Prior research on supply chain innovation incentive contracts has focused on incentivizing upstream green innovation (Yenipazarli, 2017; Liu et al., 2020), and incentive mechanism design between upstream and downstream supply chain members (Aydin & Parker, 2018; Yu et al., 2021). Among them, some studies have paid attention to innovation in upstream key components. In the context of upstream suppliers' innovation of key components that can improve performance of the final product, Wang and Shin (2015) analyze the impact of three types of contracts —wholesale price contract, quality-dependent wholesale price contract and RS contract — on coordination of the supply chain, and their findings suggest that RS contracts can play a role in coordinating the supply chain. After comparing RS and investment-sharing contracts in a supply chain consisting of a financially-constrained upstream start-up supplier that has invested in quality innovations and a downstream manufacturer that is facing demand uncertainty, Xing et al. (2022) provide a reference for the manufacturer on which contract to choose in which situation. Based on a supply chain setting between automobile manufacturers (marketers) and battery manufacturers (innovators), Yu et al. (2021) analyze four types of R&D co-operation contracts in the R&D and sales phases, including two vertical co-operation models of revenue-sharing contracts and fixed R&D payment fees, as well as two types of co-development contracts in which the marketer is in charge of the sales phase, or the innovator is in charge of the sales phase. Based on an analysis of the impact of demand uncertainty on supply chain innovation investment decisions, Liu et al. (2023) suggest that centralized decision-making has the highest investment efficiency and better value of investment choices, and that volume discount contracts are preferred when market volatility is high. Focusing on comparing the differences between various contracts in innovation promotion, these prior studies have provided valuable insights by showing that revenue-sharing contracts can indeed incentivize upstream suppliers to innovate in R&D. Building on these studies, our study focuses on revenue-sharing contracts and examines the effect of overconfidence as one of the common cognitive biases of decision makers on collaborative innovation of supply chain members.

The second stream of relevant research is the issue of overconfidence in supply chains. When making R&D decisions, or operational decisions such as production volume and wholesale price, supply chain members tend to have several behavioral characteristics, of which overconfidence is a common one and an important cause of biased decision-making. Based on a review of more than 600 papers (Moore & Healy, 2008; Moore & Schatz, 2017) on overconfidence, Moore and associates classified three types of overconfidence: over-precision — people believe that they are more accurate in making predictions about what is actually going on (Li, 2019), over-positioning — people believe that they have above-average abilities, and overestimation — people usually overestimate their actual abilities, behaviors, level of control and likelihood of success (Li, 2019). The existing literature in operations management research mainly focuses on over-precision and overestimation. Over-precision is usually portrayed as the expected distribution of market demand having the same mean as the actual value but lower variance (Kirshner & Shao, 2019, Li, 2019; Ren & Croson, 2013). Overestimation means that the mean of the expected market demand distribution is larger than the actual value (Ren & Croson, 2013). Corporate decision makers are more prone to overconfidence in overestimating the benefits generated from R&D and innovation investments in the supply chain. For example, Apple's founder Steve Jobs was happy to overestimate the expected benefits of uncertainty associated with new technologies.

A study by Hirshleifer et al. (2012) finds that decision makers who overestimate the benefits from an innovation project tend to invest more in the firm's innovation projects. In their study investigating how overconfidence with regard to overestimation of a firm's decision makers in a supply chain affects the decisions of other firms in the chain, Nelson and Schwartz (2019) find that overestimation of demand by downstream customer firms leads to increased investment by upstream suppliers. Through an empirical study in the context of durable goods manufacturing and high technology barrier industries, Phua et al. (2018) suggest that the overestimated overconfidence of decision makers of downstream manufacturers can attract more suppliers and encourage suppliers to make R&D investments. Tang et al. (2015) show that the tendency of decision makers to evaluate firms higher than the actual situation of the firms, i.e., to overestimate the expected firm performance, can positively influence the innovation activities of firms, but the strength of the influence is contingent on the external environment. In their research on overconfidence of the supply chain members, Xiao et al. (2020) have analyzed the implications of retailers' equity concerns and manufacturers' overconfidence for sustainable two-phase supply chains, and they suggest that RS contracts are the most lucrative of the three types of decentralized contract. Du et al. (2021) analyze which of the wholesale price contract and the cost-sharing contract is able to provide a better incentive for upstream suppliers to innovate when the manufacturer overestimates the ability of innovation to increase demand, and find that the wholesale price contract is not favorable to supplier innovation, and they also find that the cost-sharing contract is more favorable to supplier innovation than the centralized decision-making model. Wang et al. (2022) build a two-tier logistics service supply chain consisting of logistics service providers and logistics service integrators, analyzing the impact of logistics service integrators' overconfidence on the innovation of green logistics service products and proposing an incentive contract of "green subsidy and loss sharing". Hao et al. (2023) examine the effect of overconfidence in one or both partners on inventory responsibility allocation and profitability.

