue, NXP became the number one smart card chip company in 2013, with a 30 % market share, according to the analyst firm ABI Research, thanks in part to the company's success with its NFC solutions and embedded security technologies.

This case highlights how technological and market uncertainty drive the formation of vertical cooperation in the R&D program and how these vertical partners can establish cross-cost-sharing mechanisms to diminish the tendency of agreement deviation.

To deepen the lessons learned from the above example, we conducted semi-structured interviews with industry professionals. We interviewed technical managers and partnership managers from companies designing and manufacturing semiconductors, as well as technicians and marketing managers from semiconductor user firms in various sectors (healthcare, automotive, and agriculture). These interviews were supplemented by exchanges with political makers involved in dynamizing the microelectronics industry. They were asked to describe their perceptions of R&D collaborations, and then to specify the advantages and limitations they saw in vertical R&D cooperation. Finally, the cross-cost-sharing mechanisms and examples were presented to them, and we gathered their feedback. They emphasized that the stability of vertical cooperation serves as a crucial condition for successful collaboration in R&D programs. High-tech start-ups are, in particular, highly vulnerable to uncertainty linked to technological and market developments, as well as to the instability of the agreement. They underlined that by building up cross-cost-sharing mechanisms, firms not only share costs and uncertainty but also engage in and monitor the activities of their partners to reduce the tendency of agreement deviation. For policymakers, encouraging the formation of a cross-cost-sharing mechanism among key stakeholders improves the transparency and efficiency of local R&D program incubation and acceleration, thereby strengthening the robustness and competitiveness of local innovation ecosystems (e.g., the Important Projects of Common European Interest (IPCEI) in the European Union).

## 5. Discussion and conclusions

This paper develops a holistic framework on how vertical cooperation contributes to the success of R&D programs and addresses how to maintain the stability of vertical cooperation in the context of technological and market uncertainties. Focusing on the specific innovation ecosystem of the semiconductor industry, we apply a two-stage game theory model to a cooperation agreement between upstream and downstream firms. It allows us to highlight the strengths and weaknesses of vertical cooperation in R&D programs. Concerning the instability of vertical cooperation incurred due to agreement deviation, we introduce

the cross-cost-sharing mechanism regarding technological improvement and market acceptance. Based on the model analysis, we claim the following three major contributions in terms of theoretical development and managerial practices.

First, we reiterate the contribution of vertical cooperation to the success of R&D programs under technological and market uncertainty. Based on the analysis of the upstream-downstream relationship in the semiconductor industry, we argue that vertical cooperation contributes to the positive synergy effect on the total profit of the partners, which is higher than the sum of the separate profits of upstream and downstream players that do not collaborate (Ge et al., 2021a, 2021b; Le Roy et al., 2022; Veugelers, 1998). In addition, vertical cooperation enhances the technological and commercial success of R&D programs by limiting uncertainty (Goodale et al., 2011; Karbowski and Prokop, 2019; Letterie et al., 2008). Furthermore, overcoming the risks of key know-how exploitation and the lock-in effect in horizontal alliances, vertical partnership extends the complementary technological and market knowledge bases among partners and eliminates the potential conflicts of interests of knowledge sharing among direct competitors (Bhaskaran and Krishnan, 2009; Ge et al., 2021a, 2021b; Wang and Hu, 2020). Specifically, considering the fast-changing innovation dynamics of the semiconductor industry, we suggest that intermediary component suppliers and their buyers may tap into each other's specialized fields in vertical cooperation to reduce the impact uncertainties on the R&D program in question.

