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Contested Logistics: A Game-Theoretic Approach

Jakub Cerny, Chun Kai Ling, Darshan Chakrabarti, Jingwen Zhang, Gabriele Farina, Christian Kroer, Garud Iyengar

TL;DR

Contested Logistics (CL) formalizes a two-player zero-sum game on a graph where Blue optimizes multi-modal logistics under Red's edge interdictions. The framework defines Blue's two-stage routing and loading actions alongside Red's budgeted interdiction, with Leontief utilities governing demand satisfaction. The authors prove NP-hardness for computing equilibria but offer a practical double-oracle solver built on best-response MILPs to approximate Nash equilibria, and validate scalability on synthetic grids and real-world maps (UK and Ukraine). They also quantify robustness, showing that explicit adversarial modeling markedly improves performance over heuristic baselines and that over- or under-estimating Red's capabilities has asymmetric effects. The work advances robust logistics planning by integrating adversarial behavior into large-scale, graph-based routing with practical optimization techniques.

Abstract

We introduce Contested Logistics Games, a variant of logistics problems that account for the presence of an adversary that can disrupt the movement of goods in selected areas. We model this as a large two-player zero-sum one-shot game played on a graph representation of the physical world, with the optimal logistics plans described by the (possibly randomized) Nash equilibria of this game. Our logistics model is fairly sophisticated, and is able to handle multiple modes of transport and goods, accounting for possible storage of goods in warehouses, as well as Leontief utilities based on demand satisfied. We prove computational hardness results related to equilibrium finding and propose a practical double-oracle solver based on solving a series of best-response mixed-integer linear programs. We experiment on both synthetic and real-world maps, demonstrating that our proposed method scales to reasonably large games. We also demonstrate the importance of explicitly modeling the capabilities of the adversary via ablation studies and comparisons with a naive logistics plan based on heuristics.

Contested Logistics: A Game-Theoretic Approach

TL;DR

Contested Logistics (CL) formalizes a two-player zero-sum game on a graph where Blue optimizes multi-modal logistics under Red's edge interdictions. The framework defines Blue's two-stage routing and loading actions alongside Red's budgeted interdiction, with Leontief utilities governing demand satisfaction. The authors prove NP-hardness for computing equilibria but offer a practical double-oracle solver built on best-response MILPs to approximate Nash equilibria, and validate scalability on synthetic grids and real-world maps (UK and Ukraine). They also quantify robustness, showing that explicit adversarial modeling markedly improves performance over heuristic baselines and that over- or under-estimating Red's capabilities has asymmetric effects. The work advances robust logistics planning by integrating adversarial behavior into large-scale, graph-based routing with practical optimization techniques.

Abstract

We introduce Contested Logistics Games, a variant of logistics problems that account for the presence of an adversary that can disrupt the movement of goods in selected areas. We model this as a large two-player zero-sum one-shot game played on a graph representation of the physical world, with the optimal logistics plans described by the (possibly randomized) Nash equilibria of this game. Our logistics model is fairly sophisticated, and is able to handle multiple modes of transport and goods, accounting for possible storage of goods in warehouses, as well as Leontief utilities based on demand satisfied. We prove computational hardness results related to equilibrium finding and propose a practical double-oracle solver based on solving a series of best-response mixed-integer linear programs. We experiment on both synthetic and real-world maps, demonstrating that our proposed method scales to reasonably large games. We also demonstrate the importance of explicitly modeling the capabilities of the adversary via ablation studies and comparisons with a naive logistics plan based on heuristics.
Paper Structure (20 sections, 3 theorems, 11 equations, 11 figures, 2 tables, 1 algorithm)

This paper contains 20 sections, 3 theorems, 11 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Logistics and Routing Models
  4. Game-Theoretic Models
  5. Contested Logistics
  6. Blue's Strategy Space
  7. Red's Strategy Space
  8. Utilities
  9. Computing Solutions of Contested Logistics
  10. Best Response Complexity and Computation
  11. Approximating NE Using Strategy Generation
  12. Empirical Evaluation
  13. Quantitative Evaluation on a Synthetic Grid World
  14. Qualitative Evaluation on Real-World Maps
  15. Contested Logistics in the United Kingdom
  16. ...and 5 more sections

Key Result

proposition thmcounterproposition

It is NP-hard in terms of $|{\sf G}|$, $|{\cal C}|$, $|{\cal P}|$, and $T$ to find a NE for a contested logistics game with Leontief utilities given in Formulation lp:recourse.

Figures (11)

  • Figure 1: Physical graph ${\sf G}$ and layered graphs $\mathcal{G}_{c_1}, \mathcal{G}_{c_2}$ obtained by unrolling ${\sf G}$ over 3 steps. Connectors $c_1$ and $c_2$ start at A and B, respectively. $\mathsf{G}$ has a loop at A for $c_1$ only, taking 2 steps to cross. All the other edges can be crossed in a single timestep by either connector. Unreachable nodes are in white.
  • Figure 2: The physical graph described in Proposition \ref{['prop:eqm']}, serving as a game for the 3-SAT problem we aim to reduce from. Each node in the top layer, denoted by $x_i$, corresponds to a variable $x_i$ in the SAT formula, which contains a total of $n$ variables. The nodes in the layer below signify a positive or negative assignment. Each assignment node is connected to the clauses it satisfies. For example, in the depicted graph $C_1=x_1\lor\neg x_2$, $C_2=\neg x_1\lor x_2$, and $C_n = x_2\lor x_n$. Edges available to the assignment connector starting from $x_1$ are solid, edges of the clause connectors starting from $C_j$ are dashed. Every path of the assignment connector of length $2n$ ending in the terminal node $t$ encodes a full assignment.
  • Figure 3: The physical graph described in Proposition \ref{['prop:br_red']}, serving as a game for the set cover problem we aim to reduce from. The second and third layers are identical, containing a node for each set $S_i$. Every $u_i$ corresponds to a path going through the edges $S_j,S'_j$ of all the sets $S_j$ the $u_i$ is contained in. Red can interdict only the forward edges between the second and third layer (depicted in red color), encoding a selection of sets in the cover.
  • Figure 4: Computation times of the double oracle algorithm for grid world contested logistics scenarios with uniform edge interdiction costs.
  • Figure 5: Computation times of the double oracle algorithm for grid world contested logistics scenarios with randomly assigned edge interdiction costs.
  • ...and 6 more figures

Theorems & Definitions (5)

  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary
  • proposition thmcounterproposition
  • proof