## 2 Materials and Methods
### Agent-Based Simulation for the Design of a Spectrum Auction Policy
#### 2.3.3 Calibration Model  
The calibration model is used to identify parameter \(\lambda\) referring to the price sensitivity of an agent. Because the price sensitivity can vary according to market competitiveness (or budget in the auction), this parameter should be calibrated from historical data (i.e., auction results). Although this paper only introduces three types of agents in terms of competitiveness, the auction can include more than three bidders. As an example, there were  
\begin{table}
\begin{tabular}{c c} \hline
**Variable** & **Description** \\ \hline RoundNum & Round indicator \\ \hline bCompletion & Completion status of the auction (false: not finished; true: finished) \\ \hline bSelectedPrice & The existence of selected price (false: not selected; true: selected) \\ \hline arrBlockNumSold & Total number of spectrum blocks demanded by bidder agents over rounds \\ \hline arrFinalBlockNumSold & Total number of spectrum blocks sold to bidder agents over rounds \\ \hline arrSuggestedPrice & Suggested price over rounds \\ \hline arrBiddingBidderID & ID of a bidder agent \\ \hline arrBiddingPrice & A list of bidding prices \\ \hline arrBiddingBlockNum & A list of numbers of spectrum blocks demanded by agents \\ \hline arrSortedBidderID & Sorted IDs of bidder agents in ascending order \\ \hline arrSortedBiddingPrice & Sorted prices of bidder agents in ascending order \\ \hline arrSortedBlockNum & Sorted number of spectrum blocks of bidder agents in ascending order \\ \hline \end{tabular}
\end{table}
Table 1: Major variables used in the simulation model.  
Figure 6: Pseudo code of the selected price generation function.  
13 bidders at the 3G spectrum auction in the UK in 2000 [30]. To generalize the proposed model, let \(\lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}\), where \(n\) is the number of bidders; \(0<\lambda_{i}<1\); and \(\lambda_{i}\
eq\lambda_{j}\) for \(\forall i,j,1\leq i\leq n,1\leq j\leq n\). We would like to identify \(\lambda\) with the minimum estimation error in terms of the coefficient of determination (R\({}^{2}\)). Equation (7) is devised from the equation of R\({}^{2}\).  
\[Min\ Z=1-\sum_{i\in\mathbf{I}}\left(d_{i}-\hat{d}_{i}(\lambda)\right)/\sum_{i \in\mathbf{I}}\left(d_{i}-\overline{d}\right) \tag{7}\]  
Subject to the following:  
\[M\Big{|}n_{finalround}-\hat{n}_{finalround}\Big{|}\leq\delta \tag{8}\]  
\[|m_{i}-\hat{n}_{i}|=\varepsilon\ \ \ \text{for}\ \forall i\in\mathbf{I} \tag{9}\]  
\[\sum_{i\in\mathbf{I}}\hat{m}_{i}=M \tag{10}\]  
In Equation (7), \(\hat{d}_{i}(\lambda)\) is the final bidding price estimated by the simulation model under the given \(\lambda\), and \(\overline{d}\) is an average from the historical data of \(d_{i}\). Equation (8) is a constraint computing the difference between the number of rounds with the final decision (\(n_{finalround}\)), and the estimated number of rounds with the final decision (\(\hat{n}_{finalround}\)). Equation (9) refers to the difference between the number of spectrum blocks sold to bidder \(i\) (\(m_{i}\)), and the estimated number of spectrum blocks sold to bidder \(i\) (\(\hat{n}_{i}\)). Equation (10) regulates that the total number of spectrum blocks sold to bidders equals the number of the supplied spectrum blocks (\(M\)).