Channel coordination under Nash bargaining fairness concerns in differential games of goodwill accumulation

We study a supply chain coordination problem with an upstream manufacturer and a downstream retailer that have Nash bargaining fairness concerns. The evolution of consumer goodwill is assumed to be influenced by the final product's quality improvement level (from the manufacturer) and advertising effort level (from the retailer). We determine the optimal strategies by differential game models for centralized, fairness-neutral decentralized, and fairness-concerned decentralized channels under different power structures. A revenue- and cost-sharing contract is also developed to coordinate the fairness-concerned decentralized channels. We show that decision-makers will accept the proposed contract only if the revenue-sharing rate satisfies certain conditions. Comparison discussions, sensitivity analyses of some key parameters, and numerical studies are conducted to provide further insights. We observe that a dominant channel member's sensitivity to fairness is relatively more significant in the decision-making process and channel efficiency. Specifically, each member has a greater incentive to adjust the investment and pricing strategies in the channel that it dominates. Additionally, we find that when the supply chain members are fairness-neutral, no power structure exists that can simultaneously be the most beneficial to the channel with the greatest profit, the lowest selling price, and the most improved quality. However, when the supply chain members are fairness-concerned, conditions exist under which either the retailer- or manufacturer-dominated channel can exhibit optimum performance in all three aspects.