Greenwashing risks in environmental quality competition: Detection and deterrence

Mukherjee, Arka; Ghosh, Subhadip Article Greenwashing risks in environmental quality competition: Detection and deterrence Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Mukherjee, Arka; Ghosh, Subhadip (2025) : Greenwashing risks in environmental quality competition: Detection and deterrence, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 2, pp. 1-17, https://doi.org/10.3390/g16020014 This Version is available at: https://hdl.handle.net/10419/330128 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editors: Bruno Antonio Pansera, Gafurjan Ibragimov, Lucianna Cananà and Ulrich Berger Received: 23 October 2024 Revised: 4 February 2025 Accepted: 26 February 2025 Published: 11 March 2025 Citation: Mukherjee, A., & Ghosh, S. (2025). Greenwashing Risks in Environmental Quality Competition: Detection and Deterrence. Games, 16(2), 14. https://doi.org/10.3390/ g16020014 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article Greenwashing Risks in Environmental Quality Competition: Detection and Deterrence Arka Mukherjee and Subhadip Ghosh * Department of Decision Sciences, School of Business, MacEwan University, Edmonton, AB T5J 4S2, Canada; [email protected] *Correspondence: [email protected] Abstract: The rising prevalence of greenwashing by firms has emerged as a major concern for regulatory authorities over the past decade. This paper examines the impact of regulation on firms’ strategic decisions regarding greenwashing and environmental quality in an oligopolistic market. We model two firms that compete on environmental quality and greenwashing levels, operating under the oversight of a regulatory authority. The authority’s policy instruments include a detection mechanism and fines imposed on firms engaging in greenwashing. Using a differential game-theoretical framework, we examine the effectiveness of regulatory interventions like detection and penalties in reducing greenwashing behavior and enhancing environmental quality. Additionally, we discuss the post-detection trajectories of both firms, providing insights into the effects on consumer perceptions and market competition. We find that while regulation can reduce greenwashing as expected, it may also reduce firms’ environmental quality efforts. Indeed, when penalties are sufficiently high, the marginal returns on investment in greenwashing exceed those from actual green quality improvements. Keywords: differential game; greenwashing; detection; environmental quality; regulation 1. Introduction In recent years, firms have been pushed to make their products more environmentally friendly due to rising consumer interest in sustainable products and regulatory pressures to reduce carbon emissions. However, in the race to achieve a high level of “greenness”, firms have often resorted to exaggerating the true environmental quality of products (Boncinelli et al.,2023;Szabo & Webster,2021). This is usually referred to as greenwashing. Formally, greenwashing is “the act of misleading consumers regarding the environmental practices of a company or the environmental performance and positive communication about environmental performance” (Terrachoice,2010). Although greenwashing is not a novel phenomenon, its prevalence has increased significantly in recent years, impacting many industries, from energy to consumer goods. A recent UK Competition and Markets Authority review examined over 500 websites. It revealed that over 40 percent of employed green advertising strategies reviewed were misleading and potentially violated consumer protection laws (CMA,2021). Similarly, an investigation by the European Commission found that 42 percent of the green claims assessed were either exaggerated, false, or deceptive, potentially constituting unfair commercial practices (EC,2021). The prevalence of greenwashing has consequently led to a rising body of academic literature addressing it, most of them over the last decade. A recent comprehensive Games 2025,16, 14 https://doi.org/10.3390/g16020014 Games 2025,16, 14 2 of 17 systematic review of the greenwashing literature observed that nearly 70 