Data-Driven Models of Selfish Routing: Why Price of Anarchy Does Depend on Network Topology

We investigate traffic routing both from the perspective of theory as well as real world data. First, we introduce a new type of games: $\theta$-free flow games. Here, commuters only consider, in their strategy sets, paths whose free-flow costs (informally their lengths) are within a small multiplicative $(1+\theta)$ constant of the optimal free-flow cost path connecting their source and destination, where $\theta\geq0$. We provide an exhaustive analysis of tight bounds on PoA($\theta$) for arbitrary classes of cost functions, both in the case of general congestion/routing games as well as in the special case of path-disjoint networks. Second, by using a large mobility dataset in Singapore, we inspect minute-by-minute decision-making of thousands of commuters, and find that $\theta=1$ is a good estimate of agents' route (pre)selection mechanism. In contrast, in Pigou networks, the ratio of the free-flow costs of the routes, and thus $\theta$, is \textit{infinite}; so, although such worst case networks are mathematically simple, they correspond to artificial routing scenarios with little resemblance to real world conditions, opening the possibility of proving much stronger Price of Anarchy guarantees by explicitly studying their dependency on $\theta$. For example, in the case of the standard Bureau of Public Roads (BPR) cost model, where$c_e(x)= a_e x^4+b_e$, and for quartic cost functions in general, the standard PoA bound for $\theta=\infty$ is $2.1505$, and this is tight both for general networks as well as path-disjoint and even parallel-edge networks. In comparison, for $\theta=1$, the PoA in the case of general networks is only $1.6994$, whereas for path-disjoint/parallel-edge networks is even smaller ($1.3652$), showing that both the route geometries as captured by the parameter $\theta$ as well as the network topology have significant effects on PoA.