Among the above studies on overestimated overconfidence, the studies done by Xiao et al. (2020), Du et al. (2021), Wang et al. (2022), and Hao et al. (2023) are more relevant to the topic examined in this paper. Hao et al. (2023) examine the impact of overconfidence on the allocation of inventory responsibility on both sides of the supply chain, in contrast to this paper's focus on promoting innovation in upstream suppliers. Xiao et al. (2020), Du et al. (2021), and Wang et al. (2022) focus on overconfidence on one side of the supply chain, but do not explore the interactive effects of overconfidence on the decision-making of each member of the supply chain. In particular, given its focus on revenue-sharing contracts, the study by Xiao et al. (2020) is highly valuable to our study. Extending the study by Xiao et al., our paper further explores the situation where bilateral bargaining models determine revenue-sharing proportions. This unique focus of our study enables us to generate some new research findings. For example, our study suggests that overconfidence of a supplier (manufacturer) damages its own profits but increases the profits of the manufacturer (supplier), and that when the degree of overconfidence is under a certain threshold, the value of the damage to its own profits is smaller than the value of the increase to the other side's profits, resulting in pure benefits for the whole supply chain.

A comparison of Lemmas 1-4 leads to Corollary 1-4.

Corollary 1 $I_{SM*}>I_{SN*}(I_{NM*})>I_{NN*}$, $q_{SM*}>q_{SN*}(q_{NM*})>q_{NN*}$, $w_{SM*}>w_{SN*}(w_{NM*})>w_{NN*}$; when ${\epsilon }_s>{\epsilon }_{s1}$, $I_{SN*}>I_{NM*}$, $q_{SN*}>q_{NM*}$, when ${\epsilon }_s<{\epsilon }_{s1}$, $I_{SN*}<I_{NM*}$, $q_{SN*}<q_{NM*}$; if and only if ${\epsilon }_s>{\epsilon }_{s2}$ and $H_1>0$, $w_{SN*}>w_{NM*}$ otherwise $w_{NM*}>w_{SN*}$.

