## 3 System Model
### Mechanism Design Problem Formulation  
In this paper, multi-domain network slicing admission control is formulated as a mechanism design problem. The designed mechanism can be classified as _deterministic_ or _randomized_. In a _deterministic_ mechanism, the outcome is purely decided by the inputs. In a randomized mechanism, the outcome is decided by both the inputs and some random variable(s). A randomized mechanism can also be seen as a lottery over deterministic mechanisms. The designed mechanism can also be classified as _direct_ or _sequential_. In a  
Figure 2: The recipe tree example: (a) a recipe tree with two recipes; and (b) the values and bandwidths of the participants. direct mechanism, the participant's private information is revealed only once, e.g., when submitting the bandwidth and price information. In a sequential mechanism, the participant's private information is revealed sequentially, e.g., by answering questions about whether a price is acceptable or not. Specifically, a _deterministic direct mechanism_ aims to find a function that takes as input a bid vector \(\mathbf{b}\) of bid/ask prices and a bandwidth vector \(\mathbf{r}\) of bandwidth requests/resources and returns a trade scheme \(\mathbf{S}(\mathbf{b},\mathbf{r})\) and a price vector \(\mathbf{p}(\mathbf{b})\). For the trade scheme \(\mathbf{S}(\mathbf{b},\mathbf{r})\) to be _bandwidth-feasible_, the following constraints must be satisfied:  
\[\sum_{S:i\in S}\hat{r}_{S}=r_{i},\forall i\in\mathcal{N}_{1} \tag{1}\]  
\[\sum_{S:j\in S}\hat{r}_{S}\leq r_{j},\forall j\in\mathcal{N}_{2}\cup\mathcal{ N}_{3}\cup\mathcal{N}_{4} \tag{2}\]  
where Constraint (1) requires that the admitted users are served exactly with the requested bandwidth and Constraint (2) requires that the consumed bandwidth from the service providers cannot exceed their available resources. The desirable properties of the designed mechanism include _optimality_ (i.e., with the maximum gain from trade), _individual rationality_, _budget balance_, _truthfulness_. It is well known that these properties cannot be achieved simultaneously in a two-sided market [54]. This impossibility theorem holds for the multi-sided market considered in this paper, which becomes a two-sided market when only one category of service providers exists. A deterministic direct mechanism is _optimal_ if the gain from trade of its trade scheme is maximum over all bandwidth-feasible trade schemes, regardless of the price vector. A deterministic direct mechanism is _truthful_ if a participant bids truthfully based on their value when maximizing their own utility. Formally, the participant's utility is denoted by \(u_{i}\), the bid is \(b_{i}\) and the bids of all other participants are denoted by \(\mathbf{b}_{-i}=\{b_{1},b_{2},\dots,b_{i-1},b_{i+1}\dots\}\). Then, a mechanism is truthful if, for every participant \(i\) and any fixed bids of all the other participants, \(u_{i}(v_{i},\mathbf{b}_{-i})\geq u_{i}(b_{i},\mathbf{b}_{-i})\). A stronger propriety called _obvious truthfulness_[55] requires that the lowest utility for a participant by bidding truthfully is at least as high as the highest utility they may obtain by acting non-truthfully. Formally, a mechanism is obviously truthful if for every participant \(i\) and bid vector \(\mathbf{b}\), \(u_{i}(v_{i},\mathbf{b}_{-i})\geq u_{i}(\mathbf{b})\). In other words, the bids of all the other participants do not need to be fixed. A deterministic direct mechanism is _individually rational_ if a participant loses nothing to participate in the market. In other words, the participant's utility \(u_{i}\) should be non-negative, regardless of the bids of all other participants. For those participants that are not in any procurement set, the price should be zero. A deterministic direct mechanism is _weakly budget-balanced_ if the total amount paid by all participants together should be at least zero. Similar to the values and bids, the prices of the service providers are negative.