# Game-Theoretic Power and Rate Control in IEEE 802.11\(p\) Wireless Networks
## 1 Introduction  
Wireless networks are often constraint by Quality of Service (QoS) requirements that aim to enhance their reliability. The unstable wireless medium does not provide a stable environment for wireless parameters tuning. Issues such as link quality and rate are restrictions that need to be addressed to optimise the behaviour of a wireless network. Such QoS constraints appear in practical scenarios. Their encapsulation in an optimisation process is essential to produce efficient approaches towards optimal wireless behaviour. A major concern in wireless networks operating with the IEEE 802.11 standard is the maximisation of transmission reliability with high link quality and maximum rate, where delay limitation may be satisfied. In particular, Vehicular Ad-hoc Networks (VANETS)-based communication introduced the IEEE 802.11\(p\) standard. The difficult environment of deploying Vehicle to Infrastructure (V2I) devices is evident. Since the early days of the Vehicle to Vehicle (V2V) and V2I technologies, the necessity of a wireless standard emerged; hence, the IEEE 802.11\(p\) or Dedicated Short-Range Communications (DSRC) was established. This protocol offers a dedicated bandwidth of 75 MHz in the 5.850-5.925 frequency band, as established and allocated by the Federal Communications Commission (FCC) in the USA [1]. Furthermore, DSRC provide a high V2I of V2V data rate. In terms of Wireless Access in Vehicular Environments (WAVE), the DSRC standards comprise the IEEE 1609.\(x\) family and IEEE 802.11\(p\). Specifically, the IEEE 1609 sits on top of the IEEE 802.11\(p\) and handles upper layers from the PHY and MAC without having information about low-layer parameters. The PHY and MAC layers are specified by the IEEE 802.11\(p\) for single-channel operations. Multi-channel operations can be offered by the division of the access time to two separate channels. This is done using a trial IEEE 1609.4 on top of the IEEE 802.11\(p\)[2]. The reason behind employing game theory is that it studies mathematical models of conflict and cooperation [3] between vehicles in our example. Therefore, our meaning of the term game corresponds to any form of social interaction between two or more devices. The utilisation of a potential game approach comes as a result of the fact that in 2008, [PERSON] et al. [4] proved that finding a Nash equilibrium is PPAD-complete. Polynomial Parity Arguments on Directed graphs (PPAD) is a class of total search problems [5] for which solutions have been proven to exist; however, finding a specific solution is difficult if not intractable. There exists a class of games called \"Potential Games\" [6], which have important properties that pure Nash equilibria will always exist and best response dynamics are guaranteed to converge. This class of games consists of the exact, ordinal and weighted potential games. In this paper, we utilise exact potential games and we build a potential function that has the same behaviour as the individual utility function when a player unilaterally deviates. We take into account the physical-layer (PHY) characteristics of the IEEE 802.11\(p\) standard, and we adopt the link layer Packet Reception Ratio (PRR) measure. The PRR can be used to implicitly optimise throughput by reducing re-transmissions. To do so, we construct a model to optimise the rate and transmission power for each vehicle to enhance throughput. In contrast with the conventional methodology, we do not take into account parameters such as the Received Signal Strength Indicator (RSSI), given that for low-power transmissions, it may result into severe packet losses. Thus, we are considering the network layer parameter of PRR to model the packet reception or drop. Using this, we devise a utility function that is represented by the PRR and rate. We employ this utility function to construct a potential game, where each device can optimise its rate and transmission power in a selfish manner.