Proof $I_{SM*}-I_{SN*}=\frac{a{\epsilon }_m\left(1-\varphi \right)\left({\psi }_2\left({\psi }_2+{\epsilon }_m\right)+4K(2-\varphi )-{\epsilon }_s{\psi }_1{\varphi }^2+\left(\left({\epsilon }_s-{\psi }_2\right){\epsilon }_m+2{\psi }_2{\epsilon }_s\right)\varphi \right)}{\left(4K(2-\varphi )-{\left({\psi }_1\varphi -{\psi }_2-{\epsilon }_m\right)}^2\right)\left(4K(2-\varphi )-{\left(\varphi {\epsilon }_s+{\psi }_2\right)}^2\right)}>0$,$I_{SM*}-I_{NM*}=\frac{a\varphi {\epsilon }_s({\left(1-\varphi \right)}^2{\epsilon }_m^2+\left(1-\varphi \right)\left(\varphi {\epsilon }_s+2{\psi }_2\right){\epsilon }_m+\left({\psi }_2{\epsilon }_s-4K\right)\varphi +{\psi }_2{}^2+8K)}{((-{\psi }_1{}^2{\varphi }^2+2({\epsilon }_m^2+\left({\psi }_2-{\epsilon }_s\right){\epsilon }_m-{\psi }_2{\epsilon }_s-2K)\varphi -{\epsilon }_m^2-2{\psi }_2{\mathsf{\epsilon }}_m-{\psi }_2{}^2+8K)(4K(2-\varphi )-{\left(1-\varphi \right)}^2{\epsilon }_m^2-2{\psi }_2\left(1-\varphi \right){\epsilon }_m-{\psi }_2{}^2))}>0,$ $I_{SN*}-I_{NN*}=\frac{a\varphi {\epsilon }_s\left({\psi }_2\varphi {\epsilon }_s+{\psi }_2{}^2+4K\left(2-\varphi \right)\right)}{\left(4K\left(2-\varphi \right)-\varphi {\epsilon }_s\left(\varphi {\epsilon }_s+2{\psi }_2\right)-{\psi }_2{}^2\right)\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right)}>0$,$I_{NM*}-I_{NN*}=\frac{a{\epsilon }_m\left({\psi }_2\left(1-\varphi \right)\left({\psi }_2+{\epsilon }_m\left(1-\varphi \right)\right)+4K\left(2-\varphi \right)\left(1-\varphi \right)\right)}{\left(4K\left(2-\varphi \right)-{\left({\psi }_2+{\epsilon }_m\left(1-\varphi \right)\right)}^2\right)\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right)} >0$,$q_{SM*}-q_{SN*}=\frac{2Ka{\epsilon }_m\left(2{\psi }_2+{\epsilon }_m-\left({\epsilon }_m-2{\epsilon }_s\right)\varphi \right)\left(1-\varphi \right)}{\left(\varphi {\psi }_1\left({\epsilon }_m \left(1-\varphi \right)+\varphi {\epsilon }_s+2{\psi }_2+{\epsilon }_m\right)-{\epsilon }_m{}^2-2{\psi }_2{\epsilon }_m-{\psi }_2{}^2+4K\left(2-\varphi \right)\right)\left(4K\left(2-\varphi \right)-\varphi {\epsilon }_s\left(\varphi {\epsilon }_s+2{\psi }_2\right)-{\psi }_2{}^2\right)}>0$,$q_{SM*}-q_{NM*}=\frac{2Ka\varphi {\epsilon }_s\left(2{\epsilon }_m\left(1-\varphi \right)+\varphi {\epsilon }_s+2{\psi }_2\right)}{\left(4K\left(2-\varphi \right)-{\left({\psi }_1\varphi -{\psi }_2-{\epsilon }_m\right)}^2\right)\left(4K\left(2-\varphi \right)-{\left({\psi }_2-\varphi {\epsilon }_m+{\epsilon }_m\right)}^2\right)} >0$,$q_{SN*}-q_{NN*}=\frac{2Ka\varphi {\epsilon }_s\left(\varphi {\epsilon }_s+2{\psi }_2\right)}{\left(4K\left(2-\varphi \right)-{\left(\varphi {\epsilon }_s+{\psi }_2\right)}^2\right)\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right)} >0$,$w_{SM*}-w_{SN*}= \frac{a{\epsilon }_m((1-\varphi )(\varphi {\epsilon }_s+{\psi }_2)\left(\delta -\varphi (\beta +{\epsilon }_s)\right)(\varphi {\epsilon }_s+{\psi }_2+{\epsilon }_m(1-\varphi ))+(4(1-\varphi )(\beta {\varphi }^2+{\varphi }^2{\epsilon }_s-\delta \varphi +(2\beta +{\epsilon }_m)(1-\varphi )))K)}{(4K(2-\varphi )-{(\varphi {\epsilon }_s+{\psi }_2+{\epsilon }_m(1-\varphi ))}^2)(4K(2-\varphi )-{(\varphi {\epsilon }_s+{\psi }_2)}^2)}>0$,