Second, we highlight the sources of the instability of vertical cooperation in R&D programs. We argue that the agreement deviation tendency is rational on an individual basis but creates a new form of uncertainty for the partner companies and reduces the overall success of R&D programs. To prevent the negative impact of opportunistic behavior, both partners must monitor and anticipate each other's intent to maintain their cooperation. We show here that, without cross-cost-sharing, the stability of the vertical cooperation is dependent on trust and commitment (Dai et al., 2017; Ge et al., 2021a, 2021b; Mishra and Shah, 2009; Yu et al., 2021). From a management perspective, due to the high commitment of resources, the uncertainty of innovation performance, and the sensitivity of know-how diffusion, stable trusting relationships between partners are essential. However, the rapidly evolving industrial dynamics that are associated with the strong uncertainty linked to technological and commercial development do not allow partners to sustain long-term mutual commitment. Thereafter, business managers need to raise their awareness of the relational stability of vertical cooperation under uncertain conditions (Aubry and Renou-Maissant, 2013, 2014; Ge et al., 2021a, 2021b).

Third, we introduce the cross-cost-sharing mechanism as a solution to stabilize vertical cooperation and to prevent agreement deviation in R&D program development. Balancing their efforts and gains, the upstream firms can intervene in the market acceptance effort, whereas the downstream firm can also penetrate the technological improvement effort (Gupta et al., 2019). Moreover, the game theory analysis underlines that the cross-cost-sharing mechanism also balances the bargaining power in vertical cooperation. In response to the asymmetry of interest distribution in forming vertical cooperation, the cross-cost-sharing mechanism moderates the benefit acquisition, so that neither party can abuse the dominant bargaining power in value capture to infringe the rights of the other (Dai et al., 2017; Pi and Li, 2022).

Although the semiconductor industry provides a representative scenario of a fast-changing industry, the generalization of the conclusions should take the industrial heterogeneity into consideration in comparison with other knowledge-intensive industries (e.g., biotechnology, advanced manufacturing, artificial intelligence, high-end professional services). Moreover, the theoretical model only simulates a non-repeating, one-step game as neither party intends to collaborate again after the agreement deviation. However, if both parties repeated the agreement negotiations and established penalty mechanisms for non-collaborative behavior, the incentives and outcomes would be very
different. Whereas the span of vertical cooperation in semiconductor R&D programs can be very short in the form of non-repetitive games, other industries with longer cycles of R&D programs may adapt to multi-stage repetitive games.

Regarding future research, it would be possible to extend our model and analyze other types of agreement such as joint monopoly, for example. Instead of assuming, as before, that the price of the intermediary good is determined non-cooperatively by the upstream firm, we can imagine that the two firms agree at the second stage to maximize their joint operating profit. It would also be interesting to extend the paper by proposing an analysis of upstream and downstream competition. Confronting our results with oligopolistic situations could reinforce them and raise new questions about the relative importance of R&D program results and the balance of bargaining power of each type of firm. In addition, future studies can specify the structure of the shareholding between partners in vertical cooperation to ensure the stability of the cooperative equilibrium. Indeed, cross-shareholdings can facilitate R&D cooperation between vertically linked firms. Meanwhile, upcoming articles can introduce social welfare analysis to observe the impact of vertical cooperation on the benefits of partners under various competition scenarios (e.g., perfect competition, bilateral monopoly, oligopoly). We also suggest further analyzing the different incentive mechanisms in repetitive games in vertical cooperation negotiation in industries with longer R&D program lifespans.

## Ethic statement

Not applicable

## CRediT authorship contribution statement

**Mathilde Aubry:** Conceptualization, Formal analysis, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing, Data curation, Investigation. **Yihan Wang:** Conceptualization, Formal analysis, Validation, Writing – original draft, Writing – review & editing.

## Data availability

The data that support the findings of this study are available from the corresponding author, M. Aubry, upon reasonable request.

## Acknowledgement

Both authors contributed to the work equally. There is no conflict of interests. We would like to thank Professor Bernard Franck for his guidance and comments on this model.

This research was initiated as part of a thesis funded by the company NXP Semiconductors and then supported by the EM Normandie Business School, Métis Lab as part of the work carried out within the Digitalisation and Innovation within Organisations and Territories chair.

# Appendix A

## Appendix 1

Notations.

