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Game-Theoretic Drone Swarm Defense: A Case Study in Applied Differential Game Theory

Ross E. Allen

arXiv:2609.04394v1cs.GTcs.MAcs.RO

TL;DR

The problem is to assign guards and guide their trajectories so they intercept bandits before the latter reach high-value assets. The report develops three-phase differential-game tactics and compares them with unilateral-optimization baselines; under evasive bandits, the paired-trial analysis gives a 99.9% posterior probability that DG tactics have a higher successful-defense probability than CA tactics.

  • Problem

    Swarm defense requires assigning guards to bandits and guiding dynamically feasible trajectories to intercept them before they reach high-value assets.

    Source: §1 · p. 3

  • Method

    The report develops differential-game tactics within a shared three-phase architecture and compares them with nearest-bandit and coverage-aware unilateral-optimization baselines.

    Source: §2 · p. 5, §2.1 · p. 5

  • Results

    99.9% is the probability that DG tactics have a higher probability of successful defense than CA tactics for evasive bandits.

    Source: §5 · p. 15

  • Takeaways & Limitations

    With evasive bandits, DG tactics maintain successful-defense performance while the non-game-theoretic baselines show a modest drop.

    Source: §5 · p. 13

    The study assumes guards are faster than bandits and uses simplified aircraft and interception models, with centralized observation and communication.

    Source: §6 · p. 18

Abstract

from arXiv · show

This technical report is a study of the use of differential game (DG) theory to solve the target-assignment and midcourse guidance problems of drone swarms tasked with intercepting opposing swarms in defense of high-value assets. The game-theoretic tactics---which treat the intruder swarm as a rational agent and seek a Nash equilibrium between defenders and intruders---are compared against baseline tactics that model the defense problem as a unilateral optimization of the defenders' maneuvers. Monte Carlo simulation and Bayesian analysis show that the game-theoretic approach has a higher probability of successfully intercepting all intruders than the baseline techniques. This improvement in successful defense probability is most pronounced when the intruder swarm is capable of evasive maneuvers: relative to baseline optimization tactics, differential-game tactics increase estimated defense success from 94.6% to 96.8%, closing approximately 41% of the remaining gap to perfect defense. To add statistical credibility to this result, a paired-trial Bayesian analysis assigns a 99.9% posterior probability that differential-game tactics have a higher probability of successful asset defense than baseline tactics in this scenario.

1 Problem Formulation

The formulation treats swarm defense as assigning and guiding guards to intercept bandits before they reach high-value assets. It models the teams, their motion and information, and the distinction between unilateral optimization and mutually responsive game strategies.

1 Problem Formulation

  • Guards must intercept incoming bandits before any reaches a high-value asset, framing defense as a many-agent Lady-Bandit-Guard problem.The task includes both assigning guards to bandits and generating dynamically feasible interception trajectories.Source: §1 · p. 3
  • Defense requires both assigning guards to specific bandits and planning dynamically feasible guidance trajectories to achieve interceptions.The formulation identifies target assignment and guidance as distinct but related parts of the problem.Source: §1 · p. 3

1.1 Teams, State, and Information

  • Episode populations are fixed, while active guard and bandit rosters omit inactive vehicles from planning problems.The full environment state continues to represent inactive vehicles.Source: §1.1 · p. 3
  • Vehicles follow fixed-speed planar Dubins dynamics; altitude changes are ignored and guards are assumed faster than bandits.Each vehicle has a three-dimensional state and a scalar heading-rate control.Source: §1.1 · p. 3
  • Guards receive a common global observation, while bandit actions use a guard-blind observation and do not respond to guard state.Bandit behavior depends on bandit state, asset positions, speeds, status, and time.Source: §1.1 · p. 4

1.2 Objectives and Solution Concepts

  • The unilateral formulation minimizes the guard policy's cost for a prescribed bandit policy; the report colloquially calls feedback policies “tactics.”Game-theory literature also calls these feedback policies “strategies”.Source: §1.2 · p. 4
  • The differential-game formulation instead solves for mutually responsive guard and bandit strategies.The planned bandit strategy is adversarial rather than a prediction of the simulated guard-blind behavior.Source: §1.2 · p. 4
  • The modeled Nash equilibrium leaves neither team better off by changing its strategy alone, and receding-horizon replanning lets guards respond to observed deviations.The equilibrium definition is attributed to [2, Chapter 6].Source: §1.2 · p. 4

2 Guard/Defender Tactics

Guard tactics share a three-phase architecture, while baseline and differential-game versions differ most in how they assign targets and plan midcourse guidance. Both use proportional navigation for terminal guidance.

