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

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.

💡 Research Summary

The paper tackles the pervasive problem of sub‑optimal outcomes in interdependent networked systems where multiple self‑interested operators make design decisions in isolation. It proposes a unified game‑theoretic framework that blends non‑cooperative and cooperative elements, enabling operators to both compete and collaborate on infrastructure investments.

Non‑cooperative stage (Network Design Game – NDG).
A set of N operators I = {1,…,N} each controls a subgraph Γ_i of a directed, edge‑labeled network Γ = (V,E,ℓ). Each operator’s strategy h_i ∈ H_i consists of binary decisions (whether to build a link) and continuous decisions (capacity expansion) on its local edges. The payoff function f_i(h_i, y) depends on the design and the resulting flow vector y ∈ ℝ^{|E|}_+, which is determined by a user‑response mapping Y(h, Γ) (e.g., a user equilibrium traffic assignment). Operators are subject to budget constraints b_i(h_i) ≤ B_i. The NDG is formalized as a simultaneous‑move game; a Nash equilibrium (NE) is defined and its existence is proved under standard convexity and continuity assumptions. The NE serves as a benchmark to quantify inefficiencies relative to the socially optimal solution.

Cooperative stage (Co‑investment & Payoff‑sharing).
The authors introduce a mechanism whereby operators can jointly finance a subset of infrastructure projects. The total surplus ΔS generated by the joint investment is allocated according to a sharing rule σ_i that depends on each operator’s bargaining power β_i and a parameter θ_i capturing the risk of strategic exploitation. The rule is designed to satisfy (i) Pareto improvement over the non‑cooperative NE, (ii) a fairness criterion approximating the Shapley value, and (iii) incentive compatibility (no operator benefits by deviating from the agreement). A “guarantee deposit” clause is added to prevent free‑riding. Theoretical results demonstrate that, under mild conditions, the cooperative outcome is a Nash equilibrium of the extended game and strictly dominates the original NE.

Methodology and case studies.
Two empirical settings are examined:

1. Sioux Falls transportation network (USA). Using realistic demand data, the authors compare the baseline NDG (no cooperation) with a cooperative scenario where operators jointly fund a limited set of capacity upgrades. Results show a 12 % reduction in average travel time, an 8 % cut in CO₂ emissions, and a payback period of roughly five years for the initial joint investment.

2. Zurich–Winterthur public‑transport network (Switzerland). Real operational data, operator budgets, and demand patterns are used. The study varies β_i (reflecting the size and market power of the operators) and θ_i (degree of potential exploitation). When the dominant operator (high β_i) participates in the cooperative scheme, total social welfare rises by up to 15 %. The guarantee deposit effectively stabilizes negotiations, preventing the dominant player from imposing an unfair split. Sensitivity analyses illustrate how the distribution of benefits changes with bargaining asymmetry and budget heterogeneity.

Contributions.

1. A unified hybrid game model that simultaneously captures network‑user interactions (via the flow mapping Y) and network‑network interactions among multiple operators.
2. A rigorously designed co‑investment and payoff‑sharing mechanism that guarantees Pareto improvement, fairness, and incentive compatibility, while explicitly modeling bargaining power and exploitation risk.
3. Comprehensive validation on both synthetic (Sioux Falls) and real‑world (Zurich–Winterthur) transportation networks, providing actionable insights for policymakers and infrastructure managers.

Limitations and future work.
The current flow model assumes a static user equilibrium, ignoring dynamic demand fluctuations, multimodal transfers, and stochastic travel times. Estimating β_i and θ_i requires prior information that may be imperfect in practice, leading to potential misallocation. Future research directions include integrating stochastic user equilibrium models, employing machine‑learning techniques to infer bargaining parameters from observed behavior, and exploring blockchain‑based smart contracts for automated, transparent payoff distribution.

Conclusion.
By embedding cooperative investment and fair payoff‑sharing into the strategic design of interdependent networks, the proposed framework demonstrates that modest joint investments can generate substantial long‑term system‑wide gains. The hybrid game‑theoretic approach offers a practical decision‑support tool for multi‑operator environments, bridging the gap between competitive self‑interest and collective welfare.