percent of the 310 articles published by 2021 were concentrated in the preceding five years (2017–2021) (Santos et al.,2024). This surge in academic attention reflects the growing recognition of greenwashing as an important issue. A substantial body of empirical research, along with a relatively lower but growing number of theoretical studies, has explored various dimensions of greenwashing (Tang, 2018). These studies span a range of topics, from the drivers and consequences of greenwashing to its impact on consumer behavior, ethics, and regulatory responses. Our paper contributes to the body of theoretical research on greenwashing. Several papers that explore greenwashing using mathematical modeling consider strategic interaction between firms, of which we will cite only a few. A detailed literature survey is provided by Santos et al. (2024). One of the early works on greenwashing was by Shleifer (2004), who found that high competitive pressure can lead to unethical behavior like greenwashing, which can lower costs associated with green innovation or quality improvements. More recently, Wu et al. (2020) examined the interaction between a potential greenwasher and a genuinely green firm in direct price competition. They found that information transparency plays a critical role in markets where greenwashing occurs, influencing how consumers perceive environmental claims and make purchasing decisions. Subsequently, J. Zhang and Yang (2022) offered additional insights, showing that greenwashing firms may reduce prices while improving product quality in the presence of genuinely green competitors, thereby attracting greater market demand. Awasthy et al. (2022) noted that greenwashing by one firm negatively affects its competitors. However, when such negative “spillover” is minimal, the competing firm may still invest in positive environmental efforts, thereby increasing its green quality and prices. Using a two-period Stackelberg game for a dual-channel manufacturer–retailer setting, Zong et al. (2022) found that greenwashing can be profitable for manufacturers in a two-period game, though it may not benefit the overall supply chain. They argued that greenwashing, or “misreporting”, has a minimal impact on prices across the supply chain. However, most of these papers, and several others—for example, those by Cetin et al. (2023); Ruiz-Blanco et al. (2021); and Q. Zhang et al. (2020)—examine the effects of greenwashing performed by competing firms in a market without the presence of a regulatory authority. However, most developed countries and many emerging economies have implemented regulations to address greenwashing, with regulatory bodies imposing penalties when firms are caught engaging in it. For instance, the U.S., the EU, the UK, Australia, Canada, and China have all implemented legal frameworks that explicitly prohibit greenwashing, subjecting firms that engage in deceptive environmental claims to regulatory penalties and sanctions. The presence of regulatory authorities and the threat of detection and subsequent penalties alter the incentives of the firms to greenwash. One of the most notable cases of greenwashing detection and enforcement involved Volkswagen (VW), which faced widespread legal repercussions after the Federal Trade Commission (FTC) revealed that the company had manipulated emissions testing software in its diesel vehicles. VW falsely promoted these vehicles as low-emission and environmentally friendly, leading to substantial fines and legal actions across multiple jurisdictions (Shepardson,2016). In a more recent case from 2021, Kohl’s and Walmart were fined USD 5.5 million by the U.S. FTC for misleadingly marketing their bamboo-based textiles as eco-friendly despite the products being manufactured with environmentally harmful chemicals—violating the FTC’s Green Guides (FTC,2021). A similar incident occurred in Canada, where Keurig Canada was fined CAD 3 million by the Competition Bureau for making false environmental claims about its K-Cup pods, misleading consumers by Games 2025,16, 14 3 of 17 advertising them as recyclable in municipalities where recycling facilities did not accept the pods (Competition-Bureau-Canada,2022). Given the importance of the effect of regulation on greenwashing, we try to consider how the presence of such an agency affects the incentive of firms to greenwash. The effect of some form of regulation on greenwashing has been discussed by Huang et al. (2020); Lee et al. (2018); Sun and Zhang (2019); and Wang et al. (2022). From a supply chain perspective, Lee et al. (2018) demonstrated that regulating greenwashing in competitive supply chains can sometimes undermine firms’ CSR (Corporate Social Responsibility) efforts due to high costs. Interestingly, their findings suggest that permitting some level of greenwashing may provide incentives to certain firms to invest more in genuine environmental quality, as it allows them to balance cost pressures while still competing on green attributes. However, our focus is on the effect of regulations on the profits of the firms, the welfare and perception of final consumers, and on the level of greenwashing and environmental quality. Another noteworthy study in this field was by Sun and Zhang (2019), who employed evolutionary game theory to analyze the effectiveness of government regulations in controlling greenwashing across firms with different levels of market dominance. Their findings suggest that while government-imposed penalties are effective in deterring greenwashing among dominant enterprises, tax subsidies alone may not be sufficient to prevent greenwashing, particularly among smaller or inferior enterprises. They assumed greenwashing is always detected by regulators, which is often unrealistic in practice. In contrast, our paper assumes the probabilistic detection of greenwashing by regulatory bodies, with identical firms, making no arbitrary distinction between those engaging in greenwashing and genuinely green firms. Our framework is also different in that we use a differential game theoretic framework. Later, Huang et al. (2020) used a Stackelberg game with an incumbent and an entrant firm to investigate how competitive pricing strategies are affected by greenwashing, consumer behavior, and anti-greenwashing enforcement. They found that more lax enforcement of greenwashing can sometimes paradoxically lead to higher social welfare and that greenwashing can benefit both incumbent green firms and overall customer surplus when the market’s greenness gap is small. In contrast, we consider a differential game framework where the firms move simultaneously, and the firms are symmetric, with no arbitrary distinction between green and greenwashing firms. Another recent study that examined the effect of regulation on greenwashing was conducted by Wang et al. (2022). They employed a three-stage Bayesian game to examine how false-claims ban (FCB) regulations influence firms’ greenwashing behavior when they possess imprecise information about their products’ environmental impact. Their findings suggest that while strict FCB regulations are effective in curbing intentional greenwashing, they may also lead to uninformative non-greenwashing if the precision of greenness information acquired by firms is lower than a certain threshold level. Their model also assumes a binary state, either green or non-green. In contrast, our study assumes that the degree of greenwashing varies dynamically based on a range of model parameters. More recently, the effect of regulation on greenwashing was studied by Mukherjee and Ghosh (2024) in a differential game context. However, they considered a duopoly market with one greenwasher and one green firm. They also assumed a constant penalty for greenwashing. In contrast, we consider a scenario where no predefined distinction exists between firms, allowing both to potentially engage in greenwashing. Moreover, we assume that the penalty is a proportion of the firm’s revenue, which is how penalties are imposed on various countries, like Canada and the EU. For example, under Canada’s new Bill C-59, corporations found guilty of greenwashing may face penalties of up to CAD 10 million for a first offense, CAD 15 million for subsequent offenses, three times the value of the benefit derived from the deceptive conduct, or if that amount cannot be