2 Guard/Defender Tactics

  • The shared control flow has target assignment, midcourse guidance, and terminal guidance, with PN used in the final interception phase.Target assignments may be updated during an engagement; the baseline and differential-game tactics use different algorithms, especially for midcourse guidance.Source: §2 · pp. 4–5, Figure 2 · p. 5

2.1 Baseline Tactics

  • Nearest-bandit and coverage-aware baselines use unilateral optimization and differ in target assignment; both follow the same three-phase architecture.Nearest-bandit assigns each guard to its nearest bandit, while coverage-aware assignment balances interception ease with coverage of all bandits.Source: §2.1 · p. 5, §2 · p. 5

2.1.1 Heading Error Midcourse Guidance

  • Heading-error guidance steers each guard toward its assigned bandit's line of sight using a wrapped heading error and a positive gain.The commanded turn rate is the gain multiplied by the heading error.Source: §2.1.1 · p. 6

2.2 Differential Game Tactics

  • Differential-game tactics use soft target assignments and a finite-horizon Nash-game solution for midcourse guidance, with receding-horizon updates and PN terminal handoff.The initial assignment is a first guess; the game-theoretic midcourse guidance may lead to a different interception pairing.Source: §2.2 · p. 6, §2.2.1 · p. 6

2.2.1 Dynamic Target Soft-Assignment

  • Each guard receives one provisional bandit target, while bandits may have no assigned guards or several; a redundancy discount controls the diminishing value of additional guards.The soft assignment is an initial guess, not a fixed interception pairing.Source: §2.2.1 · pp. 6–8

2.2.2 Game Theoretic Midcourse Guidance

  • DG midcourse repeatedly solves a finite-horizon adversarial LQ game for feedback Nash control, optimizing smooth pairwise terminal-handoff distance and PN demand before replanning from updated states.Regularization keeps the range and PN-acceleration terms differentiable near zero; the soft minimum evaluates active guard–bandit pairs.Source: §2.2.2 · pp. 8–9

2.2.3 Terminal Guidance

  • Terminal guidance uses pure PN, converting its scalar acceleration command to a Dubins turn rate; the DG cost uses command magnitude as handoff maneuvering demand rather than propagating PN midcourse.Source: §2 · p. 5, §2.2.3 · pp. 9–10

3 Simulation Environment

The simulation separates guard tactics from environment-embedded bandit behavior and adjudicates interceptions with a time-step-aware effective range. It studies non-evasive and evasive bandits, which do not execute the computed equilibrium strategy.

3 Simulation Environment

  • The simulation models guards as an agent whose tactics are separable from the environment, while bandit tactics are built into the environment.The environment is implemented in Python with PettingZoo, and the differential-game linear-quadratic game is solved with PYDGENS.Source: §3 · p. 10

3.1 Interception Adjudication

  • The effective interception range expands for closing, nearly collinear, low-relative-speed encounters, addressing interceptions that large simulation time steps might otherwise miss.This adjudication tends to favor approach-from-behind configurations.Source: §3.1 · p. 10

3.2 Bandit/Intruder Tactics

  • Bandits use static target assignments and cannot observe or react to guard states; the study compares non-evasive B1 with evasive B2 midcourse behavior.B2 uses scheduled jinks that temporarily replace nominal heading-error guidance before it resumes.Source: §3.2 · pp. 10–11
  • The bandits do not actually use the game-theoretic equilibrium strategy computed for the bandit swarm.Their simulated behavior is instead represented by the environment's specified tactics.Source: §3.2 · p. 11

4 Experiment Design

The study compares guard tactics in Monte Carlo campaigns with non-evasive and evasive bandits, using matched trials and Bayesian analyses of defense and control outcomes. Its primary defense-success comparison treats each episode as a paired binary outcome across tactics.

