As one of the most fundamental concepts in transportation science, Wardrop equilibrium (WE) was the cornerstone of countless large mathematical models that were built in the past six decades to plan, design, and operate transportation systems around the world. However, like Nash Equilibrium, its more famous cousin, WE has always had a somewhat flimsy behavioral foundation. The efforts to beef up this foundation have largely centered on reckoning with the imperfections in human decision-making processes, such as the lack of accurate information, limited computing power, and sub-optimal choices. This retreat from behavioral perfectionism was typically accompanied by a conceptual expansion of equilibrium. In place of WE, for example, transportation researchers had defined such generalized equilibrium concepts as stochastic user equilibrium (SUE) and boundedly rational user equilibrium (BRUE). Invaluable as these alternatives are to enriching our understanding of equilibrium and advancing modeling and computational tools, they advocate for the abandonment of WE, predicated on its incompatibility with more realistic behaviors. Our study aims to demonstrate that giving up perfect rationality need not force a departure from WE, since WE may be reached with global stability in a routing game played by boundedly rational travelers. To this end, we construct a day-to-day (DTD) dynamical model that mimics how travelers gradually adjust their valuations of routes, hence the choice probabilities, based on past experiences.
Our model, called cumulative logit (CULO), resembles the classical DTD models but makes a crucial change: whereas the classical models assume routes are valued based on the cost averaged over historical data, ours values the routes based on the cost accumulated. To describe route choice behaviors, the CULO model only uses two parameters, one accounting for the rate at which the future route cost is discounted in the valuation relative to the past ones (the passivity measure) and the other describing the sensitivity of route choice probabilities to valuation differences (the dispersion parameter). We prove that the CULO model always converges to WE, regardless of the initial point, as long as the passivity measure either shrinks to zero as time proceeds at a sufficiently slow pace or is held at a sufficiently small constant value. Importantly, at the aggregate (i.e., link flow) level, WE is independent of the behavioral parameters. Numerical experiments confirm that a population of travelers behaving differently reaches the same aggregate WE as a homogeneous population, even though in the heterogeneous population, travelers’ route choices may differ considerably at WE.
By equipping WE with a route choice theory compatible with bounded rationality, we uphold its role as a benchmark in transportation systems analysis. Compared to the incumbents, our theory requires no modifications of WE as a result of behavioral accommodation. This simplicity helps avoid the complications that come with a “moving benchmark”, be it caused by a multitude of equilibria or the dependence of equilibrium on certain behavioral traits. Moreover, by offering a plausible explanation for travelers’ preferences among equal-cost routes at WE, the theory resolves the theoretical challenge posed by Harsanyi‘s instability problem. Note that we lay no claim on the behavioral truth about route choices. Real-world routing games take place in such complicated and ever-evolving environments that they may never reach a true stationary state, much less the prediction of a mathematical model riddled with a myriad of assumptions. Indeed, a relatively stable traffic pattern in a transportation network may be explained as a point in a BRUE set, an SUE tied to properly calibrated behavioral parameters, or simply a crude WE according to the CULO model. More empirical research is still needed to compare and vet these competing theories for target applications. However, one should no longer write off WE just because it has no reasonable behavioral foundation.