## 1 Introduction  
Satellite communication makes up for the shortage of terrestrial networks with its advantages including wide coverage and support for massive access services, and the satellite-terrestrial integrated network is proposed to achieve ubiquitous connectivity [1]. Satellite network guarantees reliable communication services for remote areas not covered by existing terrestrial networks, e.g., 4G and 5G [2]. The integration expands the network coverage area and supports continuous service for massive Internet of Things (IoT) devices and mobile terminals, e.g., ships and aircraft. With the support for broadcast and multicast services, the satellite network offers data distribution services for users with a higher communication efficiency [3]. Besides these technological benefits, the satellite-terrestrial integrated network brings some economic benefit for remote areas, in which the deployment of 5G terrestrial networks is economically prohibitive [4]. Although the satellite-terrestrial integrated network has been seen as an important part of future B5G/6G networks, there are still many technological and economic chal lenges [5]. The first challenge is the differentiated user requirements, which cannot be fulfilled within a single network domain. The second challenge is the technological compatibility when heterogeneous infrastructures have different communication standards, protocols, and systems on the ground and in the space. The third challenge is the limited resource constraint, especially when the user demand exceeds the transmission capacity [6]. The last challenge is the service provider (SP) incentive, especially when network service providers in different domains need to cooperate with each other to serve user requests, and it becomes impossible to neglect the problem of how to encourage the participation and resource sharing from different operators. As one of the key technologies in 5G, network slicing (NS) is proposed for solving the first and second challenges in the satellite-terrestrial integrated network and adopted in this study [7]. Both intra-domain and inter-domain network slices have been considered before. In this paper, the inter-domain network slice was considered with multiple routes supported and without delegating the specific domains along the route as long as the service can be provided. For example, with the deployment of low-Earth-orbit (LEO) satellite constellations, cross-continent network slicing can be fulfilled with inter-satellite links involving only one domain (i.e., the satellite network). Alternatively, network slicing can be fulfilled with multiple domains on the ground, e.g., radio access networks, wide-area networks and submarine cables. Since the network slicing service cannot be accomplished alone for the network service providers in some domains, the cooperation among several network operators is required to establish a feasible slice. When considering the admission decision for a slice request, a joint decision is made with the involvement of multiple parties with their own interests, making the admission control problem far more complex than the single-domain case. To solve the third challenge and improve the utilization efficiency of limited resources, slice admission control is introduced and formulated as a mechanism design problem, in which different combinations of users and service providers are considered, as long as the network slices can be formulated [8]. While two-sided markets (e.g., buyers and sellers) have drawn much research attention in recent decades, multi-sided markets are much less seen in the literature due to both theoretical and practical challenges [9]. The trade decision process in a two-sided market only involves a buyer and a seller, while in a multi-sided market, it becomes a combinatorial problem when more than one combination of participants can form a deal.