  Hierarchical Federated Learning (HFL) is a distributed machine learning
paradigm tailored for multi-tiered computation architectures, which supports
massive access of devices' models simultaneously. To enable efficient HFL, it
is crucial to design suitable incentive mechanisms to ensure that devices
actively participate in local training. However, there are few studies on
incentive mechanism design for HFL. In this paper, we design two-level
incentive mechanisms for the HFL with a two-tiered computing structure to
encourage the participation of entities in each tier in the HFL training. In
the lower-level game, we propose a coalition formation game to joint optimize
the edge association and bandwidth allocation problem, and obtain efficient
coalition partitions by the proposed preference rule, which can be proven to be
stable by exact potential game. In the upper-level game, we design the
Stackelberg game algorithm, which not only determines the optimal number of
edge aggregations for edge servers to maximize their utility, but also optimize
the unit reward provided for the edge aggregation performance to ensure the
interests of cloud servers. Furthermore, numerical results indicate that the
proposed algorithms can achieve better performance than the benchmark schemes.