$w_{SM*}-w_{NM*}=\frac{a{\epsilon }_m((\varphi {\epsilon }_s+{\psi }_2)\left(1-\varphi \right)(\delta -\varphi (\beta +{\epsilon }_s))\left(\varphi {\epsilon }_s+{\psi }_2+{\epsilon }_m(1-\varphi )\right)+4(1-\varphi )(\beta {\varphi }^2+{\varphi }^2{\epsilon }_s-\varphi (\delta +{\epsilon }_m)+2\beta (1-\varphi )+{\epsilon }_m)K)}{(4K(2-\varphi )-{({\psi }_2-\varphi {\psi }_1+{\epsilon }_m)}^2)(4K(2-\varphi )-{(\varphi {\epsilon }_s+{\psi }_2)}^2)}>0$,$w_{SN*}-w_{NN*}=\frac{a\varphi {\epsilon }_s\left(1-\varphi \right)\left(\beta {\psi }_2\left(\varphi {\epsilon }_s+{\psi }_2\right)-4K\left(\beta \varphi +\varphi {\epsilon }_s-2\delta \right)\right)}{\left(4K\left(2-\varphi \right)-{\left(\varphi {\epsilon }_s+{\psi }_2\right)}^2\right)\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right)}>0$,$w_{NM*}-w_{NN*}=\frac{a{\epsilon }_m\left(4\left(1-\varphi \right)\left(\beta {\varphi }^2-\delta \varphi +\left(1-\varphi \right)\left(2\beta +{\epsilon }_m\right)\right)K-\left(1-\varphi \right){\psi }_2\left(\beta \varphi -\delta \right)\left((1-\varphi ){\epsilon }_m+{\psi }_2\right)\right)}{\left(4K\left(2-\varphi \right)-{\left((1-\varphi ){\epsilon }_m+{\psi }_2\right)}^2\right)\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right)}>0$, when $I_{SN*}-I_{NM*}=0$, ${\epsilon }_s={\epsilon }_{s1}$，${\epsilon }_{s1}=\frac{{\epsilon }_m \left(1-\varphi \right)}{\varphi }$ or ${\epsilon }_s=-\frac{\left(\beta -\delta \right)\left(\beta -\delta +{\epsilon }_m \left(1-\varphi \right)\right)+4K\left(2-\varphi \right)}{\varphi \left(\beta -\delta +\left(1-\varphi \right){\epsilon }_m\right)}<0$ (discard). When $q_{SN*}-q_{NM*}=0$, ${\epsilon }_s={\epsilon }_{s1}$ or ${\epsilon }_s=-\frac{2{\psi }_2+{\epsilon }_m\left(1-\varphi \right)}{\varphi }$ (discard). When ${\epsilon }_s>{\epsilon }_{s1}$, we have $I_{SN*}>I_{NM*}$, $q_{SN*}>q_{NM*}$. When $w_{SN*}=w_{NM*}$, we have ${\epsilon }_s={\epsilon }_{s2}$, ${\epsilon }_{s2}=\frac{ -4\left(\beta \varphi -2\delta \right)K+\left({\psi }_2+{\epsilon }_m\left(1-\varphi \right)\right)\left(\beta \varphi {\epsilon }_m+\beta {\psi }_2+\beta {\epsilon }_m-2\delta {\epsilon }_m\right)-\sqrt{H_1}}{2\left(4K-\left({\epsilon }_m+\beta \right)\left({\psi }_2+{\epsilon }_m\right)\right)\varphi }$, where $\begin{array}{l}{H}_1={\beta }^2{\left({\psi }_2+{\epsilon }_m\left(1-\varphi \right)\right)}^4+\left(64\left(\varphi -1\right){\epsilon }_m^2+\left(64\left(2\beta +\delta \right)\varphi -2\beta -64\beta {\varphi }^2\right){\epsilon }_m+16{\left(\beta \varphi -2\delta \right)}^2\right)K^2\\ -8\left({\beta }^2\varphi +2\beta \varphi {\epsilon }_m-2\beta \delta -4\beta {\epsilon }_m-2\delta {\epsilon }_m-2{\epsilon }_m^2\right){\left({\psi }_2+{\epsilon }_m\left(1-\varphi \right)\right)}^{2}K\end{array}$, only if ${\epsilon }_s>{\epsilon }_{s2}$ and $H_1>0$时,$w_{SN*}>w_{NM*}$.

