## 3 Spectrum Auction Structures
### Simultaneous Multiple-Round Auction
#### 3.5.1 Exposure  
In the SMRA format, equilibrium is both present and stable (for more information, please refer to the seminal article [11]), under the assumption of mutually substitutive licenses and straightforward bidding. However, if even a single player prefers to combine items, this can disrupt the equilibrium. Table 2 presents a simplified version of this problem. Let the valuation of \(P_{1}\) for licenses A and B be \(a\) and \(b\), respectively. However, the combination AB is valued at \(V_{1}=a+b+c\). In contrast, \(P_{2}\) values each license at \(a+d\) and \(b+d\), regardless of whether you obtain both licenses or just one. Let \(\xi<d<c\). The auctioneer would prefer to assign both licenses to \(P_{1}\) at prices \(a+b+c\), as this allocation achieves the highest efficiency in terms of both distribution and revenue. However, without any guarantee of securing both licenses, \(P_{1}\) is unwilling to bid higher than \(a\) for the first item, as doing so would expose him to risk. To outb \(P_{2}\), the first player must bid \(p_{a}\geq a+d\) and \(p_{b}\geq b+d\). However, at those prices, \(P_{1}\) would prefer to abstain entirely from the auction. Consequently, there is no equilibrium in this structure. The outcome is consistently that \(P_{2}\) wins both licenses separately at prices \(a+\epsilon\) and \(b+\epsilon\), (with \(\epsilon\) representing the smallest possible increment), resulting in inefficiencies. Although the previous example illustrates that there is a breakdown in equilibrium due to combinatorial preferences, this phenomenon also occurs with items sold individually. Table 3 illustrates a scenario in which the fear of exposure causes an auction to stall, as each player's optimal strategy is to wait for another player to make a move. Let the players be restricted in their actions as follows: \(P_{1}\) is eligible to bid on both licenses, while \(V_{1A}=15\) and \(V_{1B}=30\). Player 2 is restricted to license A, and \(V_{2A}=10\). Finally, \(P_{3}\) is allowed to bid solely on license B, with varying valuation probabilities. It is assumed that the bidding increments are uniform. Each player has an incentive to wait for the other players to bid before determining their own actions. Consider \(P_{2}\) as our case study: In a scenario where \(J_{3}\) reveals a preference with a \(b_{3B}>\) USD 5, all players immediately recognize that \(V_{3B}=15\). Since \(V_{1B}>V_{1A}\), it is common knowledge that \(b_{1B}=16\). All players' budgets are standardized at USD 20; therefore, \(P_{1}\) no longer has sufficient budget to outbied \(P_{2}\) for license A, resulting in \(b_{2A}=\) USD 5 being adequate to secure the win. Player 2's payoff is then \(\pi_{2}=10-5=5\). Consequently, \(P_{2}\)'s optimal strategy is to consistently wait for the other players to bid, to maximize their payoff. This strategy applies equally to all other players. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Licenses**} \\
**Player** & **A** & **B** & **AB** \\ \hline
1 & a & b & a + b + c \\
2 & a + d & b + d & a + b + d \\ \hline \hline Adapted from [11]. & & & \\ \end{tabular}
\end{table} 
Table 2: Exposure. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Licenses**} & **Budget** \\
**Player** & **A** & **B** & \\ \hline
1 & 15 & 30 & 20 \\
2 & 10 & - & 20 \\
3 & - & 0.9\% = 15 & 20 \\  & & 0.1\% = 5 & \\ \hline \hline Adapted from [11]. & & & \\ \end{tabular}
\end{table} 
Table 3: The case for an Activity Rule. This example serves as a foundation to illustrate the necessity of an activity rule. By implementing a rule that requires each player to announce their bids each round while increasing the prices, this vulnerability can be effectively mitigated. Let the minimum bid for a given round be \(MB^{n+1}=HB^{n}+\epsilon(p\lor\hat{p})\), where \(HB^{n}\) represents the highest bid submitted in round \(n\). The second term represents the increase in the price vector, as previously defined by the established rule.