## 6 Considerations on a Proposed Auction for Brazilian 6G
### Clock Phase
In case of excess demand or market clearing for a particular item, its price for the subsequent round is raised according to Table 10. If this increment results in \(O_{i}>D_{i}\), we return to the previous round price where \(O_{i}\leq D_{i}\). If, for instance, there are three more blocks demanded than offered, then the minimum price for the subsequent round should be set following an a priori rule such as \(p_{i}^{n+1}=1.04(p_{i}^{n})\), for example. The iterative procedure continues until there is no additional excess demand. Table 10 illustrates a suggested progression of price increases (the activity rule controls price increases after each-round when there is excess demand) that must be defined a priori by the auctioneer and made broadly known. For simplicity, we applied a different activity rule in each step of the example, as needed for conceptual demonstrations. The subsequent round of the auction is represented in Table 11. Here, the rule \(p_{i}^{n+1}=2p_{i}^{n}\) is used for all items where \(D_{i}^{n}\leq O_{i}\). The minimum prices are raised from USD 3 to USD 6 for all items, as there is no excess supply in the initial round. This new activity rule shows a clear problem, as explained in Table 8. Player 3 is not willing to pay more than USD 4 for each unit of item {C}. Thus, there is a market imbalance: \(O_{\text{C}}>D_{\text{C}}^{n}\) by one unit. As explained previously, the auctioneer must return to the last-round price where \(D_{i}^{n}\leq O_{i}\). In this example, the auctioneer is forced to set \(P\{C\}=3\) for all subsequent \begin{table}
\begin{tabular}{c c} \hline \hline
**Excess Demand** **(in Number of Blocks)** & **Price Increment** \\ \hline \(<\)0 & 0\% \\
0 & 1\% \\
1 & 2\% \\
2 & 3\% \\
3 & 4\% \\ \(\geq\)4 & 5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Table of price increment. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Player** & **P {A**} & **P {B**} & **P {C**}** & **D {A**}** & **D {B**}** & **D {C**}** & **Cost** & **Cost** & **Total** & **Cost** \\  & & & & & & **[A]** & **[B]** & **[C]** & **[C]** \\ \hline
1 & 3 & 3 & 3 & 3 & 2 & 0 & 9 & 6 & 0 & 15 \\
2 & 3 & 3 & 3 & 2 & 2 & 3 & 6 & 6 & 9 & 21 \\
3 & 3 & 3 & 3 & 5 & 0 & 5 & 15 & 0 & 15 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: First-round bids and budget constraints. rounds and consider the market to be \"solved\". In other words, we observe that the critical point for item {C}--\(\
u_{3}\{C\}=4\)--is wasted due to the implementation of an inefficient activity rule. Item {B}, meanwhile, is not affected by this new activity rule: Players 1 and 2 continue to demand two units each, as the auctioneer does not reveal provisional winners. Players know only that market {B} did not clear in the initial round. Thus, they continue to demand while \(P\{B\}\leq\
u_{i}\{B\}\) for all \(i=\{1,2\}\). It is possible for more than one price combination to be considered active at the end of the clock round. This example can be observed in Table 12 in the market for product {B}. In this round, we apply the activity rule again, adding 10% to the price of the previous round, except for market {C}, which is considered solved. The demand from Player 1 is altered at this point, as \(\
u_{1}\{B\}<P\{B\}\), resulting in a demand \(2=D_{B}^{n}<O_{B}\). Player 2, in turn, continues to demand units of this good. The auctioneer, as previously established, defines \(P\{B\}^{n-1}\) according to the last clock round in which \(D_{B}^{n}\leq O_{B}\), that is, \(P\{B\}=6\) as observed in the previous round \(n-1\). The players gain an important advantage at the end of this round: it becomes common knowledge that \(6\leq(P_{B}^{n})\leq 6.6\), where \(P_{B}^{n}\) denotes the critical market clearing price for {B}, allowing them greater flexibility with their budgets if they wish to demand other available goods or greater quantities.