In order to observe the relationship $I_{SN*}$ with $I_{NM*}$, $q_{SN*}$ with $q_{NM*}$, and $w_{SN*}$ with $w_{NM*}$ more clearly, we have conducted a numerical analysis, and set $a=3,K=0.5,c=2,\beta =0.7,\delta =0.3,\varphi =0.2$. The boundary line between $I_{SN*}$ and $I_{NM*}$, $q_{SN*}$ and $q_{NM*}$ is the same as ${\epsilon }_{s1}=4{\epsilon }_m$. The relationship between $w_{SN*}$ and $w_{NM*}$ when ${\Theta }_1>0$ we have $0<{\epsilon }_m<0.073$, and ${\epsilon }_{s2}=\frac{5.77\sqrt{\left({\epsilon }_m+2.13\right)\left({\epsilon }_m+1.40\right)\left({\epsilon }_m-0.07\right)\left({\epsilon }_m-0.92\right)}-2.58-0.48{\epsilon }_m^2-0.80{\epsilon }_m}{0.8{\epsilon }_m^2+0.96{\epsilon }_m-1.72}$ under this condition, see Table 2. Under this numerical assumption, in order to ensure that the Lemmas 1-4 are established, there are ${\epsilon }_{mNMMAX}=0.895$, ${\epsilon }_{sSNMAX}=4.184$, ${\epsilon }_{mSMSEG}=0.687$, ${\epsilon }_{sSMMAX1}=-2.667{\epsilon }_m+1.667\sqrt{0.64{\epsilon }_m{}^2+0.64{\epsilon }_m+10.96}-1.333$, ${\epsilon }_{sSMMAX2}=5(-0.8{\epsilon }_{\mathrm{m}}{}^2-0.4{\epsilon }_{\mathrm{m}}+1)/{\epsilon }_{\mathrm{m}}$.

From Corollary 1, among the four models, the optimal wholesale price $w$, optimal level of R&D inputs $I$, and optimal production volume $q$ are the highest when both the supplier and the manufacturer are overconfident (model SM); the optimal wholesale price $w$, optimal level of R&D inputs $I$, and optimal production volume $q$ are the lowest when neither the supplier nor the manufacturer is overconfident (model NN); and the optimal wholesale price $w$, optimal level of R&D inputs $I$, and optimal production volume $q$ are between models SM and NN for the supplier-only or manufacturer-only overconfidence (model SN or model NM). The optimal wholesale price $w$, optimal level of R&D inputs $I$, and optimal production volume $q$ for the case where only the supplier or only the manufacturer is overconfident (Model SN or Model NM) are set between Model SM and Model NN. This suggests that overestimation of the impact of the level of R&D inputs in key components on the selling price of the product by either the supplier or the manufacturer leads to an increase in the optimal wholesale price $w$, optimal level of R&D inputs $I$, and optimal production volume $q$. When both overestimate the impact of the level of R&D inputs on the selling price of key components, the impacts are superimposed so that the optimal wholesale price $w$, the optimal level of R&D inputs $I$, and the optimal production volume $q$ are the highest.

From Corollary 1 and Table 2, it can be seen that when the overconfidence coefficient ${\epsilon }_s$ in Model SN satisfies the relationship ${\epsilon }_s>{\epsilon }_{s1}$ with the one ${\epsilon }_m$ in Model NM, the level of R&D inputs and production volume of the formulation in Model SN will be higher than that in Model NM, and in the reverse case, it will be low. The overconfidence coefficient ${\epsilon }_s$ in Model SN satisfies the relationship $H_1>0$ (i.e. ${\epsilon }_m$ is smaller) with the one ${\epsilon }_m$ in Model NM, and when ${\epsilon }_s>{\epsilon }_{s2}$, the wholesale price of key components formulated in Model SN will be higher than that formulated in Model NM, and vice versa.

