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Competition among seaports through Mean Field Games and real-world data

Charles-Albert Lehalle, Giulia Livieri

TL;DR

This paper develops a discrete-time Mean Field Game framework to model competition among seaports by treating ships as rational agents whose routing decisions depend on transport costs, expected margins, and congestion. It provides a closed-form solution for the stationary, quadratic-cost version of the maritime flow MFG and a determinant-based condition for the existence and uniqueness of the Mean Field Equilibrium (MFE). A two-step statistical inference procedure is proposed to estimate MFG parameters from real-world data, validated on the ShipFix Dry Coal dataset, with margins, congestion, and transport costs inferred from country-level flows and Searoutes distances. The work advances maritime traffic management by connecting economic signals to network-wide flow patterns and offering a data-driven approach to calibrate MFGs for policy and operations. Limitations include stationarity assumptions and coarse country-level aggregation, suggesting directions for port-level analysis and non-stationary extensions in future research.

Abstract

This paper presents a Mean Field Game (MFG) model for maritime traffic flow, treating the navigation of ships between seaports as a large-scale stochastic control problem. The MFG framework enables the modeling of agents at a microscopic level as rational decision-makers who seek to optimize their utility, thereby translating complex microscopic behaviors into macroscopic models. We build upon this MFG framework to develop a mesoscopic-scale MFG model that defines the payoff and cost functions for a coordinator at each seaport considered in our study. The coordinator determines the routes taken by ships transporting goods between ports by evaluating several key factors: transportation costs, expected profit margins from loading specific goods at the seaports and unloading them at various destinations, and a congestion term that reflects the costs associated with accessing the destination port. We derive an explicit solution for the stationary version of the model under certain approximations and establish conditions necessary to ensure the uniqueness of the corresponding Mean Field Equilibrium (MFE). Furthermore, we introduce a statistical methodology to infer the parameters of the game from real-world data, specifically focusing on costs and the components of expected commercial margins. To validate our model in a real-world context, we analyze the ShipFix dataset of daily ''Dry Coal'' shipments worldwide from 2015 to 2025. Our discussion highlights the influence of empirical traffic flow on various components of costs. We believe that this research represents a significant advancement in the application of MFGs for effective maritime traffic management and offers valuable insights for practitioners in the field.

Competition among seaports through Mean Field Games and real-world data

TL;DR

This paper develops a discrete-time Mean Field Game framework to model competition among seaports by treating ships as rational agents whose routing decisions depend on transport costs, expected margins, and congestion. It provides a closed-form solution for the stationary, quadratic-cost version of the maritime flow MFG and a determinant-based condition for the existence and uniqueness of the Mean Field Equilibrium (MFE). A two-step statistical inference procedure is proposed to estimate MFG parameters from real-world data, validated on the ShipFix Dry Coal dataset, with margins, congestion, and transport costs inferred from country-level flows and Searoutes distances. The work advances maritime traffic management by connecting economic signals to network-wide flow patterns and offering a data-driven approach to calibrate MFGs for policy and operations. Limitations include stationarity assumptions and coarse country-level aggregation, suggesting directions for port-level analysis and non-stationary extensions in future research.

Abstract

This paper presents a Mean Field Game (MFG) model for maritime traffic flow, treating the navigation of ships between seaports as a large-scale stochastic control problem. The MFG framework enables the modeling of agents at a microscopic level as rational decision-makers who seek to optimize their utility, thereby translating complex microscopic behaviors into macroscopic models. We build upon this MFG framework to develop a mesoscopic-scale MFG model that defines the payoff and cost functions for a coordinator at each seaport considered in our study. The coordinator determines the routes taken by ships transporting goods between ports by evaluating several key factors: transportation costs, expected profit margins from loading specific goods at the seaports and unloading them at various destinations, and a congestion term that reflects the costs associated with accessing the destination port. We derive an explicit solution for the stationary version of the model under certain approximations and establish conditions necessary to ensure the uniqueness of the corresponding Mean Field Equilibrium (MFE). Furthermore, we introduce a statistical methodology to infer the parameters of the game from real-world data, specifically focusing on costs and the components of expected commercial margins. To validate our model in a real-world context, we analyze the ShipFix dataset of daily ''Dry Coal'' shipments worldwide from 2015 to 2025. Our discussion highlights the influence of empirical traffic flow on various components of costs. We believe that this research represents a significant advancement in the application of MFGs for effective maritime traffic management and offers valuable insights for practitioners in the field.

Paper Structure

This paper contains 16 sections, 4 theorems, 29 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature Review
  4. Contribution of this Paper
  5. Problem statement and solution
  6. Maritime flow MFG
  7. Solution of the quadratic version of the maritime flow MFG
  8. Derivation of the MFE for the maritime flow MFG
  9. Statistical inference of the MFG parameters
  10. Real-world data application
  11. Data processing
  12. Transportation cost.
  13. The notion of margins
  14. Expected crowdedness and selection of a stationary time period.
  15. Estimation of the parameters of the maritime flow MFG via the two-steps procedure
  16. ...and 1 more sections

Key Result

Proposition 1

consider the constrained maximization problem in Equations eq::optimization_quadratic-eq:FP:MF. If the number of seaports $N$ is sufficiently large, each coordinator focuses on optimizing the problem regarding one seaport of origin and one good, treating the other flows as independent effects: Therefore, the maximization problem in Equation eq::optimization_quadratic reads as:

Figures (8)

  • Figure 1: From left to right: Total imports and exports to and from the five considered countries, expressed in log scale. SAFR stands for South Africa.
  • Figure 2: Sankey flow diagrams showing the impact of the pandemic on flow distributions, comparing pre-pandemic (left panel) and pandemic (right panel) periods. SAFR stands for South Africa.
  • Figure 3: The figure displays the ten ports categorized by country, highlighting the greater variation in Russian and US ports compared to South African (SAFR) or Japanese ports. Additionally, all routes calculated using the Searoutes library are included. The shades of grey indicate the number of routes that share the same offshore segment: the darker the shade, the more routes there are.
  • Figure 4: The dispersion of data along the main routes, organized by their averages, reveals significant differences. On the right side, we see a wide dispersion among Russian and US ports, while on the left side, South Korean and Japanese ports exhibit a much narrower concentration.
  • Figure 5: Time series of crowdedness for the selected countries.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Proposition 2