A many-to-many assignment game and stable outcome algorithm to evaluate collaborative mobility-as-a-service platforms
Introduction
There is a growing need to focus on managing the capacities, allocation, and pricing of mobility services in a Mobility-as-a-Service (MaaS) (Hensher, 2017; Djavadian and Chow, 2017) ecosystem. Under this ecosystem, city agencies play a key role as facilitators in either economic deregulation through relationships with suppliers, or through government contracting with the operators, as illustrated by the evolution from Fig. 1a and b or c (Wong et al., 2019). As such, city agencies need to be able to assess the impact on other mobility operators and travelers when a new mobility operator enters the market, or an existing one changes their service capacity, routing algorithm, or pricing mechanism. A new mobility service or change to an existing one can cause travelers to switch routes or combine the service of one operator with those of other operators to fulfill their trips. It can lead to certain routes becoming unstable to operate. Changes in algorithms (e.g. Stiglic et al., 2015) or government policies like ride surcharges (e.g. Hu, 2019) can alter the allocation of costs between travelers and operators. Any MaaS market equilibrium model developed for the public policymaker needs to be sensitive to both traveler (multimodal/multi-operator routes) and operator (service coverage, fleet, pricing) decisions.
For this purpose, classic traffic assignment models that emphasize only traveler route decision-making are not effective tools. In a MaaS setting the policy questions are not focused on congestion on the roadway, but instead on how travelers match to different combinations of fixed route public transit and/or various mobility options (e.g. bikeshare, ride-hail, microtransit, among others) considering capacities of these systems and their cost allocation policies (e.g. fares transfer dollars to service, stop locations trade-off between access time). For capacity we consider effective planning-level service capacities; i.e. a bike share service that uses a certain rebalancing algorithm would provide a certain maximum flow from one location to another.
Rasulkhani and Chow (2019) proposed such a method for evaluating unimodal trip systems, where each operator acts as a set of service routes and each traveler matches “many-to-one” to one route while ensuring the line capacities are not violated and stability conditions from the core (Shapley and Shubik, 1971) are met. The model is computationally tractable and can be solved using classic algorithms for capacitated assignment and linear programming (for the stable outcome problem). Unlike network flow games (e.g. Bird, 1976; Derks and Tijs, 1985; Fragnelli et al., 2001; Agarwal and Ergun, 2008) that form coalitions between operators, the model matches between travelers and operators so that it explicitly captures both operator and travel behavior in a network of mobility markets. The model from Rasulkhani and Chow (2019) does not handle matching of travelers’ paths to multiple operators for modeling MaaS platforms. Use cases for such models abound: city agencies may act as cyber-physical platform providers in which mobility operators and travelers match with transaction fees that depend on the design of the built environment (see Chow, 2018). MaaS policy-makers, including government agencies and transport providers, can then use the solution of such a model to make trade-offs for designing their platform under different cost allocation policies and algorithms, link capacities, and determine negotiating power of different operators for the purpose of forming coalitions or justifying subsidies between operators to match with traveler paths. We illustrate these use cases that need to be modeled in Table 1.
The challenge of matching multiple links of different traveler paths to multiple operators is a many-to-many assignment game. In such a game, the stability conditions become more complex because they need to be considered from both a user's path level as well as an operator's level in serving that user. In a MaaS market, each operator owns one or more links and may choose not to serve that link (i.e. exit the market) if there is insufficient incentive to provide service there. Each link is capacitated with planning-level service capacities. Furthermore, each operator can choose a price to charge for using their link to the users. How should competing operators sharing different legs of a traveler's trip set their prices? Blocking pairs” may form to prevent a stable path from forming, which are not trivial to model in this setting.
We propose a model for this many-to-many assignment game and show how to derive an optimal assignment flow and corresponding stable outcome space between the operators and the travelers or users. If a stable outcome space exists, it provides boundaries over which a city agency can work with competing operators to allocate costs between users and each operator to set prices, as illustrated in the use cases in Table 1. The model does not assume any specific cost allocation mechanism or policy used by each operator; it only determines the thresholds within which such mechanisms are stable. An empty stable outcome space implies the unsustainability of the platform as designed, which would warrant further planning (e.g. changing travel costs through infrastructure investments or policies, adding further capacities, or introducing additional candidate routes for operators to serve).
Section snippets
Literature review
While several studies have examined demand for MaaS services (Strömberg et al., 2018; Matyas and Kamargianni, 2019a, Matyas and Kamargianni, 2019b), these studies have not sought to quantify or structure the relationships between decisions made by operators and users in providing and consuming routes in a MaaS market. Earlier research on coexisting operators (see Chow and Sayarshad, 2014, for a review) consider noncooperative games, including a generalized Nash equilibrium for a duopolistic
Problem notation
F: set of operators, where f = 0 is a dummy operator representing no operator in the platform
N: set of nodes in the platform
A: set of links in the network
Af: a disjoint subset of A owned by operator f ∈ F
Ni( + ),Ni( − ): the set of heads (+) and tails (-) that are formed by links connected to node i ∈ N
S: set of traveler groups represented by OD from one node to another
O(s), D(s): origin-destination nodes of s ∈ S
Rs: set of feasible paths that can serve OD s ∈ S
: set of optimal paths for
Constraint generation algorithm using lexicograhpic core allocations
It is not practical to explicitly enumerate all feasible paths Rs to construct the stability constraints in Eq. (6). We develop a method similar to Bahel and Trudeau (2014) to identify only the paths that form the extreme boundaries of the shortest path problem cooperative game to determine a subset that can produce an equivalent set of constraints to Eq. (6). The algorithm leads to exact solutions.
Numerical tests
Having illustrated the model and algorithm, we test the effectiveness of the method on a larger instance, the classic 24-node Sioux Falls network (see LeBlanc et al., 1975; Stabler, 2019) as shown in Fig. 6a. In this duopoly there are two operators: Operator 1 (blue) is a bus network while Operator 2 (orange) is a set of two rail lines. Transfer links are denoted by grey links (28 additional links added, 14 nodes added, for total of 38 nodes and 104 links). Since a network of even this size is
Conclusion
With the emergence of MaaS ecosystems, public agencies need modeling tools to consider trade-offs when facilitating markets with private operators. For the most part, such modeling tools do not exist. Recent work from Rasulkhani and Chow (2019) sought to rectify this but only capture line-level interactions between operators and users without allowing users to make multimodal trips. A new modeling framework is proposed using the MCND as a capacitated link-based matching model to fully capture
Author statement
All authors worked on the research design. T. Pantelidis and J. Chow developed the model, solution algorithm, analysis, and writing.
Acknowledgements
This research was conducted with support from NSF CMMI-1634973.
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