Corollary 2 ${\Pi }_{m,SN*}>{\Pi }_{m,NN*}>{\Pi }_{m,NM*}$, ${\Pi }_{m,SN*}>{\Pi }_{m,SM*}>{\Pi }_{m,NM*}$; when ${\mathsf{\epsilon }}_m>{\mathsf{\epsilon }}_{m1}$, ${\Pi }_{m,NN*}>{\Pi }_{m,SM*}$, when ${\mathsf{\epsilon }}_m<{\mathsf{\epsilon }}_{m1}$, ${\Pi }_{m,SM*}>{\Pi }_{m,NN*}$.

Proof ${\Pi }_{m,SN*}-{\Pi }_{m,NN*}=\frac{4K^2{a}^2\left(1-\varphi \right)\varphi {\epsilon }_s\left(8\left(2-\varphi \right)\left(\varphi {\epsilon }_s+2{\psi }_2\right)K-\left(\varphi {\epsilon }_s+2{\psi }_2\right)\left({\varphi }^2{\epsilon }_s^2+2{\psi }_2\varphi {\epsilon }_s+2{\psi }_2{}^2\right)\right)}{{\left(4K\left(2-\varphi \right)-\varphi {\epsilon }_s\left(\varphi {\epsilon }_s+2{\psi }_2\right)-{\psi }_2{}^2\right)}^2{\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right)}^2}>0$,${\Pi }_{m,NN*}-{\Pi }_{m,NM*}=\frac{2a^{2}K\left(1-\varphi \right){\epsilon }_m\left(\begin{array}{l}16\left(2-\varphi \right)\left(\left({\psi }_2-{\epsilon }_m\right)\varphi +{\epsilon }_m\right)K^2+(2{\left(1-\varphi \right)}^4{\epsilon }_m^3+8{\left(1-\varphi \right)}^3{\psi }_2{\epsilon }_m^2-\\ 4{\psi }_2{}^2\left(1-\varphi \right)\left(1+\varphi \right){\epsilon }_m-8{\psi }_2{}^3)K+{\psi }_2{}^4\left({\psi }_2+\left(1-\varphi \right){\epsilon }_m\right)\end{array}\right)}{{\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right)}^2{\left(4K\left(2-\varphi \right)-{\left({\psi }_2+{\epsilon }_m(1-\varphi \epsilon )\right)}^2\right)}^2}>0$,${\Pi }_{m,SM*}-{\Pi }_{m,NM*}=\frac{2\left(\left(1-\varphi \right){\epsilon }_m^2+{\psi }_2{\epsilon }_m-2K\right)a^2\left(1-\varphi \right)K}{{\left({\left({\psi }_2+\left(1-\varphi \right){\epsilon }_m\right)}^2+4K\left(\varphi -2\right)\right)}^2}>0$,${\Pi }_{m,SN*}-{\Pi }_{m,SM*}=\frac{ 2K{a}^2\left(1-\varphi \right){\epsilon }_m\left(\begin{array}{l}16\left(2-\varphi \right)\left({\varphi }^2{\epsilon }_s+\left({\epsilon }_m(1-\varphi )-{\epsilon }_m\right)\varphi +{\epsilon }_m\right)K^2+(2{\left(1-\varphi \right)}^4{\epsilon }_m^3+8{\left(1-\varphi \right)}^3(\varphi {\epsilon }_s+\\ {\psi }_2){\epsilon }_m^2-4\left(1-\varphi \right)\left(\varphi +1\right){\left(\varphi {\epsilon }_s+{\psi }_2\right)}^2{\epsilon }_m)K+{\left(\varphi {\epsilon }_s+{\psi }_2\right)}^4\left(\varphi {\epsilon }_s+{\psi }_2+{\epsilon }_m(1-\varphi )\right)\end{array}\right)}{ {\left(4K\left(2-\varphi \right)-{\left(\varphi {\epsilon }_s+{\epsilon }_m(1-\varphi )\right)}^2\right)}^2{\left(4K\left(2-\varphi \right)-{\left(\varphi {\epsilon }_s+{\psi }_2+{\epsilon }_m(1-\varphi )\right)}^2\right)}^2}>0$.Let ${\Pi }_{m,NN*}={\pi }_{m,SM*}$, after multiplication cross, shifting to the left side of the equation gives $\begin{array}{l} -2K{\left(1-\varphi \right)}^4{\epsilon }_m^4-8K{\left(1-\varphi \right)}^3\left(\varphi {\epsilon }_s+{\psi }_2\right){\epsilon }_m^3+(-16\left(2-\varphi \right)K^2+12{\varphi }^3{\epsilon }_s^2+12{\epsilon }_s\left(2{\psi }_2-{\epsilon }_s\right){\varphi }^2+\\ 4{\psi }_2\left({\psi }_2-6{\epsilon }_s\right)\varphi +4{\psi }_2{}^{2}K-{\psi }_2{}^4)\left(1-\varphi \right){\epsilon }_m^2+(8K{\varphi }^3{\epsilon }_s^2+8K\left(2{\epsilon }_s{\psi }_2-{\epsilon }_s^2+2K\right){\varphi }^2-16K(2K+\\ {\epsilon }_s{\psi }_2)\varphi +{\psi }_2{}^2\left(8K-{\psi }_2{}^2\right))\left(\varphi {\mathsf{\epsilon }}_{\mathrm{s}}+{\psi }_2\right){\mathsf{\epsilon }}_{\mathrm{m}} +2\mathrm{K}\varphi {\mathsf{\epsilon }}_{\mathrm{s}}\left(\varphi {\mathsf{\epsilon }}_{\mathrm{s}}+2{\psi }_2\right)\left(8\left(2-\varphi \right)\mathrm{K}-{\mathsf{\epsilon }}_{\mathrm{s}}\varphi \left(\varphi {\mathsf{\epsilon }}_{\mathrm{s}}+2{\psi }_2\right)-2{\psi }_2{}^2\right)\end{array}$, let $H_2$ be equal to the above equation. Since ${\mathsf{\epsilon }}_{\mathrm{s}}>0$, when ${\mathsf{\epsilon }}_{\mathrm{m}}=0$, so $H_2=2\mathrm{K}\varphi {\mathsf{\epsilon }}_{\mathrm{s}}\left(\varphi {\mathsf{\epsilon }}_{\mathrm{s}}+2{\psi }_2\right)\left(8\left(2-\varphi \right)\mathrm{K}-{\mathsf{\epsilon }}_{\mathrm{s}}\varphi \left(\varphi {\mathsf{\epsilon }}_{\mathrm{s}}+2{\psi }_2\right)-2{\psi }_2{}^2\right)>0$, and when ${\mathsf{\epsilon }}_m={\mathsf{\epsilon }}_{\mathrm{s}}=K$, $H_2= -K\left(2K^4+\left(16{\varphi }^2+8{\psi }_2-48\varphi +32\right)K^3+\left(4{\psi }_2\left(2\varphi {\psi }_2+4{\varphi }^2-{\psi }_2-8\varphi \right)\right)K^2+{\psi }_2{}^3\left(8\varphi -8+{\psi }_2\right)K+{\psi }_2{}^5\right)<0$, so there must exist a point ${\mathsf{\epsilon }}_{\mathrm{s}}$ with respect to ${\mathsf{\epsilon }}_m={\mathsf{\epsilon }}_{m1}$ which makes ${\pi }_{m,SM*}-{\pi }_{m,NN*}=0$.

In order to observe the relationship between ${\Pi }_{m,NN*}$ with ${\Pi }_{m,SM*}$ more clearly, we make a graph, see Figure 2(a), with the same parameter values as above $a=3,$$K=0.5,$$c=2,$$\beta =0.7,$$\delta =0.3,$$\varphi =0.2$.

