The growing awareness of our environmental responsibilities and the impact of climate change on human lives has led to a greater demand for eco-friendly practices throughout supply chains. This study focuses on the relationship between a Third-Party Logistics provider (3PL) and its upper hand stages, such as a manufacturer, within a supply chain. We've identified four potential scenarios based on the strategies employed by the manufacturer (whether they use marketing or not) and the 3PL (whether they offer green delivery or not). We've examined these scenarios using Stackelberg game assumptions, with the 3PL as the leader of the game. In addition to the traditional game analysis, we've applied Evolutionary Game Theory (EGT) to explore the dynamic interaction between the manufacturer and the 3PL, without making the traditional game assumptions. Our findings indicate that manufacturers tend to gravitate towards adopting marketing strategies as their Evolutionary Stable Strategy (ESS), while 3PLs tend to favour green delivery strategies as their ESS. This tendency is particularly pronounced when customers place a higher importance on both marketing efforts and eco-friendly delivery options. We've conducted a comprehensive numerical analysis to illustrate how the system behaves under various parameter values, considering both classical and evolutionary game assumptions.