designing mobility-as-a-service subscription bundles under a joint bundle-path equilibrium with continuous trip-frequency heterogeneity

1. The Challenge: Modeling Real-World MaaS Adoption

Mobility-as-a-Service (MaaS) is transforming urban transport by integrating various modes like public transit, ride-hailing, and bike-sharing into a single platform. A key feature of MaaS is the use of subscription bundles, which offer users convenience and cost certainty. However, designing these bundles is incredibly complex.

Existing models often fail to capture a critical feedback loop:

1. The prices of bundles influence which ones travelers adopt.
2. Traveler adoption patterns determine the traffic flow on different modes.
3. Traffic flow creates network congestion, which changes travel times.
4. These new travel times alter the perceived value of each bundle, circling back to influence pricing and adoption.

Current frameworks lack the tools to model this dynamic interplay, especially when dealing with a population of travelers who are all different. This paper addresses this gap by introducing a novel equilibrium framework to design more effective, welfare-oriented MaaS policies.

2. Core Contribution 1: The Dual-Heterogeneous User Equilibrium (DHUE)

The cornerstone of this research is a new equilibrium concept called the Dual-Heterogeneous User Equilibrium (DHUE). It is designed to be behaviorally sound by simultaneously capturing two fundamental types of user heterogeneity:

1. Continuous Heterogeneity in Trip Frequency: Travelers have different travel needs. Some are daily commuters, while others travel infrequently. The model represents this as a continuous probability distribution of monthly trip frequency for each traveler.
2. Discrete Choice of MaaS Bundles: Faced with a menu of tiered subscription bundles (e.g., Pay-as-you-go, Basic, Premium), travelers make a discrete choice.

The Choice Mechanism

A rational traveler will choose the bundle that minimizes their expected total monthly cost. This cost is a combination of the fixed monthly subscription fee (τ) and the variable per-trip costs (π), which depend on network congestion.

As shown in Figure 2, this creates a natural market segmentation. Infrequent travelers prefer bundles with low subscription fees, while frequent travelers are attracted to bundles with higher fees but larger per-trip discounts. The model finds the “indifference points” that define the boundaries between these market segments.

• Figure 2: The bundle choice mechanism. (a) Travelers select the bundle on the lower cost envelope. (b) This partitions the user base into distinct adoption segments based on their monthly trip frequency.

Theoretical Reformulation

The central analytical challenge is the circular dependency: the per-trip costs (π) that drive bundle choice are themselves determined by the resulting traffic flows. To solve this, the authors reformulate the entire problem—from user choice principles to a tractable mathematical program.

This complex, infinite-dimensional problem is proven to be equivalent to a finite-dimensional Variational Inequality (VI). This reformulation (summarized in Figure 3) is a critical theoretical leap, making the problem solvable and allowing for the analysis of its properties, such as the existence and uniqueness of a solution.

• Figure 3: The logical reformulation of the DHUE problem, from behavioral principles to a solvable mathematical formulation.

3. Core Contribution 2: A Bilevel Framework for Policy Design

To demonstrate the DHUE’s practical relevance, it is embedded as the lower-level problem within a bilevel optimization framework. This structure models the strategic interaction between a MaaS authority and its users.

• Upper-Level Problem: A public-sector authority designs the MaaS bundles. Its goal is to maximize social welfare, which is defined as the total passenger surplus minus the total monetized system travel time. The authority decides the key policy levers:

  • τ: The monthly subscription fee for each bundle.
  • δ: The per-trip discount factors for each mode within each bundle. This optimization is subject to real-world constraints, such as a total subsidy budget.

• Lower-Level Problem: For any given policy (τ, δ) set by the authority, the travelers react. They choose their bundle and travel routes according to the DHUE model. The system settles into a new equilibrium, determining the bundle adoption rates, traffic flows, congestion levels, and overall social welfare.

This bilevel structure provides a powerful tool for data-driven policy analysis, allowing policymakers to design and test bundle strategies to achieve system-wide objectives before deployment.