Π: Gross profit (in uppercase)

π: Net profit (in lower case)

A: Producer of intermediary semiconductor components
Upstream firm (firm A)

B: Producer of final products incorporating semiconductor components (e.g., cellphones, automobiles, family electronic appliances, aircrafts)
Downstream firm (firm B)

k₁: Number of intermediary goods necessary to produce a unit of the final product if there is no technological improvement

k₂: Number of intermediary goods necessary to produce a unit of the final product if there is a technological improvement

pA: Price of the intermediary good

pB: Price of the final product

qA: Quantity of intermediary goods

qB: Quantity of final products

cmA: Marginal cost of the intermediary good

cmB: Marginal cost of the final product

e₁: Expenditure on technological improvement

e₂: Expenditure on market acceptance

E: Total expenditure on technological improvement and market acceptance E = e₁(ρ) + e₂(η)

ρ: Probability of the success of a technological improvement program if firm A invests e₁

η: Probability of success of a market acceptance program if firm B invests e₂

a₁: Constant term in the demand function if the final product is rejected by consumers (no market acceptance)

a₂: Constant term in the demand function if the final product is accepted by consumers (market acceptance)

ΠA(1): Gross profit of the upstream firm when there is no technological improvement

ΠB(1): Gross profit of the downstream firm when there is no technological improvement

ΠA(2,1): Gross profit of the upstream firm when there is a technological improvement, but the final product is rejected by the market (no market acceptance)

ΠB(2,1): Gross profit of the downstream firm when there is a technological improvement, but the final product is rejected by the market (no market acceptance)

ΠA(2,2): Gross profit of the upstream firm when there is a technological improvement and market acceptance

ΠB(2,2): Gross profit of the downstream firm when there is a technological improvement and market acceptance

δI(1): Difference in gross profit between the situation in which there is a technological improvement, but it is rejected by the market and the situation in which there is no technological improvement

δI(2): Difference in gross profit between the situation in which there is a technological improvement, and it is accepted by the market, and the situation in which there is a technological improvement, but it is rejected by the market

πA, πB: Net profits of each firm if there is no cooperation

πA, πB: Net profits of each firm if there is no cooperation but cross-cost-sharing

πA+B: Net joint profit of the upstream and downstream firms if the firms cooperate
## I.1. Production and prices analysis

The gross profits of firm B are as follows:

$$ \Pi_B^{(1)} = (p_B^{(1)} - k_1 p_A^{(1)}) q_B^{(1)} $$

$$ \Pi_B^{(2,1)} = (p_B^{(2,1)} - k_2 p_A^{(2,1)}) q_B^{(2,1)} $$

$$ \Pi_B^{(2,2)} = (p_B^{(2,2)} - k_2 p_A^{(2,2)}) q_B^{(2,2)} $$

Knowing

$$ p_B^{(1)} = a_1 - b q_B^{(1)} $$

$$ p_B^{(2,1)} = a_1 - b q_B^{(2,1)} $$

$$ p_B^{(2,2)} = a_2 - b q_B^{(2,2)} $$

firm B establishes the price of the final products maximizing its profit according to the price of the intermediary goods.

$$ \text{We obtain } p_B^{(1)} = \frac{a_1 + p_A^{(1)} k_1}{2} $$

$$ p_B^{(2,1)} = \frac{a_1 + p_A^{(2,1)} k_2}{2} $$

$$ p_B^{(2,2)} = \frac{a_2 + p_A^{(2,2)} k_2}{2} $$

$$ \text{and } q_B^{(1)} = \frac{(a_1 - k_1 p_A^{(1)})}{2b} $$

$$ q_B^{(2,1)} = \frac{(a_1 - k_2 p_A^{(2,1)})}{2b} $$

$$ q_B^{(2,2)} = \frac{(a_2 - k_2 p_A^{(2,2)})}{2b} $$

The gross profits of firm A are as follows:

$$ \Pi_A^{(1)} = (p_A^{(1)} - cm_A) q_A^{(1)} \quad \Pi_A^{(2,1)} = (p_A^{(2,1)} - cm_A) q_A^{(2,1)} $$

