## 3 System Model
### Model Description  
In this paper, a basic structure of the multi-domain network slicing service provision market is described and used, as shown in Figure 1. Both users and network service providers are seen as participants. Users send network resource requests and network service providers serve user requests. The service providers from different domains are seen as different categories. In the example of Figure 1, there are four categories of participants \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(G\) & The participant type set \\ \hline \(\mathcal{N}\) & The participant set \\ \hline \(N\) & The size of a participant set \\ \hline \(K\) & The domain number \\ \hline \(M\) & The service provider number \\ \hline \(r_{i}\) (\(r_{j}\)) & The resource request (amount) of user \(i\) (service provider \(j\)) \\ \hline \(b_{i}\) (\(b_{j}\)) & The unit bid (ask) price of user \(i\) (service provider \(j\)) \\ \hline \(p_{i}\) (\(p_{j}\)) & The paid (received) price of user \(i\) (service provider \(j\)) \\ \hline \(v_{i}\) (\(v_{j}\)) & The unit value of user \(i\) (service provider \(j\)) \\ \hline \(u_{i}\) (\(u_{j}\)) & The utility of user \(i\) (service provider \(j\)) \\ \hline \(S\) & The procurement set \\ \hline \(p_{S}\) & The served bandwidth amount of \(S\) \\ \hline \(S\) & The trade scheme \\ \hline \(\mathcal{R}\) & The recipe tree \\ \hline \(V\) & The large value as the initial price \\ \hline \(\Delta\) & The unit quantity as the price increase step \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The description of major mathematical symbols. and the participant type set \(G\) is denoted as \(G=\{1,2,3,4\}\). In Figure 1, the user set is denoted as \(\mathcal{N}_{1}\), which contains all users. The service provider set in the satellite network domain is denoted as \(\mathcal{N}_{2}\), which contains all satellite network service providers. The service provider set in the radio access network (RAN) domain is denoted as \(\mathcal{N}_{3}\), which contains all RAN service providers. And, the service provider set in the wide area network (WAN) domain is denoted as \(\mathcal{N}_{4}\), which contains all WAN service providers. We assume that the participant sets of different categories are pairwise-disjoint, e.g., a single service provider cannot appear in the satellite network domain and the RAN domain simultaneously. In more complex cases, the number of participant categories could be more than four. A group of users with size \(N\) (i.e., \(|\mathcal{N}_{1}|=N\)) aims to request the network slicing service between the source and destination domains. Two alternative routes are established to provide this network slicing service. The first route (the upper route in Figure 1) is named a _service shortcut route_, which only goes through a single domain. The second route (the lower route in Figure 1) is named a _service chain route_, which passes through \(K\) different domains (e.g., \(K=2\) in Figure 1). It is worth mentioning that complex structures with more routes can also be described in a similar approach to the concepts we define in this section. In a single timeslot, user \(i\) submits a network slice bandwidth request \(r_{i}\geq 0\) and a unit bid price \(b_{i}\geq 0\) for using the network service. The unit value \(v_{i}\geq 0\) of using the service is the user's private information and \(b_{i}=v_{i}\) if the user bids truthfully. We assume that the user's utility \(u_{i}\) is quasi-linear, which is the value \(v_{i}\) minus the price \(p_{i}\) the user pays, multiplied by the bandwidth, i.e., \(u_{i}=v_{i}\times r_{i}-p_{i}\times r_{i}\). We also assume that a binary admission decision is made for a user, i.e., either the user is admitted and served with the bandwidth amount \(r_{i}\), or the user is rejected without service. The user may behave strategically to maximize their own utility. We assume that the strategic behavior of users only exists in the submitted bid price information instead of the requested bandwidth information.