4. Core Contribution 3: A Scalable Solution Algorithm

Solving the bilevel framework is computationally intensive because the upper-level’s objective function is implicit—its value can only be found by solving the lower-level DHUE equilibrium.

To tackle this, the authors develop a robust, two-part solution methodology:

1. Modified Frank-Wolfe Algorithm: An efficient, customized algorithm to solve the lower-level DHUE problem. It is specifically adapted to handle the model’s unique path-based costs and layered network structure. A key innovation is a two-speed adaptive step-size strategy that enhances stability and accelerates convergence.

2. Alternating Simultaneous Perturbation Stochastic Approximation (ASPSA): A powerful, derivative-free algorithm for the upper-level policy search. The ASPSA treats the entire lower-level solver as a “black box,” iteratively proposing and evaluating candidate policies (τ, δ) to navigate the complex policy space and find the one that maximizes social welfare.

• Figure 4: The flowchart of the ASPSA solution algorithm, which alternates between optimizing bundle fees (τ) and discount structures (δ).

This approach is proven to be computationally tractable and scalable, capable of solving the problem on large-scale, realistic transport networks like the Winnipeg benchmark.

5. Numerical Results: Optimizing MaaS in Sioux Falls

The framework was tested on the benchmark Sioux Falls network, expanded into a full multimodal system. Three policy scenarios were compared:

1. No-MaaS (PAYG): A baseline with no subscription options.
2. Heuristic Policy: An intuitive but unoptimized set of four tiered bundles.
3. Optimal Policy: The welfare-maximizing bundles found by the bilevel framework.

Key Findings

The results demonstrate the immense value of optimization. The Optimal Policy dramatically outperforms the Heuristic Policy, increasing social welfare from 6,445 to 13,473 units.

This massive gain is driven almost entirely by a 20.1% increase in passenger surplus. The model discovers that the most effective use of a public subsidy budget is to create deep, valuable discounts that align with traveler preferences, rather than chasing minor reductions in network travel time.

The optimization fundamentally restructures the market. Under the heuristic policy, the premium B4 bundle is a market failure, capturing only 2.0% of users. The optimal policy makes B4 the dominant choice, with 43.6% adoption, by strategically adjusting its fees and discounts.

• Table 4: A selection of assignment results, highlighting the dramatic shift in bundle adoption and the resulting decrease in private car use in the optimal scenario.

This market restructuring is visualized in the heatmaps below, showing the change in bundle choices across all origin-destination (OD) pairs. The optimized policy creates a more rational and effective market where different bundles appeal to different trip types, from short, simple journeys to long, complex ones.

• Figure 6: Bundle adoption per OD pair. The heuristic policy (a) leads to a flawed market. The optimal policy (b) creates a far more effective and differentiated market structure.

6. Conclusion and Policy Implications

This research makes significant theoretical and practical contributions to the field of transportation science. By developing the DHUE framework, it provides the first model to endogenously capture the full feedback cycle between MaaS bundle pricing, heterogeneous user choice, and network congestion.

The key policy insights derived from the model are profound:

• Maximizing Social Welfare ≠ Minimizing Congestion: An optimal MaaS strategy may rationally prioritize huge gains in passenger surplus (through deep discounts) over marginal improvements in travel times, even if it leads to slightly longer journeys on average.
• Intelligent Subsidy is Crucial: Naively designed or “common sense” bundle policies are likely to be highly suboptimal. The bilevel framework provides a tool to strategically deploy a limited public budget to achieve the greatest welfare gains.
• Market Design is Highly Contextual: The optimal MaaS bundle structure is highly contingent on factors like the available budget and the network’s topology. A one-size-fits-all approach will not work.

Ultimately, this work provides a rigorous and powerful framework for policymakers to design and analyze innovative MaaS subscription policies, helping to ensure that the next generation of mobility solutions is not only technologically advanced but also economically efficient and socially equitable.