$$ \Pi_A^{(2,2)} = (p_A^{(2,2)} - cm_A) q_A^{(2,2)} $$

$$ \text{where } q_A^{(1)} = k_1 q_B^{(1)} $$

$$ q_A^{(2,1)} = k_2 q_B^{(2,1)} $$

$$ q_A^{(2,2)} = k_2 q_B^{(2,2)} $$

Firm A establishes the price of the intermediary goods, maximizing its profit according to this price:

$$ \text{The prices are as follows: } p_A^{(1)} = \frac{a_1 + cm_A k_1}{2k_1} $$

$$ p_A^{(2,1)} = \frac{a_1 + cm_A k_2}{2k_2} $$

$$ p_A^{(2,2)} = \frac{a_2 + cm_A k_2}{2k_2} $$

Replacing $p_A$ in $q_B$ and $p_B$, we obtain

$$ q_B^{(1)} = \frac{a_1 - cm_A k_1}{4b} $$

$$ q_B^{(2,1)} = \frac{a_1 - cm_A k_2}{4b} $$

$$ q_B^{(2,2)} = \frac{a_2 - cm_A k_2}{4b} $$

Thus,

$$ p_B^{(1)} = \frac{3a_1 + cm_A k_1}{4} $$

$$ p_B^{(2,1)} = \frac{3a_1 + cm_A k_2}{4} $$

$$ p_B^{(2,2)} = \frac{3a_2 + cm_A k_2}{4} $$

Therefore,

$$ q_B^{(2,2)} - q_B^{(2,1)} = \frac{a_2 - a_1}{4b}, \text{ and thus, } q_B^{(2,2)} > q_B^{(2,1)} \text{ since } a_2 > a_1 $$

$$ q_B^{(2,1)} - q_B^{(1)} = \frac{cm_A k_1 - cm_A k_2}{4b}, \text{ and thus, } q_B^{(2,1)} > q_B^{(1)} \text{ since } k_1 > k_2 $$

$$ q_B^{(2,2)} - q_B^{(1)} = \frac{(a_2 - a_1) + (cm_A k_1 - cm_A k_2)}{4b}, \text{ and thus, } q_B^{(2,2)} > q_B^{(1)} \text{ since } a_2 > a_1 \text{ and } k_1 > k_2 $$

$$ \text{Finally, } q_B^{(1)} < q_B^{(2,1)} < q_B^{(2,2)} $$

As $q_A^{(1)} = k_1 q_B^{(1)}$, $q_A^{(2,1)} = k_2 q_B^{(2,1)}$, $q_A^{(2,2)} = k_2 q_B^{(2,2)}$, we can conclude that $q_A^{(2,1)} < q_A^{(2,2)}$ since $q_B^{(2,1)} < q_B^{(2,2)}$. However, we cannot reach a conclusion about the relationship between $q_A^{(2,2)}$ or $q_A^{(2,1)}$ and $q_A^{(1)}$ since $k_2 < k_1$. The evolution of the quantities of the intermediary goods depends on the relative importance of the technological improvement ($k_2 - k_1$) and of the demand enhancement ($a_2 - a_1$).

The expected quantities of the final products produced by B, called $E(q_B^N)$ if a cooperation agreement is not reached or fails and $E(q_B^I)$ if both firms agree to cooperate and coordinate their investment, are given by the following:

$$ E(q_B^N) = q_B^{(1)} + \rho^N (q_B^{(2,1)} - q_B^{(1)}) + \rho^N \eta^N (q_B^{(2,2)} - q_B^{(2,1)}) $$

$$ E(q_B^I) = q_B^{(1)} + \rho^I (q_B^{(2,1)} - q_B^{(1)}) + \rho^I \eta^I (q_B^{(2,2)} - q_B^{(2,1)}) $$

$$ \text{As } \rho^N < \rho^I \text{ and } \eta^N < \eta^I, E(q_B^N) < E(q_B^I) \text{ since } q_B^{(1)} < q_B^{(2,1)} < q_B^{(2,2)}. $$

