## 4 Theoretical Efficiency Comparison Between Combinatorial Auction Formats and Empirical Data
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Own** & **Gain** & **Auction Type** & **\% Efficiency** \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{0.81} & \multirow{3}{*}{Yes} & CCA & 100, 100, 100 \\ \cline{3-5}  & & & & CRA & 78, 79, 78 \\ \cline{3-5}  & & & & SMRA & 59 \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{0.81} & \multirow{3}{*}{No} & CCA & 100, 100, 100 \\ \cline{3-5}  & & & & CRA & 97, 79 \\ \cline{3-5}  & & & & SMRA & 63 \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{0.70} & \multirow{3}{*}{Yes} & CCA & 100, 100, 100, 100 \\ \cline{3-5}  & & & & CRA & 100, 100 \\ \cline{3-5}  & & & & SMRA & 70 \\ \hline \multirow{3}{*}{2a} & \multirow{3}{*}{0.80} & \multirow{3}{*}{Yes} & CCA & 100, 100, 99, 100, 99, 100 \\ \cline{3-5}  & & & & CRA & 99, 99, 99, 95, 94, 95, 95 \\ \cline{3-5}  & & & & SMRA & 100, 99, 95, 95 \\ \hline \multirow{3}{*}{2b} & \multirow{3}{*}{0.94} & \multirow{3}{*}{Yes} & CCA & 100, 100, 100 \\ \cline{3-5}  & & & & CRA & 91, 94, 94 \\ \cline{3-5}  & & & & SMRA & 100 \\ \hline \multirow{3}{*}{2b} & \multirow{3}{*}{0.94} & \multirow{3}{*}{No} & CCA & 100, 100, 100 \\ \cline{3-5}  & & & & CRA & 95, 95 \\ \cline{3-5}  & & & & SMRA & 100 \\ \hline \multirow{3}{*}{2b} & \multirow{3}{*}{0.80} & \multirow{3}{*}{Yes} & CCA & 100, 100, 100 \\ \cline{3-5}  & & & & CRA & 100, 91 \\ \cline{3-5}  & & & & SMRA & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Allocative efficiency. Figure 4: Auctions by format per year. CCA generally achieves higher allocative efficiency, as indicated by the revenue generated per auction type. This phenomenon is visible when the revenue per auction is compared, evaluating USD/MHz/population/year as shown in Figure 5.  
In 2021, Brazil auctioned national licenses in the 700 MHz, 3.5 GHz, and 26 GHz bands, using a sealed bid format. The Brazilian regulatory agency, Anatel, reported that the revenue generated was USD 5.285 million (values converted via USD/BRL rates of 5.6417, as the average yearly exchange rate). Revenue per band was USD 612 million for the 26 GHz band, with 1200 MHz of spectrum sold. The observed price/MHz/pop/year observed was USD 0.011. The 3.5 GHz band, which consists of 300 MHz of spectrum, generated USD 22,796 million in revenue. Thus, the price/MHz/pop/year was USD 6.2733. Finally, the remaining 20 MHz of the 700 MHz band was sold at USD 14.7406 cents/MHz/pop/year Figure 6.  
The values set by the Brazilian authorities were comparable to those observed in the rest of Latin America but significantly lower than those achieved by the United States and Canada, which employed the CCA format for their auctions. The revenue of each nation can be further analyzed by evaluating the auction format it is using. This is illustrated in Figure 7.  
Figure 5: Average price/MHz/pop/year by auction format. Figure 6: Average revenue by country. It is important to highlight Canada's exceptional SMRA performance. This achievement is indicative of the 2008 auction, where 90 MHz of spectrum was sold for nearly USD 4 billion, resulting in USD 13.1071 cents/MHz/pop/year, much higher than any other observed auction. Further research is necessary to comprehend the intricacies involved in such an auction and its remarkable outcomes.