Games 2025,16, 14 4 of 17 reasonably determined, 3 percent of the corporation’s annual worldwide gross revenues (Competition-Bureau-Canada,2024). Before delving into our model, we would like to point out that, in a broader sense, our research problem examines how to control pollution under the risk of a change in regime. Although this field has a rich and extensive body of research, we will discuss only a few studies for the sake of brevity. One of the earlier works in this field was by Tsur and Zemel (1998), who, using a novel h λ method, examined how the threat of environmental catastrophes, ranging from reversible to irreversible events, affects decisions about pollution control. They found that reversible events consistently lead to greater conservation (lower pollution) as planners mitigate risks. In contrast, irreversible events will lead to lower pollution only under certain specific conditions. Nkuiya (2015) also contributed to the literature by examining how the risk of increased environmental damage influences the strategic emissions decisions of countries. Using a non-cooperative dynamic game framework, this study demonstrated that international cooperation in pollution control yields substantial benefits. Furthermore, it highlighted the importance of policymakers accounting for the possibility of sudden regime shifts when formulating and implementing environmental policies. In a more recent article, Gromov et al. (2024) examined optimal pollution control problems in the context of varying environmental conditions. They provided an excellent review of the existing literature on this stream of research. In addition, they found that optimal solutions converge to a hybrid limit cycle, which balances profit maximization with environmental preservation. Their findings suggest that adjusting the discount rate can be a regulatory tool to encourage sustainable practices among firms. In this paper, we consider two similar firms competing in a differential game framework with respect to environmental quality and the greenwashing level. The firms operate in a market governed by a regulatory body with an enforcement mechanism. If the agency detects greenwashing by a firm, it must pay a penalty in proportion to its revenue earned in the first period. Under such a situation, we examine the effect of greenwashing on each firm’s strategies, outcomes, and consumer perception. Our objective was to investigate the following research questions: 1. What are the equilibrium environmental quality and greenwashing efforts of the firms? 2. How do the environmental quality and greenwashing levels of each firm i vary with the penalty multiplier ρiand likelihood of detection χ? 3. How do the firms’ decisions vary with consumer trust? 4. Is return on investment on environmental quality higher than that on greenwashing? In addressing our research questions, a differential game is the most appropriate because it captures the dynamic evolution of perceived quality determined by each firm’s ongoing decisions about environmental quality and greenwashing. This framework models the strategic competition over time, where each firm’s actions impact not only its own demand function but also the competitor’s. Piece-wise constant strategies reflect the reality of firms adjusting decisions periodically rather than continuously. Our findings suggest that a high risk of exposure or a high penalty may reduce greenwashing and stop it altogether, but deteriorate the environmental quality of the products as well. However, a low penalty and market with conscious consumers can result in higher environmental quality. Lack of consumer trust may also reduce the environmental quality post greenwashing. Our main contributions are the following: • As opposed to static games, we model a differential game that captures the dynamic nature of perceived quality and its evolution over time, being influenced by the firms’ environmental quality and greenwashing. Games 2025,16, 14 5 of 17 • While the previous literature has focused on regulations on greenwashing, we propose and solve a model where firms compete while anticipating the risk of exposure and incorporate this risk in their decisions. • Contrary to intuition and comporting with Lee et al. (2018) we show that too stringent regulations on greenwashing in an evolving market may be detrimental to the overall environmental quality of the firms’ products. • We also show that the return on investments for greenwashing may be higher than that of environmental quality under certain conditions. In the next section, we define the model. Thereafter, in Section 3, we discuss the equilibrium strategies and their implications. Section 4is dedicated to numerical analysis and some important observations. We conclude the paper in Section 5. 