## 1 Introduction
To solve the fourth challenge and encourage the participation and cooperation among different service providers, a multi-sided ascending-price auction mechanism is further proposed as a game theory-based solution for slice admission control and resource allocation, in which multiple strategic service providers maximize their own utilities by trading bandwidth resources. Existing solutions for the multi-domain network slicing admission control problem are mainly based on game theory, for example, matching games [10], especially when the incentives of both users and network service providers are non-negligible. However, these existing solutions are not flexible, when only a single and fixed combination of associated users and service providers is permitted, e.g., with one-to-one matching between a user and a service provider, without allowing multiple potential routes for provisioning the network slicing service. In this paper, we propose a multi-sided ascending-price mechanism for sliced network service provision with bandwidth requirements, in which the network resource to allocate is the network bandwidth represented as a positive quantity. We design a novel optimal trade scheme calculation method and a novel ascending-price auction mechanism, with the consideration for the continuous value space, instead of the binary value space as in the previous study [11]. We also prove that, in the new and continuous setting, our proposed optimal trade scheme has a polynomial time complexity and performs as an upper bound for the auction mechanism, which is proven to be strongly budget-balanced, individually rational, and obviously truthful. Simulation experiments based on real-world data are designed to validate the effectiveness of the proposed ascending-price auction mechanism for the satellite-terrestrial integrated network. The main contributions of this paper are summarized as follows:  
* The multi-domain network slicing admission control is formulated as a multi-sided mechanism design problem when multiple routes can be used simultaneously in the satellite-terrestrial integrated network. * A polynomial-time optimal trade mechanism is proposed as an upper bound that calculates the optimal trade scheme for the mechanism design problem. * A multi-sided ascending-price auction mechanism is proposed as a strongly budget-balanced, individually rational, and obviously truthful solution for the mechanism design problem, and its asymptotically optimal performance is validated with simulation experiments based on real-world data. The rest of this paper is organized as follows. The relevant studies are reviewed and discussed in Section 2. The system model and the multi-sided mechanism design formulation are defined in Section 3. The proposed optimal trade mechanism and proof of its polynomial time complexity are presented in Section 4. The proposed multi-sided ascending-price auction mechanism and the proof for its desirable properties are presented in Section 5. Numerical experiments and results are discussed in Section 6. Section 7 concludes this paper with some potential future research directions. The acronyms for terminologies used in this study are summarized in Table 1 for reference.