Co-Investment with Payoff-Sharing Mechanism for Cooperative Decision-Making in Network Design Games

TL;DR

The paper tackles the challenge of coordinating multiple self-interested network operators by introducing a coopetitive two-stage framework that combines non-cooperative network design with cooperative co-investment and payoff sharing. It formalizes a unified network design game that captures both network–network and network–user interactions, and then adds a co-investment mechanism and a Nash bargaining-based payoff-sharing scheme to promote mutually beneficial collaboration. Theoretical results establish tractability, equilibrium existence in the relaxed regime, and conditions for feasible, unique payoff allocations, while case studies on Sioux Falls and Zurich-Winterthur demonstrate substantial system-wide gains, with heterogeneity across regions acting as an opportunity under proper coordination. The work provides actionable decision-support for coordinated, user-centered improvements in interdependent, multi-operator networks, with implications for transportation, energy, and communications infrastructures.

Abstract

Network-based systems are inherently interconnected, with the design and performance of subnetworks being interdependent. However, the decisions of self-interested operators may lead to suboptimal outcomes for users and the overall system. This paper explores cooperative mechanisms that can simultaneously benefit both operators and users. We address this challenge using a game-theoretical framework that integrates both non-cooperative and cooperative game theory. In the non-cooperative stage, we propose a network design game in which subnetwork decision-makers strategically design local infrastructures. In the cooperative stage, co-investment with payoff-sharing mechanism is developed to enlarge collective benefits and fairly distribute them. To demonstrate the effectiveness of our framework, we conduct case studies on the Sioux Falls network and real-world public transport networks in Zurich and Winterthur, Switzerland. Our evaluation considers impacts on environmental sustainability, social welfare, and economic efficiency. The proposed framework provides a foundation for improving interdependent networked systems by enabling strategic cooperation among self-interested operators.

Figures (10)

• Figure 1: The interactive network design framework, featuring a non-cooperative, as well as a cooperative phase.
• Figure 2: Multimodal mobility network for Region 1 and Region 2.
• Figure 3: The proposed network design approach incorporates a cooperative network design stage involving co-investment and payoff sharing, which follows the non-cooperative . Here, operators can decide whether to invest jointly in the network, determine their individual contributions, and agree on the resulting payoff allocation.
• Figure 4: Sioux Falls network, subdivided between Region 1 and 2.
• Figure 5: Public transport networks in Zurich and Winterthur.

Theorems & Definitions (20)

• Definition 1: Network Design Game
• Definition 2: $\text{Nash Equilibrium}$ of
• Remark : Generality of the framework
• Remark : Solving the strategic local optimization problem
• Remark : Problem complexity
• Definition 3: Two-Stage Coopetitive NDG
• Remark : Comparison with non-cooperative NDG
• Remark : Problem complexity
• Remark : Interdependence of design stages
• Definition 4: Continuous Relaxation of the