2. Model 2.1. Demand Functions and State Variables We consider a duopoly market where two firms compete on the environmental quality of their products. In addition, the firms greenwash to enhance the perceived quality of their products to the consumers, thereby gaining a competitive advantage. We model the above situation using a continuous-time differential game model. The firms decide on the environmental quality qi(t) of the products and greenwashing level wi(t) , where i∈ { 1, 2 } is the index of the firm. The environmental quality of a product can be justified as a control variable in a differential game model because firms have direct influence over it through decisions such as investing in cleaner technologies or sourcing sustainable materials. For example, a firm can increase environmental quality by adopting eco-friendly packaging or by reducing emissions in the production process, actions that directly impact the firm’s cost dynamics and competitive positioning over time. More specifically, qij(t) denotes the efforts the firms make to improve environmental quality (for example pollution control), and wij(t) denotes the greenwashing efforts (for example, misleading advertising). A similar use and interpretation of these decisions can be found in the works by Dockner and Van Long (1993) and Xue and Wang (2024). Our demand function is similar to that of Mukherjee and Chauhan (2021). Specifically, the demand function of firm iis expressed as follows: Dij(t) = αi+βij(Gij(t)−G(3−i)j(t)) (1) where αi is the market potential, j is the index for the time period, and βij denotes the sensitivities to the product differentiation in terms of environmental attribute. The decision variables for the two firms in the two periods are environmental quality qij(t) and the greenwashing efforts wij(t) , i∈ { 1, 2 } . The state variables of our model are the perceived environmental qualities of the firm given by Gij(t) , i , j∈ { 1, 2 } . The state variables are positively affected by the environmental quality decisions qij(t) and the greenwashing decisions wij(t) . In essence, the state variable reflects consumers’ perception of a product’s overall environmental quality. This perception is shaped by both genuine efforts to improve environmental quality and deceptive practices (greenwashing) by the firms, which create a misleading impression of environmental quality. The evolution of the state variables are given by the following differential equations: dG1j dt =l1jw1j(t) + k1jq1j(t)−δ1jG1j(t),j∈ {1, 2},G11(0) = G10 dG2j dt =l2jw2j(t) + k2jq2j(t)−δ2jG2j(t),j∈ {1, 2},G21(0) = G20. (2) Games 2025,16, 14 6 of 17 In the above equations, kij and lij are the marginal effects of environmental quality and greenwashing on the perceived environmental quality evolution of firm i in regime j . δij is the decay in the perceived green quality of firm iin regime j. Detection of greenwashing: The firms anticipate that there is a risk of being exposed, and if detected at a random time τ , the regulatory body will impose a penalty. We assume that the time of detection is a random variable following a exponential distribution. Thus, we have an associated hazard rate χ, which is defined by lim dt→0 Prnt≤τ<t+dt|τ≥to dt =χ. (3) If the firms do not greenwash, then χ= 0 and w1j(t) = 0, and therefore, there is only one regime. This detection time splits the decision horizon into two regimes— [ 0, τ]∪(τ , ∞) , where [ 0, τ] is the period when the greenwashing is not detected, and (τ , ∞) is the period when it is detected and the penalty is imposed. If greenwashing is detected, the firms do not greenwash in the second regime. 