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We model the oligopoly competition in a dockless bike-sharing (DLB) market as a bilevel game. Each DLB operator is first committed to an action tied to a specific objective, such as maximizing profit. Then, the operators play a lower-level game to achieve their individual goals and finally reach a subgame perfect Nash equilibrium by making tactic decisions (e.g., pricing and fleet sizing). We define a Nash equilibrium under either weak or strong preference to characterize the likely outcomes of the bilevel game and formulate the demand-supply equilibrium of a DLB market that accounts for key operational features and mode choice. Using the oligopoly game model calibrated with empirical data, we show that if an operator seeks to maximize its market share with a budget constraint, all other operators must either respond in kind or be driven out of the market. When all operators compete for market dominance, even a slight efficiency edge gained by one operator can significantly shift the outcome, which signals high volatility. Moreover, even if all operators agree to focus on making money rather than ruinously seeking dominance, profitability still plunges quickly with the number of operators. Taken together, the results explain why an unregulated DLB market is often oversupplied and prone to collapse under competition. We also show that this market failure may be prevented by a fleet cap regulation, which sets an upper limit on each operator’s fleet size. Funding: This research was supported by the U.S. National Science Foundation’s Civil Infrastructure System (CIS) Program under the award CMMI no. 2225087. K. Zhang received financial support from the Swiss National Science Foundation [Grant 219232]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0846 . 

Abstract Midlatitude weather extremes such as blocking events and Rossby wave breaking are often related to large meridional shifts in the westerly jet stream. Numerous diagnostic methods have been developed to characterize these weather events, each emphasizing different yet interrelated aspects of circulation waviness, including identifying large-amplitude ridges or persistent anomalies in geopotential height. In this study, we introduce a new metric to quantify the circulation waviness in terms of effective time scale. This is based on the Rossby wave packet from the one-point correlation map of anomalous meridional wind, applicable to jet waviness involving multiple wavenumbers. Specifically, we estimate the intrinsic frequency of Rossby waves and decay time scale of wave amplitude in the reference frame moving at the local time mean zonal wind. The resulting effective time scale, derived from linear theory, serves as a proxy for the eddy mixing time scale in jet meandering. Remarkably, its spatial distribution roughly resembles that of circulation waviness in the Northern Hemisphere winter as depicted by local wave activity (LWA). In the high-latitude regions characterized by weak zonal winds, the long time scale in waviness aligns with large values in LWA. By contrast, short waviness time scales in subtropical jet regions correspond to the suppressed amplitude in waviness despite large values in eddy kinetic energy (EKE). Furthermore, the effective time scale in waviness largely captures the interannual variability of LWA in observations and its projected future changes in climate model simulations. Thus, this relation between the waviness time scale and zonal wind provides a physical mechanism for understanding how zonal wind changes impact regional weather patterns in a changing climate. Significance StatementThe purpose of this study is to better understand what controls weather extremes in midlatitude regions such as blocking events and Rossby wave breaking. We introduce a novel concept, the effective time scale of jet stream meandering, which sheds light on these phenomena. Through analyzing Rossby waves in the reference frame moving at the local time mean zonal wind, we derive a scaling relation between circulation waviness and eddy mixing time scale. Our findings reveal that this time scale closely mirrors the spatial distribution of circulation waviness in the Northern Hemisphere winter. Importantly, it captures interannual variability and climate change responses. These insights provide a physical basis for understanding how changes in zonal wind impact regional weather patterns in observations and climate models. 

As one of the most fundamental concepts in transportation science, Wardrop equilibrium (WE) has always had a relatively weak behavioral underpinning. To strengthen this foundation, one must reckon with bounded rationality in human decision-making processes, such as the lack of accurate information, limited computing power, and suboptimal choices. This retreat from behavioral perfectionism in the literature, however, was typically accompanied by a conceptual modification of WE. Here, we show that giving up perfect rationality need not force a departure from WE. On the contrary, WE can be reached with global stability in a routing game played by boundedly rational travelers. We achieve this result by developing a day-to-day (DTD) dynamical model that mimics how travelers gradually adjust their route valuations, hence choice probabilities, based on past experiences. Our model, called cumulative logit (CumLog), resembles the classical DTD models but makes a crucial change; whereas the classical models assume that routes are valued based on the cost averaged over historical data, our model values the routes based on the cost accumulated. To describe route choice behaviors, the CumLog 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 and the other describing the sensitivity of route choice probabilities to valuation differences. We prove that the CumLog model always converges to WE, regardless of the initial point, as long as the behavioral parameters satisfy certain mild conditions. Our theory thus upholds WE’s role as a benchmark in transportation systems analysis. It also explains why equally good routes at equilibrium may be selected with different probabilities, which solves the instability problem posed by Harsanyi. Funding: This research is funded by the National Science Foundation [Grants CMMI #2225087 and ECCS #2048075]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0132 . 

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, is often used to address the challenge. Built on a recently proposed day-to-day discrete-time dynamical model called cumulative logit (CumLog), this study provides a new behavioral underpinning for the maximum-entropy user equilibrium (MEUE) route flow. It has been proven that CumLog can reach a UE state without presuming that travelers are perfectly rational. Here, we further establish that CumLog always converges to the MEUE route flow if (i) travelers have no prior information about routes and thus, are forced to give all routes an equal initial choice probability or if (ii) all travelers gather information from the same source such that the general proportionality condition is satisfied. Thus, CumLog 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 CumLog model through an iterative route discovery scheme. We also examine the discrete-time versions of four popular continuous-time dynamical models and compare them with CumLog. 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. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: This research was funded by the United States National Science Foundation’s Division of Civil, Mechanical and Manufacturing Innovation [Grant 2225087]. The work of J. Xie was funded by the National Natural Science Foundation of China [Grant 72371205]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0525 . 

Abstract The relative roles of upper- and lower-level thermal forcing in shifting the eddy driven jet are investigated using a multi-level nonlinear quasi-geostrophic channel model. The numerical experiments show that the upper-level thermal forcing is more efficient in shifting the eddy-driven jet. The finite-amplitude wave activity diagnostics of numerical results show that the dominance of the upper-level thermal forcing over the lower-level thermal forcing can be understood from their different influence on eddy generation and dissipation that affects the jet shift. The upper-level thermal forcing shifts the jet primarily by affecting the baroclinic generation of eddies. The lower-level thermal forcing influences the jet mainly by affecting the wave breaking and dissipation. The former eddy response turns out to be more efficient for the thermal forcing to shift the eddy-driven jet. Furthermore, two quantitative relationships based on the imposed thermal forcing are proposed to quantify the response of both eddy generation and eddy dissipation, and thus to help predict the shift of eddy-driven jet in response to the vertically non-uniform thermal forcing. By conducting the overriding experiments in which the response of barotropic zonal wind is locked in the model and a multi-wavenumber theory in which the eddy diffusivity is decomposed to contributions from eddies and mean flow, we find that the eddy generation response is sensitive to the vertical structure of the thermal forcing and can be quantified by the imposed temperature gradient in the upper troposphere. In contrast, the response of eddy diffusivity is almost vertically independent of the imposed forcing, and can be quantified by the imposed vertically-averaged thermal wind.