## 3 System Model
### Model Description
In other words, if the submitted bandwidth is more than the user's actual need, the extra part if served brings no excess benefit to the user, and if the submitted bandwidth is less than the user's actual need, the network slicing service cannot be fulfilled as the user expects. In each domain, we assume \(M\) service providers exist, each of which has the ability to provide the cross-domain network service. The same number of service providers is used for model description simplicity, i.e., \(|\mathcal{N}_{2}|=|\mathcal{N}_{3}|=|\mathcal{N}_{4}|=M\) (in practice,  
Figure 1: The basic structure of the multi-domain network slicing service provision market. (**a**) The physical infrastructure; and (**b**) the logical structure. if the numbers of service providers from different domains are not equal, we can add dummy service providers with zero bandwidth resources to make this assumption hold). For the time period of interest, we assume that a service provider \(j\) submits a network slice bandwidth resource amount \(r_{j}\geq 0\) and a unit ask price \(b_{j}\leq 0\) for providing the network service. Similarly, the unit value \(v_{j}\leq 0\) of providing the service is the service provider's private information and \(b_{j}=v_{j}\) if the service provider bids truthfully. Note that the same mathematical symbols of the bandwidth and bid are used for the users and service providers. The value and price of the service providers are typically negative for representing the cost of deploying the infrastructure. A similar quasi-linear utility format is assumed for the service provider, which is \(u_{j}=v_{j}\times r_{j}-p_{j}\times r_{j}\). While \(v_{j}\) and \(p_{j}\) are both negative, utility, as their difference, is typically required as a positive value for the service provider to participate willingly. Different from the binary admission decision for a user, a service provider may use all or part of its bandwidth resource. We also assume that the strategic behavior of the service provider is limited to price information, e.g., by imposing a prohibitory penalty if the service provider fails to fulfill the submitted bandwidth resource in practice. Following the terminology borrowed from [11], each possible route is formulated as a _recipe_, which is a binary projection from the category set to a vector, e.g., the two routes in Figure 1 are mapped as \((1,1,0,0)\) and \((1,0,1,1)\), respectively. For example, in recipe \((1,1,0,0)\), a user is served by a satellite network service provider. Since we assume there is no loop in the routes, each recipe \(R\) can be represented by a path in a tree, as shown in Figure 1(a). The user category is used as the root node for convenience so that each path from the root to a leaf is a feasible recipe. An example of the values and bandwidths of the participants is shown in Figure 1(b) (note that the values and bandwidths are not necessarily integers in this paper. Integer values are used in examples for simplicity). A _procurement set_\(S\) can be generated by following a recipe and adding the served bandwidth value. For example, by following the recipe \((1,1,0,0)\), the user with a value of \(17\) can be served by the satellite network service provider with a value of \(-4\), for the served bandwidth amount of \(2\). The _gain from trade (GFT)_ of the procurement set \(S\) is defined as the sum of the values of all participants in \(S\), i.e., \(GFT(S):=\sum_{i\in S}v_{i}\times\hat{r}_{S}\), where \(\hat{r}_{S}\) is the served bandwidth amount, which satisfies \(\hat{r}_{S}\leq r_{i}\), \(\forall i\in S\). A _trade scheme_\(\mathbf{S}=\{S_{1},S_{2},...,S_{k}\}\) is a collection of procurement sets, with a gain from trade as the sum of the GFT of all procurement sets, i.e., \(GFT(\mathbf{S}):=\sum_{S\in S}GFT(S)\).