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Adversarial Graph Traversal

David Banks, Elvan Ceyhan, Leah Johnson, Li Zhou

TL;DR

The paper addresses sequential graph traversal under adversarial payoff manipulation by a Bayesian traveler who models costs/payoffs with a Normal-Inverse-Wishart prior and an opponent via a level-$k$ (Type 0/Type 1) framework. It derives the induced multivariate $t$ distribution after marginalizing the NIW precision and employs a Beta prior on the opponent type, using a Bayesian learning process to update beliefs as traversal unfolds. An adaptive Uncertainty Policy guides path selection by balancing computational load and information gain, and the approach is shown to outperform naive fixed-path and myopic strategies, especially when adversary learning is incorporated. Empirical results from a 6×6 grid and an ANOVA analysis demonstrate the robustness and efficiency of the method, with clear gains over baselines and insights into how adversarial presence shapes optimal routing decisions in convoy-like settings.

Abstract

Suppose a Bayesian agent seeks to traverse a graph. Each time she crosses an edge, she pays a price. The first time she reaches a node, there is a payoff. She has an opponent who can reduce the payoffs. This paper uses adversarial risk analysis to find a solution to her route selection problem. It shows how the traveler is advantaged by having an accurate subjective distribution over the costs/payoffs and by having a Bayesian prior for her opponent's strategic choices. The results are relevant to military convoy routing, corporate competition, and certain games.

Adversarial Graph Traversal

TL;DR

The paper addresses sequential graph traversal under adversarial payoff manipulation by a Bayesian traveler who models costs/payoffs with a Normal-Inverse-Wishart prior and an opponent via a level- (Type 0/Type 1) framework. It derives the induced multivariate distribution after marginalizing the NIW precision and employs a Beta prior on the opponent type, using a Bayesian learning process to update beliefs as traversal unfolds. An adaptive Uncertainty Policy guides path selection by balancing computational load and information gain, and the approach is shown to outperform naive fixed-path and myopic strategies, especially when adversary learning is incorporated. Empirical results from a 6×6 grid and an ANOVA analysis demonstrate the robustness and efficiency of the method, with clear gains over baselines and insights into how adversarial presence shapes optimal routing decisions in convoy-like settings.

Abstract

Suppose a Bayesian agent seeks to traverse a graph. Each time she crosses an edge, she pays a price. The first time she reaches a node, there is a payoff. She has an opponent who can reduce the payoffs. This paper uses adversarial risk analysis to find a solution to her route selection problem. It shows how the traveler is advantaged by having an accurate subjective distribution over the costs/payoffs and by having a Bayesian prior for her opponent's strategic choices. The results are relevant to military convoy routing, corporate competition, and certain games.
Paper Structure (8 sections, 3 theorems, 25 equations, 4 figures, 1 table)

This paper contains 8 sections, 3 theorems, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Optimization Problem
  3. Two Models
  4. Navigation in the Absence of an Adversary
  5. Navigation Against an Adversary
  6. Fixed-Path vs. Adaptive Choice
  7. Search and Simulation
  8. Conclusion

Key Result

Theorem 1

The Normal-Inverse-Wishart ($\mathcal{NIW}$) distribution is closed under conditioning.

Figures (4)

  • Figure 1: An illustration of the three situations (No Adversary, Ignoring Adversary, and Incorporating Adversary) in our 6-by-6 grid simulation.
  • Figure 2: Three significant interaction plots from the fully crossed ANOVA model showing how adversary type interacts with decision strategy, traversal policy, and belief model in predicting average net reward.
  • Figure 3: Significant policy-strategy interaction from the ANOVA model using adversary-present runs, illustrating the different reward gains for the Uncertainty Policy heuristic, the $H$-path, and the myopic policy.
  • Figure 4: Simulation time comparison across adversarial type, illustrating computational efficiency differences between the Uncertainty Policy, the $H$-path, and the myopic policy.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3