# Evolutionary Game for Content Cache in a mm-Wave-Based Vehicular Fog
## 1 Introduction
To say, a vehicle can decode a content if it can retrieve a sufficient number of coded blocks from any surrounding vehicles even in dynamic vehicular topology. However, vehicles may not be able to cooperate for sharing local cache in the vehicle fog because it costs not only the storage, but opportunity for data reception; it has to spend time for serving contents. As the number of the free-riders increases, total utility as throughput will decrease due to severe inter-beam interference in the dense vehicular fog. In this paper, we apply an evolutionary game theory for the content cache in mm-Wave-based vehicular fog to solve the free-rider problem. This evolutionary game enables vehicles to learn the best strategy from repeated games and choose one of two strategy, cooperation or defection in order to maximize their own profit. For this, we propose a replicator dynamics equation to update the strategy of each vehicle and show that such behaviour dynamics can lead vehicles to  
Figure 1: Vehicular fog architecture with directional mm-Wave communication links. converge into an evolutionarily stable strategy (ESS). To derive the utility of each vehicle, first, we establish a communication model of mm-Wave-based vehicular fog based on 2D-Poisson Point Process (PPP) in a multi-lane road, where the road is sectorized by beam-width of a directional antenna and interference is calculated by different beam-forming of neighbor vehicles according to content locations. Second, we apply one of two different algorithms, global and local replicator dynamics, to decide caching probability depending on whether the vehicles have global strategy information or not. From experiments for the global dynamics, we can observe that vehicles try to cooperate for caching when the cooperators are few because payoff from the cooperation is more than the cost. However, the connectivity gain from diversification of beam directions is limited as the proportion of cooperators becomes high. Accordingly, the cooperator proportion of the ESS is the point at which the payoff of cooperation and defection is the same. The local dynamics allow vehicles to decide their own strategy based on their past experience and converge into 1 or 0 for cooperation probability. Thereby, overall utility of the vehicular fog can have a stable state even for the heterogeneous players, but which can be varying with the initial probability of cooperation. The rest of this paper is organized as follows. First, we introduce a concept of the vehicular fog and explain how to store contents in coded cache briefly in Section 2. We review 5G mm-Wave communications and discuss recent works on mm-Wave-based V2X communications. We establish a mode of mm-Wave V2X communications at multi-lane roads in Section 3, where we first introduce a propagation model of mm-Wave and a radiation pattern of a directional antenna, build a model of sectorized beams and content caches in a multi-lane road, and calculate connectivity of a V2V link. In Section 4, the evolutionary game is proposed for content cache in the vehicular fog. In Section 5, we evaluate performance on the mm-Wave V2X communication with distributed content caches, and discuss experimental outcomes. We conclude this paper in Section 6.