A differential game of ingredient branding supply chain with Nash bargaining fairness concern

Many companies, such as Hermes and Apple or Dell and Intel, engage in ingredient branding to form strategic partnerships. A distinctive feature of these collaborations is the visible presence of the ingredient brand on the end product, creating “brand halo” effects where one brand’s reputation influences the other as a cumulative and long-term process. However, when partners contribute and benefit unequally, fairness concerns about “free-riding” arise. While ingredient branding promotes mutual gain, fairness concerns may lead to distributional conflict. Yet how this tension shapes supply chain performance remains unclear. To address this issue, we develop a cooperative supply chain framework involving a component supplier and an end-product manufacturer, incorporating Nash bargaining fairness concerns and the Nerlove–Arrow dynamic goodwill model. Our findings reveal that while fairness concerns tend to reduce market demand and brand goodwill in an ingredient branding supply chain, they also help offset the disadvantages of being a Stackelberg follower. In a profit–cost sharing supply chain, higher supplier fairness concerns degree requires greater subsidy for coordination in the supplier-dominated setting, while higher manufacturer fairness concerns degree increases the likelihood for Pareto improvement in the manufacturer-dominated setting. In the extended model with advertising and quality improvement, we find that suppliers should invest in quality improvement initially, while advertising is preferred later only if its efficiency is sufficiently high. Consumers benefit from lower prices when the Stackelberg leader exhibits a low degree of fairness concern. Finally, we use numerical analysis to examine the robustness of our model by considering nonlinear fairness concerns and budget constraints. Our results show that, despite minor fluctuations in equilibrium decisions under the nonlinear fairness-minded supply chain, the overall trend is consistent with that in the linear case. Furthermore, imposing budget constraints on the Stackelberg leader leads to greater fluctuation in overall supply chain utility than imposing them on the followers.