2.2. The Profit Maximizing Problems of the Firms In noting that j is the index for the time period, the instantaneous profits of firm i in regime j are given by πij(t) = Dij(t)mij −Cij(t) , where mij is the unit profit margin (difference between price and production cost), and Cij(t) denotes the other costs of firm i , like the costs of greenwashing, costs of environmental quality, or the penalty cost if the firm is caught for greenwashing. The term mij is constant for every regime j . It can happen that mi1=mi2. Penalty (Pi) :The penalty for greenwashing can be in several forms. According to Forbes (McGowan,2024), such a penalty can be a percentage of gross revenue, a fixed constant (like USD 10 million), or a multiple of the revenue made from the sales of the items with misleading advertising. While most of the literature considers a fixed penalty, in this model, we consider that the penalty is a proportion of the first period’s revenue of firm i , and this is denoted by ρi∈[ 0, 1 ] . Therefore, the penalty Pi=ρimi1Rτ 0Di1(t) , where mi1 is the profit margin of firm i in regime 1. We highlight that if the firms are exposed, in the second regime, τ will be known to the firms and the values of Di1(t) are also known. Therefore, the penalty Pi=ρimi1Rτ 0Di1(t) is just a number and is treated merely as a constant term for the solution of the second period’s game. The instantaneous profit functions of firm iin regime j(πij(t)) is πij(t) = Dij(t)mij −µ 2q2 ij(t)−ω 2wij(t)2,wi2=0. (4) We consider an infinite horizon regime switching differential game where there are two regimes. The first regime is when the firms greenwash, and regime 2 starts when the firms are exposed at random time τ . We are interested in the long-term decisions and profits of the firms. Such games are widely used in the applications of management science and economics (Dockner et al.,2000). One standard way of solving such a game is to use backward induction as we do here. We start with the second regime’s problem. The second regime’s long-term profits are given by Ji2(G12,G22) = Z∞ τe−rtπi2(t)dt −Pi, given Gi2(τ) = ˆ Gi(5) Games 2025,16, 14 7 of 17 and the profit maximization problem of the firms are given by Vi2(G12,G22) = Max qi2Z∞ τe−rtπi2(t)dt −Pi, given Gi2(τ) = ˆ Gi(6) where (6) is subject to the state equations given by (2) . Here, πij(t) is the instantaneous profit of firm i in regime j , Jij(t) is the long-term profit of firm i in regime j , and Vij(Gij , G(3−i)j) is the value function of firm iin regime j. Derivation of first period’s profit: At the beginning of the first period, the possible time τ of the detection of greenwashing is random, and the firms do not know ex ante the date τ∈[ 0, ∞) . Consequently the net present value of profit J1 is also a random variable, and the expected value of profit at the beginning of the planning horizon (t= 0) is given by Ji1(G11,G21) = EZτ 0e−rtπi1(t)dt +e−rτ(Ji2(G12,G22)−Pi)(7) where the expectation E[ . ] is over the stochastic process of the detection of greenwashing. Ji1(G11,G21) = EZt 0e−rsπ1(s)ds +e−rtJi2−ρimi1Zt 0Di1(t)dt =Z∞ 0Zt 0e−rsπi1(s)ds +e−rtJi2−ρimi1Zt 0Di1(t)dtχe−χtdt =Z∞ 0χe−χZt 0e−rsπi1(s)dsdt +χZ∞ 0e−(r+χ)tJi2dt −ρ1mi1Z∞ 0χe−χtZt 0Di1(t)dt ≜I1+χI2−ρimi1I3. (8) To evaluate the three integrals, we integrate by parts. Starting with I1 and letting U=Rt 0e−rsπi1(s)ds and V=−e−χt, we obtain dU =e−rtπi1(t)dt,dV =χe−χt, and I1=Z∞ 0Zt 0e−rtπi1(s)dsχe−χtdt, = (−Ue−χt)|∞ 0−Z∞ 0(e−χt)(e−rtπi1(t))dt, =0+Z∞ 0e−(r+χ)tπi1(t)dt. The above results follows from (−Ue−χt)|∞ 0= 0 since limt→∞e−χt= 0 and U(0) = R0 0e−rsπi1(s)ds =0. Similarly, we have I3=Z∞ 0e−(r+χ)tDi1(t)dt. Therefore, (8) becomes Ji1=I1+χI2−ρimi1I3 =Z∞ 0e−(r+χ)tπi1(t)dt +χZ∞ 0e−(r+χ)tJi2dt −χρimi1Z∞ 0e−(r+χ)tDi1(t)dt =Z∞ 0e−(r+χ)t(πi1(t) + χ(Ji2−ρimi1Di1(t)))dt, Games 2025,16, 14 8 of 17 Consequently, the overall optimization problem is as follows: Vi1(G11,G21) = max qi1,wi1Z∞ 0e−(r+χ)tDi1(t)mi1(t)−µ 2q2 i1(t)−ω(t) 2w2 i1(9) +χVi2(G12,G22)−ρimi1Di1(t))dt Subject to ˙ Gi1(t) = ki1qi1(t) + li1wi1(t)−δi1Gi1(t), and Gi1(0) = G0 Note that the first period’s state variable contains greenwashing wi1(t) along with environmental quality qi1(t). 