To maintain the sustainable development of a platform's economy, e-commerce platforms put forward various subsidy programs to retailers selling on them during COVID-19. This paper investigates an e-commerce platform's decision on subsidizing a retailer selling on it with logistics constraints during an epidemic scenario, with a focus on the role of power structure and altruistic preference. By constructing two Stackelberg game models, the research obtains the optimal subsidy under two power structures (i.e., the dominant platform and the weaker platform), respectively. The comparison between them shows that the conditions of the dominant platform giving subsidies (both altruistic preference and logistics constraints should be higher enough) are stricter than the weaker platform. Considering the same altruistic preference and logistics constraints, the optimal subsidy provided by the weaker platform should always be not less than the dominant platform. However, the weaker platform, surprisingly, can get more utility by lowering its altruistic preference voluntarily when the commission fee is low. No matter what the power structure is, the optimal subsidy increases with the logistics service coefficient and altruistic preference, and the dominant member's profit/utility is not less than the weaker one, which confirms "the first mover advantage". Finally, more managerial implications to the platform-retailer systems are discussed.