## 2 Problem Statement
### Cache Placement Model  
Defining \(\mathcal{F}=\{1,\dots,F\}\) as a collection of \(F\) files, the local popularity of the files is predicted using the Zipf distribution. The temporal dynamics of content popularity distribution exhibit a lower rate of change over time, so it is reasonable to assume that the content popularity is static. Given that content popularity conforms to the Zipf distribution [21], the popularity of any file \(f\) can be represented as:  
\[P_{f}=\frac{f^{-\varsigma}}{\sum_{j=1}^{F}j^{-\varsigma}} \tag{11}\]  
where \(\varsigma\) is the Zipf distribution parameter and \(0<\varsigma<1\) denotes the skewness of the popularity distribution considering a time duration of \(T\), during which each user independently requests a single piece of content. Thus, for user \(UE_{n,m}\), the probability of requesting file \(f\) is defined as \(q_{n,m,f}=\frac{f^{-\varsigma}}{\sum_{j=1}^{F}j^{-\varsigma}}\), and \(\sum_{f\in\mathcal{F}}q_{n,m,f}=1\).  
Assume that each file in the library has the same size, denoted \(Q\). In practical scenarios, files may not necessarily have the same size, but they can be partitioned into equally sized file blocks, treating each file block as a special file of the same size. Therefore, the aforementioned assumption remains applicable in real situations. Due to the limited cache capacity \(S\), UAV can only cache a subset of the content collection \(\mathcal{F}\) from the file library. Let \(C=\left\{c_{\kappa,f}\right\}\), \(\kappa\in\{1,2\}\), and \(f\in\mathcal{F}\) denote the cache placement vector sets for UAV regarding the content of the file library, where \(c_{\kappa,f}\in\{0,1\}\) and \(c_{\kappa,f}=1\) represent UAV caching files \(f\) and \(c_{\kappa,f}=0\) from the operator \(CP_{\kappa}\), respectively, while not caching.  
According to the UAV caching system model, there exists a competitive relationship of interests between content providers and the UAV. Specifically, content providers seek to elevate file rental fees to augment their revenue, while the UAV aims to rent and cache files at appropriate prices to reduce round-trip link costs. When caching fees are excessively high, UAVs experience diminished profits, consequently reducing their demand for file rentals and thereby causing a reduction in revenue for content providers. Hence, a Stackelberg game model is established between content providers and UAVs, with content providers assuming the role of leaders and UAVs as followers. Followers formulate corresponding caching strategies based on the file rental fees determined by leaders. As a single UAV obtains cached files from two mobile content providers, and considering the differential yet homogeneous nature of products offered by these providers, operators will encounter price competition. In essence, the two content providers compete through dynamic pricing of popular files to supply cached files to the UAV.  
Therefore, the revenue of the content provider \(CP_{\kappa}\) can be represented as:  
\[U_{cp_{\kappa}}=\sum_{f\in\mathcal{F}}c_{\kappa,f}\upsilon_{\kappa,f} \tag{12}\]  
where \(c_{\kappa,f}\) represents the caching placement variable for the UAV and \(\upsilon_{\kappa,f}\) denotes the pricing of files \(f\) by content providers \(CP_{\kappa}\).