On Dynamical Cournot Game on a Graph

Cournot dynamical game is studied on a graph. The stability of the system is studied. Prisoner’s dilemma game is used to model natural gas transmission.

💡 Research Summary

The paper investigates the application of dynamic Cournot competition to the transportation of natural gas, electricity, and water across an international network that is represented as a graph. The graph’s vertices are of three types: producers (firms), markets (consumers), and transit countries through which pipelines or transmission lines pass. The profit of each firm (f_j) is modeled by a quadratic function that incorporates market‑specific price parameters (\alpha_i) (intercept) and (\beta_i) (slope) as well as a firm‑specific cost parameter (\gamma_j). Total quantities supplied to each market ((s_i)) and total output of each firm ((s_j)) are defined as sums over the relevant edges, thereby capturing the interdependence among firms and markets.

To capture bounded rationality, the authors adopt a continuous‑time adjustment process: each firm changes its output proportionally to the marginal profit, i.e., (\dot q_{ij}=b_j\partial\Pi_j/\partial q_{ij}), where (b_j) is a constant adjustment speed. Expanding the derivative yields a system of coupled nonlinear differential equations (Equation 4). For analytical tractability the authors focus on a minimal network consisting of two firms and two markets. Firm 1 supplies both markets, while Firm 2 supplies only market 2. After appropriate scaling the dynamics reduce to a three‑dimensional system:

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