Analysis of Price of Total Anarchy in Congestion Games via Smoothness Arguments

Efficiency loss may exist at every stage of repeated play. With the rapid development of our society, how to improve the overall efficiency in repeated games becomes increasingly crucial. This paper studies the performance of a sequence of action profiles generated by repeated play in a multiple origin-destination network. To analyze the overall efficiency of these action profiles, the price of total anarchy (POTA), defined as the worst-case ratio of the average total latency over a period of time and the total latency of any optimal strategy, is adopted. First of all, it is shown that the sequence of action profiles generated by best response principle with inertia possesses almost sure no-regret property. Then, we identify the upper bound of POTA via smoothness arguments for both linear and nonlinear latency cases. After that, we analyze the effect of road pricing on POTA in traffic networks with linear latency function. It is shown that the road price we designed can reduce the upper bound of POTA compared with that without road price. In addition, how the inaccurate parameter information of the latency function affects the POTA is discussed for the linear latency case. Moreover, in a traffic network with heterogeneous players, we show that the upper bound of POTA is the same as that in a network with homogeneous players as time goes to infinity. Finally, the results are applied to a traffic routing problem based on the real traffic data of Singapore.