Corollary 2 compares the profitability of the manufacturer in the four models, where supplier overconfidence alone must increase the manufacturer's profitability and manufacturer overconfidence alone must reduce the manufacturer's profitability compared to neither model (overestimating the impact of the level of R&D inputs in key components on the selling price of the product). The manufacturer's profits are higher when the supplier is overconfident than when only the manufacturer is overconfident. The manufacturer's profits will be higher when only the supplier is overconfident compared to when both are overconfident. Corollary 2 illustrates that manufacturer overconfidence decreases the manufacturer's profits because the manufacturer overestimates the impact of the level of R&D inputs in key components on the selling price of the product and makes decisions accordingly, but does not actually achieve this utility, thus harming its own profits. The supplier's overconfidence leads the supplier to set higher levels of R&D inputs for key components, which increases customer acceptance, which in turn leads the manufacturer to produce more products, and therefore increases the manufacturer's profits.

When manufacturer's and supplier's overconfidence act at the same time, the effects of both will be superimposed, see Corollary 2 and Figure 2(a). The relationship between manufacturer's and supplier's overconfidence satisfies the condition ${\mathsf{\epsilon }}_m>{\mathsf{\epsilon }}_{m1}$, ${\Pi }_{m,NN*}>{\Pi }_{m,SM*}$ and vice versa ${\Pi }_{m,NN*}<{\Pi }_{m,SM*}$. This is because when the manufacturer's overconfidence is greater than the supplier's overconfidence, the manufacturer's overconfidence dominates the manufacturer's own profit reduction, resulting in a lower ${\Pi }_{m,SM*}$ compared to ${\Pi }_{m,NN*}$. When the manufacturer's overconfidence is small compared to the supplier's overconfidence, the supplier's overconfidence plays a dominant role in increasing the manufacturer's profits, resulting in ${\Pi }_{m,SM*}$ being higher compared to ${\Pi }_{m,NN*}$.

Corollary 3 ${\Pi }_{s,NM*}>{\Pi }_{s,NN*}>{\Pi }_{s,SN*}$, ${\Pi }_{s,NM*}>{\Pi }_{s,SM*}>{\Pi }_{s,SN*}$.When ${\epsilon }_m>{\epsilon }_{m2}$， ${\Pi }_{s,SM*}>{\Pi }_{s,NN*}$; when ${\epsilon }_m<{\epsilon }_{m2}$, ${\Pi }_{s,NN*}>{\Pi }_{s,SM*}$.

Proof ${\Pi }_{s,NM*}-{\Pi }_{s,NN*}=\frac{a^{2}K{\epsilon }_m\left(1-\varphi \right)\left(2{\psi }_2+{\epsilon }_m(1-\varphi )\right)}{(4K\left(2-\varphi \right)-{({\psi }_2+{\epsilon }_m(1-\varphi ))}^2)(4K(2-\varphi )-{\psi }_2{}^2)}>0$,${\Pi }_{s,NN*}-{\Pi }_{s,SN*}=\frac{a^{2}K{\varphi }^2{\epsilon }_s^2\left(\left(\varphi {\epsilon }_s+{\psi }_2\right)\left(\varphi {\epsilon }_s+3{\psi }_2\right)+4K\left(2-\varphi \right)\right)}{\left(4K\left(2-\varphi \right)-{\psi }_2{}^2\right){\left(4K\left(2-\varphi \right)-{\left(\varphi {\epsilon }_s+{\psi }_2\right)}^2\right)}^2}>0$,${\Pi }_{s,NM*}-{\Pi }_{s,SM*}=\frac{a^{2}K{\varphi }^2{\epsilon }_s^2\left(4K\left(2-\varphi \right)+\left(\varphi {\epsilon }_s+{\psi }_2+{\epsilon }_m(1-\varphi )\right)\left(\varphi {\epsilon }_s+3{\psi }_2+3\left(1-\varphi \right){\epsilon }_m\right)\right)}{\left(4K\left(2-\varphi \right)-{\left({\psi }_2+{\epsilon }_m(1-\varphi )\right)}^2\right){\left(4K\left(2-\varphi \right)-{\left(\varphi {\epsilon }_s+{\psi }_2+{\epsilon }_m(1-\varphi )\right)}^2\right)}^2}>0$,