## 6 Considerations on a Proposed Auction for Brazilian 6G
### Clock Phase  
We use the simplified game structure presented in Table 8 to illustrate the process. The quantities offered for each item are limited to four units, except for C, which can be divided into two pairs, namely, two blocks related to the 708-713 range and two blocks related to the 713-718 MHz range. This information is not revealed to the bidders during the _clock_ phase; it is kept by the auctioneer. The bidders only know that C contains four blocks of 5 MHz each. The table illustrates the preferences of the players as follows: A (USD), B (USD), and C (USD) represent the maximum price each player is willing to pay (in monetary units) for each item \(i\), and \(D\{A\},D\{B\}\), and \(D\{C\}\) represent the maximum amount demanded by each player for each item. Taking the first player as an example, \(v_{1}\{A\}=10\) and \(D_{1}\{A\}=3\), which means that Player 1 is willing to demand up to three units of item A, for all \(p_{A}^{n}\leq 10\). All players are subject to the same budget constraint of 50 monetary units. In the first phase, the generic spectrum blocks are offered at a minimum price \(p_{i}^{0}\) for each block \(i\) set by the auctioneer. Each block represents a license to exploit spectrum within a specific range. The players must decide whether to demand each block, together with the desired quantity, while honoring their bids in case the round results in _market clearing_. That is, \(D_{i}^{n}=O_{i}\), where \(D_{i}^{n}\) is the total demand in the \(n\)th round and \(O_{i}\) is the quantity offered for each block \(i\). Table 9 illustrates a scenario in which \(p_{i}^{0}=3\forall i\). Taking Player 1 as an example in the first row, we can observe that given \(p_{A}^{0}\leq v_{1}\{A\}\), the player will demand units for this item, subject to their budget constraint. The same player is interested in demanding up to three units, as represented by \(D_{1}\{A\}\) in Table 8. The total amount spent on item A is represented in the column Cost {A}, and the sum of expenses for all items is subject to the player's budget constraint. Thus, Player 1 forms the package \(\{3A,2B\}_{1}=15\), which is considered valid as it does not violate their budget. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Player**} & **A** & \multirow{2}{*}{**D {A**}**} & \multirow{2}{*}{**B (USD)**} & **D {B**} & **C** & **Budget** & **(USD)** \\  & & & & & & **(USD)** & **D {C**} & **(USD)** \\ \hline
1 & 10 & 3 & 6 & 2 & 0 & 0 & 50 \\
2 & 8 & 2 & 8 & 2 & 7 & 3 & 50 \\
3 & 15 & 5 & 0 & 0 & 4 & 5 & 50 \\ \hline \end{tabular}
\end{table} 
Table 8: Individual prices (\(v_{i}\)) and quantities demanded. If the quantity demanded for a block \(D_{i}^{n}\) exceeds the quantity offered \(O_{i}\), a new round is opened, in which \(p_{i}^{1}>p_{i}^{0}\), where \(p_{i}^{1}\) is the price of block \(i\) in round 1, and this process continues until _market clearing_ for all the blocks offered. The quantities demanded by the players must obey the Relaxed Generalized Axiom of Revealed Preferences (RRP), which means that the quantities must be consistent (the axiom and its mechanisms can be found in [10]) from round to round. An increase in the price of a particular block cannot imply an increase in its demand. For example, if Player 1 demanded \(x\) units at price \(p_{i}^{n}\), they could not demand \(x+1\) units at price \(p_{i}^{n+1}>p_{i}^{n}\), as this would be contradictory; if they were interested in \(x+1\) units, then it should already have been demanded in the previous round. This forces players to behave according to their real demands: if one wishes to be a whale at the end of the process, one must behave like so from the beginning. The excess demand, in turn, is determined by the number of requested blocks minus the number of offered blocks. In the example of Table 9, excess demand for item {A} is \(4-10=-6\), indicating excess supply for item {A}. Item {B} shows \(4-4=0\), indicating market clearing, while {C} shows a deficit of four units.