The expected quantities of the intermediary goods produced by firm A, called $E(q_A^N)$ if a cooperation agreement is not reached or fails and $E(q_A^I)$ if both firms agree to cooperate and coordinate their investment, are given by the following:

$$ E(q_A^N) = q_A^{(1)} + \rho^N (q_A^{(2,1)} - q_A^{(1)}) + \rho^N \eta^N (q_A^{(2,2)} - q_A^{(2,1)}) $$

$$ E(q_A^I) = q_A^{(1)} + \rho^I (q_A^{(2,1)} - q_A^{(1)}) + \rho^I \eta^I (q_A^{(2,2)} - q_A^{(2,1)}) $$

## Appendix 2. Illustration with the case of a double monopoly (Part 1): analysis of profits, quantities produced, and prices.
$E(q_A^N) < E(q_A^I)$ if the demand enhancement in situation (2, 2), is significant and the technological improvement, in situations (2, 1) and (2,2), is relatively limited.

If concerning the prices of the intermediary goods, the analysis is clear: $p_A^{(1)} < p_A^{(2,1)} < p_A^{(2,2)}$ (since $q_B^{(1)} < q_B^{(2,1)} < q_B^{(2,2)}$), it is more ambiguous for the final products. Indeed, on the one hand, the decrease in the need for intermediary goods between situations (1) and (2, 1) causes lower prices; on the other hand, the increase in the demand between situations (2, 1) and (2, 2) causes higher prices.

The prices of the final products are as follows:

$$p_B^{(1)} = \frac{3a_1+cm_A k_1}{4} \qquad p_B^{(2,1)} = \frac{3a_1+cm_A k_2}{4} \qquad p_B^{(2,2)} = \frac{3a_2+cm_A k_2}{4}$$

$$p_B^{(2,2)} - p_B^{(2,1)} = \frac{3(a_2-a_1)}{4}, \text{ and thus, } p_B^{(2,2)} > p_B^{(2,1)} \text{ since } a_2 > a_1$$

$$p_B^{(2,1)} - p_B^{(1)} = \frac{cm_A(k_2-k_1)}{4}, \text{ and thus, } p_B^{(2,1)} < p_B^{(1)} \text{ since } k_2 < k_1$$

$$p_B^{(2,2)} - p_B^{(1)} = \frac{3(a_2-a_1)+cm_A(k_2-k_1)}{4}, \text{ and knowing that } a_2 > a_1 \text{ and } k_2 < k_1. \text{ It is impossible to reach a conclusion here.}$$

Thus, $p_B^{(2,1)} < p_B^{(1)}$ and $p_B^{(2,2)} > p_B^{(2,1)}$.

The relationship between $p_B^{(1)}$ and $p_B^{(2,2)}$ depends on the relative importance of the technological improvement and the demand enhancement: if the impact of the R&D program in terms of increasing the demand parameter $a$ is high and the impact of the R&D program in terms of reducing $k$ is low, then $p_B^{(2,2)} > p_B^{(1)}$.

The expected prices of the final products produced by firm B, called $E(p_B^N)$ if a cooperation agreement is not reached or fails and $E(p_B^I)$ if both firms agree to cooperate and coordinate their investment, are as follows:

$$E(p_B^N) = p_B^{(1)} + \rho^N(p_B^{(2,1)} - p_B^{(1)}) + \rho^N \eta^N (p_B^{(2,2)} - p_B^{(2,1)})$$

$$E(p_B^I) = p_B^{(1)} + \rho^I(p_B^{(2,1)} - p_B^{(1)}) + \rho^I \eta^I (p_B^{(2,2)} - p_B^{(2,1)})$$

$$p_B^{(2,2)} - p_B^{(2,1)} > p_B^{(2,1)} - p_B^{(1)} \text{ only if } p_B^{(2,2)} > p_B^{(1)} \text{ (see above). In this case, } E(p_B^N) \text{ may be inferior to } E(p_B^I).$$

