## 4 Auction Game
### Auction Design  
From a cooperative perspective, the optimal relay here is the one that provides the most benefit to the source transmission under the relay QoS constraint. Vickery auction has the advantage of assigning the item for sale to the bidder who values it most.  
The Vickery auction can be described as follows:  
* _Information:_ We assume all potential relays could use the maximum power \(P_{k}\) to achieve as much capacity as possible. The publicly available information includes the noise density \(\sigma_{0}^{2}\) and \(\sigma_{1}^{2}\). Each relay, acted as a bidder, could acquire the channel gain \(G_{R_{k}C_{k}}\), \(G_{R_{k}D}\) and \(G_{SR_{k}}\), which is not available for other bidders. Hence, the incomplete information prevent bidders from lying, and each bidder would offer the true value for utilizing the shared spectrum.
* _Bids:_ The value of potential relay for the satellite is the assistance transmitted capacity. All potential relay would send the bids \(R_{AF_{k}-bids}\) or \(R_{DF_{k}-bids}\) to the satellite.
* _Allocation:_ The satellite chooses the highest bids as the winner. In addition, the satellite broadcasts all the bids to all potential relays and informs the selected secondary relay the highest non-winning bids. Once the satellite is lying, all potential relay quit as a penalty. The potential Relays are physically isolated from each other, so avoiding collusion.
* _Payoffs:_ It can be aware of that the PU obtain the highest non-winning bids \(R_{pri}\) as payoffs. The chosen relay \(R_{k}\) reconfigures the sub-time slots allocation by the highest non-winning bids. For AF relay protocols, the first two sub-time slots are reconfigured as \[t^{\prime}{}_{1}=t^{\prime}{}_{2}=R_{AF_{k}}^{-1}\left(R_{pri}\right).\] (12)For DF relay protocols, the first two sub-time slots are reconfigured as  
\[\left({t^{\prime}}_{1},{t^{\prime}}_{2}\right)=R_{DF_{k}}^{-1}\left(R_{pri}\right). \tag{13}\]  
In addition, the first two sub-time slots still obey (10), which means the \({t^{\prime}}_{1},{t^{\prime}}_{2}\) can be obtained. The third sub-time slot is \({t^{\prime}}_{3}=t-{t^{\prime}}_{1}-{t^{\prime}}_{2}\). The payoffs of the relay is \(R_{R_{k}C_{k}}\left({t^{\prime}}_{3}\right)\) After the sub-time slot reconfiguration, the chosen relay informs the satellite the time slot configuration and starts the transmission. It is worth to note that if there are bidders with the same bid, the primary network would randomly select one as the cognitive relay, and the payment will be paid at the highest price.  
According to the auction mechanism proposed for relay selection and the system assumptions, the truthful bidding is a NE of the Vickery auction for the relay selection in a distributed manner.  
It can be seen from the above analysis that the satellite does not need to obtain the global CSI, and the calculation complexity that needs to be paid is to select the highest one from all the bids. Therefore, the proposed relay selection mechanism has a low complexity.  
Besides, the proposed auction-based mechanism makes sure the minimum transmission QoS requirements, i.e., \(R_{min}\), which is the embodiment of fairness and facilitates the completion of the cooperation of primary network and the secondary network. The design of this mechanism guarantees the long-term cooperation.