# Coverage Strategy for Small-Cell UAV-Based Networks in IoT Environment
## 3 Problem Formulation
### Game Formulation  
Game theory is a branch of mathematics that studies decision making in situations where multiple agents or players are involved, each with their own objectives and strategies. It is used in a wide range of fields, including economics, political science, and psychology, to model and analyze interactions between people and organizations.  
Nash equilibrium is a central concept in game theory, named after mathematician [PERSON]. It is a solution concept for non-cooperative games that describes a set of strategies in which no player can improve their outcome by unilaterally changing their strategy, assuming that all other players also remain with their current strategies.  
In other words, a Nash equilibrium is a situation where each player is playing the best strategy they can, given the strategies of the other players. It is important to note that a Nash equilibrium does not necessarily result in the best possible outcome for all players, but rather a stable and self-enforcing outcome. The beaconing-scheduling game involves two UAVs acting as independent players, each selecting a strategy that maximizes their respective payoffs. UAV \(i\) chooses its beaconing period duration, denoted by \(\tau_{i}\), within the range from zero to \(T\). A \(\tau_{i}\) value of 0 implies that the UAV abstains from ground user detection throughout the entire activity schedule. Conversely, \(\tau_{i}=T\) indicates that the UAV maintains active beaconing for mobile users continuously. This scheduling of beaconing periods can be formally framed as a game where \(\mathcal{U}\) represents the set of UAVs, and each UAV \(i\) has an action set \(A_{i}=[0,T]\) defining its beaconing period duration. Notably, if \(\tau_{i}\) represents the beaconing period duration for UAV \(i\), its idle period extends for \(T-\tau_{i}\).  
Payoff \(u_{i}\) for UAV \(i\) comprises a reward component and a cost component. The reward hinges on the probability of successfully establishing the initial contact with ground-based mobile users during the beaconing period. Meanwhile, costs are associated with energy consumption per slot for beacon transmission and transceiver state switching. For a successful first contact, it must occur within the specified beaconing period.  
We denote \(Ps_{i}(\tau_{i},\tau_{j})\) as the probability of both drones selecting beaconing durations \(\tau_{i}\) and \(\tau_{j}\), respectively. Only the first UAV that successfully encounters mobile users during its beaconing period serves as an airborne access-point base station. Consequently, the beaconing period duration of each UAV influences the payoff of the other.  
When examining the perspective of an individual UAV, there exists a trade-off between encounter rate and throughput. On one hand, an increase in beaconing duration leads to a higher encounter rate (\(Ps\)). On the other hand, throughput is directly proportional to the beaconing period duration, denoted by \((T-\tau_{i})\). It is important to note that we assume UAVs to be in an awake (listening) state, and energy consumption is solely attributed to the beaconing rate.  
To control the data rate, we introduce the term \((T-\tau_{i})\); the utility function varies accordingly with the variation in beaconing duration.