The expected prices of the intermediary goods produced by firm A, called $E(p_A^N)$ if a cooperation agreement is not reached or fails and $E(p_A^I)$ if both firms agree to cooperate and coordinate their investment, are as follows:

$$E(p_A^N) = p_A^{(1)} + \rho^N(p_A^{(2,1)} - p_A^{(1)}) + \rho^N \eta^N (p_A^{(2,2)} - p_A^{(2,1)})$$

$$E(p_A^I) = p_A^{(1)} + \rho^I(p_A^{(2,1)} - p_A^{(1)}) + \rho^I \eta^I (p_A^{(2,2)} - p_A^{(2,1)})$$

If it is clear $E(p_A^I) > E(p_A^N)$, $E(p_B^I) > E(p_B^N)$ only if the demand enhancement is important and the technological improvement relatively low.

## 1.2. Profits analysis

The gross profits of firms B and A are as follows:

$$\Pi_B^{(1)} = \frac{(a_1-cm_A k_1)^2}{16b} \qquad \Pi_B^{(2,1)} = \frac{(a_1-cm_A k_2)^2}{16b} \qquad \Pi_B^{(2,2)} = \frac{(a_2-cm_A k_2)^2}{16b}$$

$$\Pi_A^{(1)} = \frac{(a_1-cm_A k_1)^2}{8b} \qquad \Pi_A^{(2,1)} = \frac{(a_1-cm_A k_1)^2}{8b} \qquad \Pi_A^{(2,2)} = \frac{(a_2-cm_A k_2)^2}{8b}$$

$$\text{We notice that } \frac{\Pi_B^{(1)}}{\Pi_A^{(1)}} = \frac{1}{2} \qquad \frac{\Pi_B^{(2,1)}}{\Pi_A^{(2,1)}} = \frac{1}{2} \qquad \frac{\Pi_B^{(2,2)}}{\Pi_A^{(2,2)}} = \frac{1}{2}$$

$$\delta_A^{(2)} = \frac{(a_2-cm_A k_2)^2}{8b} - \frac{(a_1-cm_A k_2)^2}{8b} \qquad \delta_B^{(2)} = \frac{(a_2-cm_A k_2)^2}{16b} - \frac{(a_1-cm_A k_2)^2}{16b}$$

$$\delta_A^{(1)} = \frac{(a_1-cm_A k_2)^2}{8b} - \frac{(a_1-cm_A k_1)^2}{8b} \qquad \delta_B^{(1)} = \frac{(a_1-cm_A k_2)^2}{16b} - \frac{(a_1-cm_A k_1)^2}{16b}$$

We know that $a_2 - a_1 > 0$ and $k_1 - k_2 > 0$.

These results confirm that $\Pi_B^{(1)} < \Pi_B^{(2,1)} < \Pi_B^{(2,2)}$ and $\Pi_A^{(1)} < \Pi_A^{(2,1)} < \Pi_A^{(2,2)}$ thus $\delta_A^{(1)} > 0$, $\delta_A^{(2)} > 0$, $\delta_B^{(1)} > 0$, $\delta_B^{(2)} > 0$.

# Appendix 2. (continued).

# Appendix 3

## Non-cooperative and cooperative equilibria: proof of Proposition 1.1.

**If there is no cooperation**, partners choose the optimal level of expenditures for the R&D program, maximizing their expected net profits, noted respectively $E(\pi_A^N)$ and $E(\pi_B^N)$, to $\rho$ for firm A and $\eta$ for firm B.

For firm A, the first-order condition for a maximum is as follows:

$$\frac{\partial E(\pi_A^N)}{\partial \rho} = \delta_A^{(1)} + \eta \delta_A^{(2)} - e_1'(\rho) = 0 \quad (A1)$$

From this follows the best response function for firm A: $\rho = \rho(\eta, \Pi_A^{(1)}, \Pi_A^{(2,1)}, \Pi_A^{(2,2)})$.

