## 3 Spectrum Auction Structures
### Vickrey Auction or Second Price Auction  
[PERSON] [8] proposed an auction structure designed to eliminate the winner's curse as well as underbidding, winning the 1996's Nobel Prize in Economics. In general terms, the Vickrey auction can be described as follows. Let the set of players be defined as \(I=1,\ldots,n\). The action space is defined within \(\mathbb{R}_{+}\), that is, bids are strictly nonnegative and there is no upper limit to the possible bid amount. Therefore, the possible bids are given by the Cartesian product \(\mathbb{S}=\{S_{1},\times\cdots\times S_{n}\}\). Each player's bid is mapped through \(b_{i}\). The two highest bids are considered when defining the winner and the final price of the auction. \[b^{\text{max}}=\max\{b_{1},\ldots,b_{n}\}\quad\text{and}\quad b^{\text{2 nd}}= \max\big{\{}\{b_{1},\ldots,b_{n}\}\setminus\{b^{\text{max}}\}\big{\}}, \tag{2}\]  
where \(b^{max}\) is the maximum among all bid values and \(b^{\text{2 nd}}\) the maximum of the remaining set, excluding \(b^{max}\). In a Vickrey auction, the bids are sealed and unknowable to other parties. The highest bidder is awarded the prize. However, the price is equal to the second highest bid received. As expounded by [4], supposing the absence of moral risks, collusion, and certain axioms characterizing the preferences of the players, it is possible to represent each player's preferences \(i\in I\) by the following utility function:  
\[U_{i}= \begin{cases}V_{i}-p^{*},&\text{if }i=\hat{i},\\ 0,&\text{if }i\
eq\hat{i}.\end{cases}  \tag{3}\]  
The utility function allows us to map each player's payoff:  
\[\pi(b_{1},\ldots,b_{n})= \begin{cases}V_{i}-b^{\text{2 nd}},&\text{if }i=\hat{i}(b_{1},\ldots,b_{n}),\\ 0,&\text{if }i\
eq\hat{i}(b_{1},\ldots,b_{n}).\end{cases}  \tag{4}\]  
This specific game structure leads to the conclusion that, in every scenario, each player's optimal strategy is to bid their true valuation of the item. In other words, each player \(i\) has an incentive to bid \(b_{i}=V_{i}\). This format ensures efficiency and mitigates the curse of the winner. It may seem counterintuitive to consider the concept of the second price. However, in both continuous- and multiple-round English auctions, where infinitesimally small bid increments are possible, participants will continue to raise their bids by such amounts until the auction concludes. This occurs precisely when the losing bidder withdraws, which corresponds to the exact amount of the second price. Since English and Dutch auctions are strategically equivalent, a Vickrey auction is equally efficient in terms of allocation for all parties involved. The second-price mechanism was extensively utilized in the early 2000s due to rapid technological advancements and widespread Internet access. The website eBay successfully negotiated bids of up to USD 200 million monthly, according to [14]. The users were asked to enter the maximum value they were willing to pay for an asset (\(V_{i}\)), which remained confidential to all other participants. The website used an algorithm to discreetly place bids, in small increments, up to the maximum bid amount (\(b_{i}=V_{i}\)). The process involved comparing all the values submitted \(V_{i}\), selecting the two highest offers. The highest bid received the award, while the second highest bid determined the final price paid. The work by [PERSON] (2018) illustrates the second-price auction mechanism through a mock oil reserve auction. In this example, bids are made in increments of ten million dollars, and a tie-breaking rule favors Player 2 (\(P_{2}\)). Table 1 presents the payoffs (\(\pi_{i}\)) received by each player. The value of the oil license is USD 30 million. The first line in the table shows that, in all cases, a bid of USD 10 by \(P_{1}\) is insufficient to win, regardless of \(b_{2}\). In all other cases, should \(b_{2}=10\), \(P_{1}\) wins the dispute, while its payoff is calculated as \(\pi_{1}=V_{1}-b_{2}\). The payoff for Player 1 would result in \(\pi_{1}=30-10=20\), regardless of \(p_{1}\).