In recent years, the rise in the collaborative consumption of fashion products has propelled many fashion brands to tap into rental markets and offer rental services. Although many studies have investigated consumer perceptions and attitudes toward fashion rental services, analytical models that help businesses make more informed supply chain decisions are scarce. Consequently, we develop game-theoretical models to study the fashion market with two firms, i.e. a fashion manufacturer and its rental platform partner. The models consider four green-advertising investment situations: no firm invests, only the fashion manufacturer invests, only the rental platform invests, or both firms invest in green advertising for fashion rental services. We compare pricing and green advertising decisions under different power structures between the two firms across the four green-advertising investment situations. Our results demonstrate when and how the power structure affects the key decision variables. We reveal that it is least profitable for both firms if neither contributes to green advertising, while both firms obtain the highest profits when they both invest in green advertising. In the two situations when only one firm invests in green advertising, we note that it is not necessarily unfavorable for firms to accept the green-advertising cost proactively. Specifically, the Stackelberg follower benefits from investing in green advertising. Furthermore, in these two situations, whether or not the Stackelberg leader covers the green-advertising cost is more important than their identity. Our results and analysis reveal important managerial implications that can assist firms in excelling in the fashion rental market.