Price of anarchy of traffic assignment with exponential cost functions

The rapid evolution of technology in connected automated and autonomous vehicles offers immense potential for revolutionizing future intelligent traffic control and management. This potential is exemplified by the diverse range of control paradigms, ranging from self-routing to centralized control. However, the selection among these paradigms is beyond technical consideration but a delicate balance between autonomous decision-making and holistic system optimization. A pivotal quantitative parameter in navigating this balance is the concept of the "price of anarchy" (PoA) inherent in autonomous decision frameworks. This paper analyses the price of anarchy for road networks with traffic of CAV. We model a traffic network as a routing game in which vehicles are selfish agents who choose routes to travel autonomously to minimize travel delays caused by road congestion. Unlike existing research in which the latency function of road congestion was based on polynomial functions like the well-known BPR function, we focus on routing games where an exponential function can specify the latency of road traffic. We first calculate a tight upper bound for the price of anarchy for this class of games and then compare this result with the tight upper bound of the PoA for routing games with the BPR latency function. The comparison shows that as long as the traffic volume is lower than the road capacity, the tight upper bound of the PoA of the games with the exponential function is lower than the corresponding value with the BPR function. Finally, numerical results based on real-world traffic data demonstrate that the exponential function can approximate road latency as close as the BPR function with even tighter exponential parameters, which results in a relatively lower upper bound.