Totally differentiating Eq. (A1) with respect to $\rho$ and $\eta$, we obtain $\frac{d\rho}{d\eta} = \frac{\delta_A^{(2)}}{e_1'(\rho)}$.

The best response function $\rho(\eta)$ has a positive slope since $e_1$ is a convex function of $\rho$.

The same reasoning holds for firm B. The first-order condition for a maximum is as follows:

(continued on next page)
## Appendix 3 (continued)

$$ \frac{\partial E(\pi_B^N)}{\partial \eta} = \rho \delta_B^{(2)} - e_2'(\eta) = 0 \quad (A2) $$

From this follows the best response function for firm B: $\eta = \eta(\rho, \Pi_B^{(1)}, \Pi_B^{(2,1)}, \Pi_B^{(2,2)})$.

Totally differentiating Eq. (A2) with respect to $\rho$ and $\eta$, we obtain $\frac{\partial \eta}{\partial \rho} = \frac{\delta_B^{(2)}}{e_2'}$.

The best response function $\eta(\rho)$ has a positive slope since $e_2$ is a convex function of $\eta$.

The stability condition is respected if and only if $(\frac{d\eta}{d\rho})_A > (\frac{d\eta}{d\rho})_B$, thus $(\frac{d\rho}{d\eta})_A (\frac{d\eta}{d\rho})_B < 1$.

The probabilities of success are functions of technological improvement and market acceptance expenditures respectively. They are strategic complements. Thus, an increase in technological improvement expenditures brings about an increase in market acceptance expenditures and vice versa.

The values of $\rho^N$ and $\eta^N$ corresponding to the non-cooperative equilibrium are obtained as follows using Eqs. (A1) and (A2):

$$ e_1'(\rho^N) = \delta_A^{(1)} + \eta^N \delta_A^{(2)} \quad (A3) $$

$$ e_2'(\eta^N) = \rho^N \delta_B^{(2)} \quad (A4) $$

If the firms cooperate, partners choose the optimal level of expenditures for the R&D program, maximizing their expected net joint profit, noted $E(\pi_{A+B}^j)$ to $\rho$ for firm A and $\eta$ for firm B.

The first-order condition for a maximum, with respect to $\rho$, is Eq. (A5) and, with respect to $\eta$, Eq. (A6):

$$ \frac{\partial E(\pi_{A+B}^j)}{\partial \rho} = \frac{\partial E(\pi_A)}{\partial \rho} + \frac{\partial E(\pi_B)}{\partial \rho} - e_1'(\rho) \quad (A5) $$

$$ \frac{\partial E(\pi_{A+B}^j)}{\partial \eta} = \frac{\partial E(\pi_A)}{\partial \eta} + \frac{\partial E(\pi_B)}{\partial \eta} - e_2'(\eta) \quad (A6) $$

The values of $\rho^j$ and $\eta^j$ corresponding to the cooperative equilibrium are obtained as follows using Eqs. (A5) and (A6):

$$ e_1'(\rho^j) = (\delta_A^{(1)} + \delta_B^{(1)}) + \eta^j (\delta_A^{(2)} + \delta_B^{(2)}) \quad (A7) $$

$$ e_2'(\eta^j) = \rho^j (\delta_A^{(2)} + \delta_B^{(2)}) \quad (A8) $$

Graphically, Eqs. (A3) and (A4) are represented by curves A (for firm A) and B (for firm B) in Fig. 2.1 in the main text.

Similarly, Eqs. (A7) and (A8) are represented by curves A* (for firm A) and B* (for firm B) in Fig. 2.2 in the main text.

The cooperative equilibrium corresponds to point J, while the non-cooperative equilibrium corresponds to point N.

Eqs. (A3)–(A4) and (A7)–(A8) highlight that, as $\delta_B^{(1)} > 0$ and $\delta_B^{(2)} > 0$, curve A* is on the right of curve A, because for a fixed $\eta$, $\rho$ is higher for curve A* than for curve A, and curve B* is above curve B (for the same reason).

Therefore, $\rho^j > \rho^N$ and $\eta^j > \eta^N$.

