## 3 Spectrum Auction Structures
### Generalized Vickrey Auction--Gloves-Clarke Auction  
As previously discussed, the second-price auction is considered the most effective format for auctioning individual items. In practice, however, goods often exhibit certain complementaries. This characteristic has prompted some authors to extend the Vickrey auction into a generalized form, which allows for greater flexibility while preserving the core features of second-price auctions, such as preventing collusion and exposure, and ensuring the bidding process remains truthful.  
The results of the Gloves-Clarke auction from these attempts successfully achieved most of the proposed goals. However, if several synergistic items are sold simultaneously, the computational burden required to analyze each possibility and combination of revenue becomes prohibitively high. Consequently, this format is beyond the scope of this article; however, it remains tangentially relevant due to its efficiency in resource allocation,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Licensees**} & \\
**Player** & **A** & **B** & **Budget** \\ \hline
1 & 10 & 10 & 20 \\
2 & 10 & 10 & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Collusion.  
serving as a theoretical benchmark against which all other combinatorial auctions can be compared [11; 16; 17].  
In a Generalized Vickrey Auction (GVA), the auctioneer must evaluate all possible combinations. Let \(L\) represent a set of licenses and \(P\) the set of allocations; thus, \(P\) is an indexed subset of \(L\). For any allocation \(S\subset P\), the element \(S_{i}\) signifies the set allocated to player \(i\). The rules of a GVA stipulate that each player submits a bid for a nonempty set that is a partition of \(L\). The auctioneer then needs to analyze and compare all submitted bids, ultimately selecting an allocation \(S^{*}\) that maximizes efficiency.  
In a scenario with \(L\) licenses, the total number of possible combinations is \(2^{L}-1\). In the Netherlands, an auction featuring 18 licenses generated over 260,000 combinatorial options to be evaluated. Auctions such as Auction 101, conducted by the FCC in 2019, began with 3072 licenses, while Auction 102, held in the same year, offered 2909 distinct licenses for sale.  
[PERSON] [11] proposed a mechanism known as Adaptive User Selection Mechanism (AUSM) to help alleviate the computational demands of GVAs when implemented in the SMRA format.  
By enabling users to enhance other players' bids and consolidate them into a single offer, this mechanism expedites the selection process by streamlining the algorithm utilized by the auctioneer.  
Table 6 illustrates a scenario in which four licenses are auctioned. The reader may note that, even with such small numbers, the number of combinations that the auctioneer would need to map would be 15, were it not for the AUSM.  
Player 1 is interested in the ABCD combination and is not interested in single licenses. Player 2 values A at USD 12, while Player 3 desires CD, valuing it at USD 10. All players have a minimal interest in B but would still prefer to have it rather than not.  
In round 1 of this auction, players submit bids as previously described. Player 2, however, notices \(P_{3}\)'s bid of USD 10 and realizes that this bid could be elevated to a new combined offer of \(b_{\{23\}\{ACD\}}=12+10=22\) since the interests of both \(P_{2}\) and \(P_{3}\) show no conflict. Upon receiving this offer, the auctioneer would then calculate USD 22 \(-\) USD 20 = USD 2. Since the combined offer exceeds \(b_{\{1\}\{ABCD\}}\), the licenses would be redistributed to \(P_{2}\) and \(P_{3}\), while B would be freed.  
Some additional rules need to be established to ensure the continuity of the process; however, these are relatively benign in comparison. For a more detailed explanation, the reader can refer to [11].