3. Discussion of Equilibrium Strategies We use the Hamilton–Jacobian–Bellman (HJB) equations to solve the above problems. HJB Equations (11) and (12) are linear quadratic and such linear quadratic games have been extensively studied in the literature under different contexts. It can be shown that the value functions of such games are linear quadratic (Dockner et al.,2000). We posit that the value functions will be of the form Vij(Gij,Gij) = XijG1j+YijG2j+Zij (10) where Xij , Yij , and Zij are the constant coefficients. The second period’s HJB equations are rV12(G12,G22) = max q12 hD12(t)m12(t)−µ 2q2 12(t) + ∂V12 ∂G12 ˙ G12(t) + ∂V12 ∂G22 ˙ G22(t) −ρ1m12 Zτ 0D11(t)dt] rV22(G12,G22) = max q22 hD22(t)m22(t)−µ 2q2 22(t) + ∂V22 ∂G12 ˙ G12(t) + ∂V22 ∂G22 ˙ G22(t) −ρ2m21 Zτ 0D21(t)dt](11) From Equation (9), the first period’s HJB equations are given by (r+χ)V11(G11,G21) = max q11,w11"D11(t)m11(t)−µ 2q2 11(t)−ω 2w2 11(t) + ∂V11 ∂G11 ˙ G11(t) + ∂V11 ∂G21 ˙ G21(t) + χV12(G12,G22)−ρ1m11D11(t) (r+χ)V21(G11,G21) = max q21,w21"D21(t)m21(t)−µ 2q2 21(t)−ω 2w2 21(t) + ∂V22 ∂G11 ˙ G11(t) + ∂V22 ∂G21 ˙ G21(t) + χV22(G12,G22)−ρ2m21D21(t)#(12) From the above equations, it is easy to see that the structure of the game is linear quadratic and the solution of the game will yield piece-wise constant strategies in the two regimes. The illustration and importance of such games can be found in Dockner et al. (2000). 3.1. Equilibrium Analysis We assume that if there is a second regime, after the firms are possibly exposed, the firms do not greenwash anymore. Therefore, the firms’ only decisions in the second regime are the environmental qualities. Games 2025,16, 14 15 of 17 Data Availability Statement: The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. Conflicts of Interest: The authors declare no conflicts of interest. Appendix A. Proof of Proposition 1 We want to prove that the equilibrium quality decisions of the two firms in the regime period are q12 =k12m12β12 µ(r+δ12) q22 =k22m22β22 µ(r+δ22)(A1) and the value functions are given by V12 =X12G12 +Y12G22 +Z12 and V22 =X22G12 + Y22G22 +Z22, where X12 =m12β12 r+δ12 ;Y12 =−m12β12 r+δ22 ; Proof. We assume that the value functions are of the form V12(G12,G22) = X12G12 +Y12G22 +Z12 V22(G12,G22) = X22G12 +Y22G22 +Z22 (A2) From (11), the second period’s HJB equations are rV12(G12,G22) = max q12 hD12(t)m12(t)−µ 2q2 12(t) + ∂V12 ∂G12 ˙ G12(t) + ∂V12 ∂G22 ˙ G22(t)−P1] rV22(G12,G22) = max q22 hD22(t)m22(t)−µ 2q2 22(t) + ∂V22 ∂G12 ˙ G12(t) + ∂V22 ∂G22 ˙ G22(t)−P2](A3) By noting that Pi is a number, substituting the demand functions, the state variables, and the derivatives in Equation (A3), and using the first-order conditions of the decision variables qi2, we obtain −µq12(t) + X12k12 =0=⇒q12(t) = k12(X12) µ −µq22(t) + Y22k22 =0=⇒q22(t) = k22(Y22) µ(A4) Replacing the value functions in (A3) with their linear forms in (A2), we obtain r(Xi2Gi2+Yi2G(3−i)2+Zi2) = max qi2hDi2(t)mi2(t)−µ 2q2 i2(t) + ∂Vi2 ∂G12 ˙ G12(t) + ∂Vi2 ∂G22 ˙ G22(t)] (A5) Substituting the demand functions, the state variables, and the derivatives in Equation (A5), we obtain the following two equations. Remark A1. For brevity, we are dropping the time index (t) in the proofs. Games 2025,16, 14 16 of 17 r(X12G12 +Y12G22 +Z12) = α1+β12(G12 −G22)m12 −(k12X12 µ)2+ (X12)(k12(k12X12 µ)−δ12G12) + (Y12(k22 k22X22 µ−δ22G22)) r(X22G12 +Y22G22 +Z22) = α2+β22(G22 −G21)m22 −(k22X22 µ)2+ (X22)(k22(k12X12 µ)−δ12G12) + (Y22(k22 k22X22 µ−δ22G22)) Equating the coefficients of state variables G12 and G22 and the constant terms from both sides of the above equations, we obtain the following set of six equations: rX12 =m12β12 −X12δ12 (A6) rY12 =−m12β12 −Y12δ22 (A7) rZ12 =1 2−2P1+2m12α1+m12β12k2 12m12β12 µ(r+δ12)2−2k2 22m22β22 µ(r+δ22)2) (A8) rX22 =m12β12 −X12δ12 (A9) rY22 =−m12β12 −Y12δ22 (A10) rZ22 =1 2−2P2+2m22α2+m22β22k2 22m22β22 µ(r+δ22)2−2k2 12m12β12 µ(r+δ12)2) (A11) From (A6), we obtain X12 =m12β12 (r+δ12). From (A7), we obtain Y12 =−m12β12 (r+δ22). From (A9), we obtain X22 =−m22β22 (r+δ12). From (A10), we obtain Y22 =m22β22 (r+δ22). 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