One large metropolis might have several different train systems, from local intercity lines to commuter trains to longer regional lines.

When designing a system of train tracks, stations, and schedules in this network, should rail operators assume each entity operates independently, seeking only to maximise its own revenue? Or that they fully cooperate all the time with a joint plan, putting their own interest aside?

In the real world, neither assumption is very realistic.

Researchers from ETH Zurich and MIT have developed a new planning tool that mixes competition and co-operation to help operators in a complex, multi-regional network strategically determine when and how they should work together.

Their framework is unusual because it incorporates co-investment and payoff-sharing mechanisms that identify which joint infrastructure projects a stakeholder should invest in with other operators to maximise collective benefits.

The tool can help mobility stakeholders, such as governments, transport agencies, and firms, determine the right time to collaborate, how much they should invest in cooperative projects, how the profits should be distributed, and what would happen if they withdrew from the negotiations.

“It might seem counterintuitive, but sometimes you want to invest in your opponent so that, at some point, this investment will come back to you. Thanks to game theory, one can formalise this intuition to give rise to an interesting class of problems,” says Gioele Zardini, the Rudge and Nancy Allen Assistant Professor of Civil and Environmental Engineering at MIT, a principal investigator in the Laboratory for Information and Decision Systems (LIDS), an affiliate faculty with the Institute for Data, Systems, and Society (IDSS), and senior author of a paper on this planning framework.

Shared infrastructure projects

Numerical analysis shows that, by investing a portion of their budget into some shared infrastructure projects, independent operators can earn more revenue than if they operated completely non-cooperatively.

In the example of the rail operators, the researchers demonstrate that co-investment also benefits users by improving regional train service. This win-win situation encourages more people to take the train, boosting revenues for operators and reducing emissions from cars, says Mingjia He, a graduate student at ETH Zurich and lead author.

“The key point here is that transport network design is not a zero-sum game. One operator’s gain doesn’t have to mean the others’ loss. By shifting the perception from isolated, self-optimisation to strategic interaction, co-operation can create greater value for everybody involved,” she says.

Beyond transportation, this planning framework could help companies in a crowded industry or governments of neighbouring countries test co-investment strategies.

He and Zardini are joined on the paper by ETH Zurich researchers Andrea Censi and Emilio Frazzoli. The research will be presented at the 2025 American Control Conference (ACC), and the paper has been selected as a Student Best Paper Award finalist.

Mixing co-operation and competition

Building transportation infrastructure in a multi-regional network typically requires a huge investment of time and resources. Big infrastructure projects have an outsized impact that can stretch far beyond one region or operator.

Each region has its own priorities and decision makers, such as local transportation authorities, which often results in the failure of co-ordination.

“If local systems are designed separately, regional travel may be more difficult, making the whole system less efficient. But if self-interested stakeholders don’t benefit from co-ordination, they are less likely to support the plan,” says He.

To find the best mix of co-operation and competition, the researchers used game theory to build a framework that enables operators to align interests and improve regional co-operation in a way that benefits all.

For instance, last year the Swiss government agreed to invest €50m to electrify and expand part of a regional rail network in Germany, with the goal of creating a faster rail connection between three Swiss cities.

The researchers’ planning framework could help independent entities, from regional governments to rail operators, identify when and how to undertake such collaborations.

The first step involves simulating the outcomes if operators do not collaborate. Then, using the co-investment and payoff-sharing mechanisms, the decision maker can explore co-operative approaches.

To identify a fair way to split revenues from shared projects, the researchers design a payoff-sharing mechanism based on a game theory concept known as the Nash bargaining solution. This technique will determine how much benefit operators would receive in different co-operative scenarios, taking into account the benefits they would achieve with no collaboration.

The benefits of co-investment

Once they had designed the planning framework, the researchers tested it on a simulated transportation network with multiple competing rail operators. They assessed various co-investment ratios across multiple years to identify the best decisions for operators.

In the end, they found that a semi-cooperative approach leads to the highest returns for all stakeholders. For instance, in one scenario, by co-investing 50% of their total budgets into shared infrastructure projects, all operators maximised their returns.

In another scenario, they show that by investing a mere 3.3% of their total budget in the first year of a multiyear co-operative project, operators can boost outcomes by 30% across three metrics: revenue, reduced costs for customers, and lower emissions.

“This proves that a small, upfront investment can lead to significant long-term benefits,” says He.

When they applied their framework to more realistic multi-regional networks where all regions were not the same size, this semi-cooperative approach achieved even better results.

However, their analyses indicate that returns do not increase in a linear way – sometimes increasing the co-investment ratio does not increase the benefit for operators.

Success is a multifaceted issue that depends on how much is invested by all operators, which projects are chosen, when investment happens, and how the budget is distributed over time, explains He.

“These strategic decisions are complex, which is why simulations and optimisation are necessary to find the best co-operation and negotiation strategies. Our framework can help operators make smarter investment choices and guide them through the negotiation process,” she says.

The framework could also be applied to other complex network design problems, such as in communications or energy distribution.

In the future, the researchers want to build a user-friendly interface that will allow a stakeholder to easily explore different collaborative options.

They also want to consider more complex scenarios, such as the role policy plays in shared infrastructure decisions or the robust co-operative strategies that handle risks and uncertainty.