## 2 Materials and Methods
### Behavior in Vickrey-Nearest-Core Pricing  
In the CCA (or the CA with an ascending rule), prices are determined according to the Vickrey-nearest core (VNC) pricing, which is known as the ideal pricing rule for truthful bidding [22]. This is because a winner pays the social opportunity cost at each round, so that the rule can maximize the total social opportunity cost at the final round [27]. [PERSON] et al. [28] addressed that a bidder only shifts to relatively cheaper packages (or blocks). This behavior can be illustrated in Equation (1). \[q_{t^{\prime}}(p_{t^{\prime}}-p_{t})\leq q_{t}(p_{t^{\prime}}-p_{t}) \tag{1}\]  
where \(q_{t}\) is a package at time \(t\); \(q_{t}\) is a suggested price at time \(t\); and \(t^{\prime}>t\). Moreover, Equation (2) represents a condition of revised package bid \(b(q_{t})\) at time \(t\). \[b(q)\leq b(q_{t})+(q-q_{t})p_{t} \tag{2}\]  
This implies that all supplementary bids (i.e., \(b(q)\)) should be less profitable than \(b(q_{t})\). Each bid for \(q_{t}\) is improved in the supplementary round. This is known as the revealed preference rule, in which all bidders try to select the most profitable package in the CCA [25]. Equation (2) can be rewritten as Equation (3) with the final round (\(f\)) [22]. \[b(q)\leq b\left(q_{f}\right)+\left(q-q_{f}\right)p_{f} \tag{3}\]  
From Equation (3), two propositions can be derived as follows: (1) the final assignment of spectrum blocks is identical to the clock assignment when there is no excess supply; (2) the winning price should have its maximum value at the final round when there exists excess supply. This is because in the auction, \(p_{t}\) increases over time \(t\) (or multiple rounds). The question then is how a bidder increases his/her bidding price over multiple rounds. In fact, the behavior of four bidders in the German 4G spectrum auction in 2010 can answer this question. In 2010, the market shares of T-Mobile, Vodafone, O\({}_{2}\), and E-Plus were 34%, 30%, 19%, and 17%, respectively [29]. During the auction, each company acted differently based on its bargaining power. For example, E-Plus (i.e., the weakest bidder in terms of the market share) reduced its demand on spectrum blocks at 800 MHz, and O\({}_{2}\) took over E-Plus from KPN with 68.55 billion in 2013 after the auction. On the other hand, the three other bidders tried to buy more blocks for their business (see [29] for more detail). The bargaining power that is heavily influenced by the market share of a company affects the results of the auction. Similar to the German 4G spectrum auction, in the 3G spectrum auction in the UK in 2000, only five companies, TIW, Vodafone, BT3G, One2 One, and Orange, could win in auctions for five licenses over 150 rounds from 6 March 2000 to 27 April 2000 [30]. The market shares of Vodafone, BT3G (or BT Cellnet), Orange, and One2 One were 37.3%, 30.1%, 17.2%, and 15.4%, respectively (N. M. Rothschild & Sons, 1999). TIW is one of another nine overseas firms that joined in the auction. As a result, the company that has the most bargaining power is likely to win the auction. Even though a big company may not win the auction, it can take other companies with spectrum licenses (see the German 4G spectrum auction). The budget constraint cannot be ignored in the auction, because of the high selling price. Moreover, according to [30], the bidding in early rounds is arbitrary. Although there was significant increase in the auction price during the latter rounds, both auctions in Germany and the UK followed the SMRA (or SAA). This implies that the bid sniping issue in both auctions can be resolved by adopting the CCA [22]. In fact, similar to the SMRA (or SAA), bidders in the CCA have their preferences on certain spectrum blocks for the success of their business. Under the budget constraint, they try to maximize their utility in the future. If a bidding price is discounted, a bidder tries to buy more spectrum blocks to achieve a competitive advantage [31].