## Appendix 4

Risk of deviation: proof of Proposition 2.

Suppose that firm B honors the agreement and chooses a level of market acceptance expenditure equal to $e_2'(\eta^j)$. Firm A has a choice between two possible behaviors.

Either it respects anyway the agreement and chooses its level of effort maximizing the net joint profit. Then, the level of expenditure is determined by $e_1'(\rho^j) = (\delta_A^{(1)} + \delta_B^{(1)}) + \eta^j (\delta_A^{(2)} + \delta_B^{(2)})$.

Or it chooses to take advantage of the fact that firm B cannot check the level of effort. It chooses its level of effort maximizing its net individual profit. Then, the level of expenditure is determined by $e_1'(\rho^N) = \delta_A^{(1)} + \eta^j \delta_A^{(2)}$.

If firm A acts in its own interest, the level of expenditure is lower than the optimal amount because $e_1'(\rho^j) > e_1'(\rho^N)$.

Moreover, the gross profit of each is positively linked to its partner's effort: $\frac{\partial E(\Pi_A)}{\partial \eta} > 0$ and $\frac{\partial E(\Pi_B)}{\partial \rho} > 0$. If it does not respect the agreement firm A harms firm B and vice versa.

## Appendix 5

The non-cooperative equilibrium with cross-cost sharing.

With cross-cost-sharing, if there is no cooperation, firms A and B choose the optimal level of expenditure for their project, maximizing their expected net profits to $\rho$ for firm A and $\eta$ for firm B.

For firm A, with cross-cost-sharing the first-order condition for a maximum is:

$$ \frac{\partial E(\pi_A^{NJ})}{\partial \rho} = \delta_A^{(1)} + \eta \delta_A^{(2)} - \alpha e_1'(\rho) = 0 \quad (A9) $$

The same reasoning holds for firm B.

The first-order condition for a maximum is:

$$ \frac{\partial E(\pi_B^{NJ})}{\partial \eta} = \rho \delta_B^{(2)} - \beta e_2'(\eta) = 0 \quad (A10) $$

The values of $\rho^{NJ}$ and $\eta^{NJ}$ corresponding to the non-cooperative equilibrium with cross-cost-sharing are obtained using

(continued on next page)
## Appendix 5 (continued)

Eqs. (A9) and (A10).

$$ \alpha e'_{1}(\rho^{NJ}) = \delta_{A}^{(1)} + \eta^{NJ}\delta_{A}^{(2)} \quad (A11) $$

$$ \rho e'_{2}(\eta^{NJ}) = \rho^{NJ}\delta_{B}^{(2)} \quad (A12) $$

## Appendix 6

Illustration with the case of a double monopoly (Part 2): cross-cost-sharing.

As demonstrated in Appendix 2, in a monopoly situation, $\frac{\Pi_{B}^{(1)}}{\Pi_{A}^{(1)}} = \frac{\Pi_{B}^{(2,1)}}{\Pi_{A}^{(2,1)}} = \frac{\Pi_{B}^{(2,2)}}{\Pi_{A}^{(2,2)}} = \frac{\delta_{B}^{(1)}}{\delta_{A}^{(1)}} = \frac{\delta_{B}^{(2)}}{\delta_{A}^{(2)}} = \frac{1}{2}$

Here, $\mu = \frac{1}{2}$, and thus, $\rho = \frac{1}{3}$ and $\alpha = \frac{2}{3}$.

With $E_A$, the efforts supported by firm A are as follows: $E_A = \frac{2}{3}(e_1(\rho) + e_2(\eta))$, and $E_B$, the efforts supported by firm B

are as follows: $E_B = \frac{1}{3}(e_1(\rho) + e_2(\eta))$.

The upstream third should bear two-thirds of the total expenditure of the R&D program, regardless of the functions relating expenditures to success probabilities.

The efforts required by each firm compared with their profits are the same because $\frac{\Pi_A}{\Pi_B} = 2$ and $\frac{E_A}{E_B} = 2$.

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