Stability analysis of a departure time choice problem with atomic vehicle models

TL;DR

The paper addresses the global stability of departure-time equilibria in a single-bottleneck network by modeling atomic users in a DTC game and introducing an $\epsilon$-Nash equilibrium to guarantee the existence of pure equilibria. It proves the DTC game is weakly acyclic via an ordering property, enabling a convergent better-response path from any initial state and motivating convergence mechanisms. Leveraging these insights, the authors construct evolutionary dynamics that achieve global convergence to the $\epsilon$-Nash equilibrium and relate this equilibrium to the fluid-model bottleneck solution, with $m(P-1)=Q$ linking the atomic and continuum frameworks and $\epsilon \to 0$ as $m\to 0$. Numerical experiments corroborate global convergence from both special and general initial profiles, illustrating the practical viability of the proposed dynamics and highlighting factors that influence stability in related dynamics. Overall, the work offers a principled, atomic-view stability analysis for DTC problems, with implications for extensions to more complex networks and hypercongestion phenomena.

Abstract

In this study, we analyse the global stability of the equilibrium in a departure time choice problem using a game-theoretic approach that deals with atomic users. We first formulate the departure time choice problem as a strategic game in which atomic users select departure times to minimise their trip cost; we call this game the 'departure time choice game'. The concept of the epsilon-Nash equilibrium is introduced to ensure the existence of pure-strategy equilibrium corresponding to the departure time choice equilibrium in conventional fluid models. Then, we prove that the departure time choice game is a weakly acyclic game. By analysing the convergent better responses, we clarify the mechanisms of global convergence to equilibrium. This means that the epsilon-Nash equilibrium is achieved by sequential better responses of users, which are departure time changes to improve their own utility, in an appropriate order. Specifically, the following behavioural rules are important to ensure global convergence: (i) the adjustment of the departure time of the first user departing from the origin to the corresponding equilibrium departure time and (ii) the fixation of users to their equilibrium departure times in order (starting with the earliest). Using convergence mechanisms, we construct evolutionary dynamics under which global stability is guaranteed. We also investigate the stable and unstable dynamics studied in the literature based on convergence mechanisms, and gain insight into the factors influencing the different stability results. Finally, numerical experiments are conducted to demonstrate the theoretical results.

Figures (19)

• Figure 1: Network and schedule delay cost function
• Figure 2: Schematic of reducing queueing delay cost without changing the destination arrival time.
• Figure 3: Description of the pattern of the departure and destination arrival times in $\mathbf{s}^{e}$; the interval between destination arrival times is $m/\mu$ and users experience corresponding queueing and schedule delay costs, of which the sum is $\rho$.
• Figure 4: Schematics of a better response of $n+1$st user to $s_{n+1}^{e}$; each dot represents the relationship between a departure time and corresponding trip cost of each user.
• Figure 5: Schematics of eventual time profiles when $C^{r}\neq \rho$.

Theorems & Definitions (36)

• Definition 1
• Theorem 1
• Proof
• Proposition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 2
• Proof
• Lemma 1