The demand for advanced air mobility (AAM) operations is expected to be at a much larger scale than conventional aviation. Additionally, AAM flight operators are likely to compete in providing a range of on-demand services in congested airspaces, with varying operational costs. These characteristics motivate the need for the development of new traffic management algorithms for advanced air mobility. In this paper, we explore the use of traffic management protocols (“rules-of-the-road” for airspace access) to enable efficient and fair operations. First, we show that it is possible to avoid gridlock and improve efficiency by leveraging the concepts of cycle detection and backpressure. We then develop a cost-aware traffic management protocol based on the second-price auction. Using simulations of representative advanced air mobility scenarios, we demonstrate that our traffic management protocols can help balance efficiency and fairness, in both the operational and the economic contexts.