# Evolutionary Game for Content Cache in a mm-Wave-Based Vehicular Fog
## 4 Evolutionary Content Cache Game  
The content cache policy for a vehicular fog can be considered as a non-cooperative game, where the vehicles as game players play in selfish, only to request contents but not to share theirs. At last, vehicles can face the situation that no one stores coded blocks of contents and delivers them. Or, the vehicles can experience excessive interference with only a few cache nodes. For this, we apply the evolutionary game theory to the content caching, which leads the players to a robust equilibrium strategy by repeated strategic interactions. In detail, players change their strategies by comparing with others strategies' payoff and eventually reach a dominant strategy that all players adopt and no mutant strategy can invade, called Evolutionarily Stable Strategy (ESS). In the model of road sectorization, a group of vehicles located on a same sector is regarded as a single player in this evolutionary game, denoted by \(S=\{s_{1},...,s_{K}\}\). Each player has a strategy space \(\{C,D\}\), cooperation (\(C\)) and defection (\(D\)); for cooperation, caching vehicles share coded blocks of contents and spend time to deliver them to clients as servers. Otherwise, vehicles only request contents as clients without caching. When a client node \(i\) has neighbor caches in \(S\) sectors and has \(k\) cooperators among them, a payoff function of defection can be  
\[U_{D}(i|k,S,T)=\delta_{i}P_{c}(k,S,T) \tag{29}\]  
where the payoff is instantaneous throughput derived by connectivity \(P_{c}\) with SINR threshold, \(T\) and data rate \(\delta\). We assume that topology and interference are not changed during transmission time. A payoff function of cooperation substrates costs from the payoff of defection. \[U_{C}(i|k,S,T)=U_{D}(i|k,S,T)(1-\frac{\tau}{k}), \tag{30}\]  
where \(\tau\) is a ratio of uploading data rate for serving content to the downloading data rate \(\delta\) as a cost factor, and this cost is divided by total cooperators, \(k\). For instance, \(\tau\) can be 0.1 if the downloading rate is 1 Mbps and the uploading rate is 100 kbps. In other words, serving duration per vehicle becomes shorter as the number of cooperators increases. When the fraction of cooperation for the content cache is \(p_{c}\), the average payoff of cooperation and defection can be calculated as below. \[\overline{\mathbb{U}_{\mathbb{D}}}(p_{c})=\sum_{k=1}^{N-1}\binom{N-1}{k}\ p_{c}^{k}(1-p_{c})^{N-1-k}U_{D}(k) \tag{31}\]  
\[\overline{\mathbb{U}_{\mathbb{C}}}(p_{c})=\sum_{k=1}^{N-1}\binom{N-1}{k}\ p_{c}^{k}(1-p_{c})^{N-1-k}U_{C}(k+1). \tag{32}\]Using Equations (31) and (32), we can derive an overall payoff \(U(p_{c})=p_{c}\overline{\mathbb{U}_{\mathbb{C}}}(p_{c})+(1-p_{c})\overline{ \mathbb{U}_{\mathbb{D}}}(p_{c})\). Since this payoff is determined by the \(p_{c}\), we are interested in the population state, \(p_{c}\) when the content caching is at equilibrium state by the ESS. If the \(p_{c}^{*}\) is given by the ESS, \(U(p,\epsilon p+(1-\epsilon)p_{c}^{*})\leq U(p_{c}^{*},\epsilon p+(1-\epsilon)p _{c}^{*})\) is satisfied for all mutant strategies \(p\
eq p_{c}^{*}\) as equilibrium condition. In our evolutionary game, two different scenarios are considered for a centralized and distributed algorithm; vehicles exchange their own strategy information or not. For the centralized algorithm, vehicles can choose own strategy based on others' strategies. For the distributed algorithm, vehicles decide their own strategy based on their past experience. For the evolution of the population state, replicator dynamics need to be designed carefully in order to reach the ESS. In this study, we define the replicator dynamics as  
\[\dot{p}_{c}=\eta p_{c}(1-p_{c})[U(c,p_{-c})-U(p)], \tag{33}\]  
where the \(p_{c}\) grows up with the rate of differential payoffs between pure cooperation strategy, \(c\) and mixed strategy (cooperation or defection), and \(\eta\) is a growth rate. These replicator dynamics can be applied to the centralized algorithm and show \(p_{c}^{*}\) by the ESS.