The lack of a unique user equilibrium (UE) route flow in traffic assignment has posed a significant challenge to many transportation applications. The maximum-entropy principle, which advocates for the consistent selection of the most likely solution as a representative, is often used to address the challenge. Built on a recently proposed day-to-day (DTD) discrete-time dynamical model called cumulative logit (CULO), this study provides a new behavioral underpinning for the maximum-entropy UE (MEUE) route flow. It has been proven that CULO can reach a UE state without presuming travelers are perfectly rational. Here, we further establish that CULO always converges to the MEUE route flow if (i) travelers have zero prior information about routes and thus are forced to give all routes an equal choice probability, or (ii) all travelers gather information from the same source such that the so-called general proportionality condition is satisfied. Thus, CULO may be used as a practical solution algorithm for the MEUE problem. To put this idea into practice, we propose to eliminate the route enumeration requirement of the original CULO model through an iterative route discovery scheme. We also examine the discrete-time versions of four popular continuous-time dynamical models and compare them to CULO. The analysis shows that the replicator dynamic is the only one that has the potential to reach the MEUE solution with some regularity. The analytical results are confirmed through numerical experiments.

In this paper, we develop an approximation scheme for solving bilevel programs with equilibrium constraints, which are generally difficult to solve. Among other things, calculating the first-order derivative in such a problem requires differentiation across the hierarchy, which is computationally intensive, if not prohibitive. To bypass the hierarchy, we propose to bound such bilevel programs, equivalent to multiple-followers Stackelberg games, with two new hierarchy-free problems: a $T$-step Cournot game and a $T$-step monopoly model. Since they are standard equilibrium or optimization problems, both can be efficiently solved via first-order methods. Importantly, we show that the bounds provided by these problems -- the upper bound by the $T$-step Cournot game and the lower bound by the $T$-step monopoly model -- can be made arbitrarily tight by increasing the step parameter $T$ for a wide range of problems. We prove that a small $T$ usually suffices under appropriate conditions to reach an approximation acceptable for most practical purposes. Eventually, the analytical insights are highlighted through numerical examples.