4.1 Monte Carlo Trials

  • Monte Carlo campaigns evaluate differential-game tactics against baselines in separate non-evasive B1 and evasive B2 bandit scenarios.Trial parameters are randomized; B2 shares the B1 parameters and adds starred jink parameters.Source: §4 · p. 11, §4.1 · p. 11, Table 1 · p. 12
  • Coverage-aware and differential-game campaigns use identical ordered trial realizations, enabling direct comparison of their Bayesian performance posteriors.The campaigns share the same random seed for trial parameters.Source: §4.1 · p. 11

4.2 Performance Metrics & Bayesian Analysis

  • Defense-success is true only when all bandits are intercepted and no assets are reached; a Beta-Binomial model estimates its success probability.The model uses a uniform prior.Source: §4.2 · p. 11
  • HVA-reached and bandits-intercepted rates are binned for Dirichlet-Multinomial analysis, while control saturation measures the share of time steps with any guard control clipped.Table 2 reports per-trial performance metrics and Bayesian analyses with central 95% credible intervals.Source: §4.2 · pp. 11–12, Table 2 · p. 13
  • Bandits-intercepted rate and defense-success can give different impressions: intercepting 99 of 100 bandits in every episode can coexist with zero defense-success.Source: §4.2 · pp. 11–12
  • Paired trials are classified by whether both tactics succeed, only differential-game succeeds, only coverage-aware succeeds, or both fail, then used to estimate their joint outcome probabilities.This paired-outcome posterior supports estimating the probability that differential-game tactics have a higher defense-success probability than coverage-aware tactics.Source: §4.2 · p. 12

5 Results & Discussion

Differential-game tactics achieve the highest probability of successful asset defense, with the advantage most pronounced against evasive bandits. Paired-trial Bayesian analysis gives 99.9% probability that they outperform coverage-aware tactics in that setting.

5 Results & Discussion

  • DG maintains successful-defense performance when bandits become evasive, while both non-game-theoretic baselines show a modest drop.Tables 3 and 4 summarize the non-evasive and evasive environments, respectively.Source: §5 · p. 13, Table 3 · p. 14, Table 4 · p. 14
  • 96.8% defense-success probability for DG versus 94.6% for CA under evasive bandits closes 40.7% of CA's 5.4% gap to perfect defense.This comparison concerns successful defense of all high-value assets, not simply the share of bandits intercepted.Source: §5 · p. 13, Table 4 · p. 14
  • 99.9% is the probability that DG has a higher successful-defense probability than CA for evasive bandits.For non-evasive bandits, the corresponding probability is 85.8%.Source: §5 · p. 15
  • Figures 3–4 show tactic-specific median and worst B2 episodes, but individual trajectories do not make CA/DG behavior differences readily discernible.The report treats the statistical results as more salient than these illustrative rollouts.Source: §5 · p. 15, Figure 3 · p. 16, Figure 4 · p. 17

6 Limitations & Future Work

The study’s real-world scope is limited by centralized information and communication assumptions, simplified vehicle and interception models, and restricted intruder behavior and speed settings.

6 Limitations & Future Work

  • The study assumes centralized access to all drone and asset states and unlimited communication, while simplifying aircraft dynamics and interception adjudication.Camera and accelerometer sensor models are identified as needed to improve interception-dynamics simulation.Source: §6 · p. 18
  • Evasive bandits are only pre-scripted and open-loop; the study does not simulate bandits that perceive guards and maneuver reactively.The omitted behavior would allow bandits to maneuver to better evade interception.Source: §6 · p. 18
  • Experiments exclude bandits faster than guards because high-relative-velocity interception simulations would rely on unrealistic adjudication assumptions.The report says realistic study of these cases would require extensive, high-fidelity platform and sensor models.Source: §6 · p. 18

A.1 Discounted-Redundancy Assignment Reformulation

The discounted-redundancy objective is reformulated as a rectangular linear assignment problem with virtual slots for assigning guards to bandits.

A.1 Discounted-Redundancy Assignment Reformulation

  • Each active bandit receives NGA virtual assignment slots, with the kth guard assigned to that bandit represented by marginal value vjβk−1.Binary variable yi,j,k indicates whether guard i is assigned to slot (j, k).Source: §A.1 · p. 19
  • The assignment formulation includes the constraint yi,j,k ≤1 for every active bandit j and slot k.Source: §A.1 · p. 19
  • The linear-program relaxation is integral because the bipartite assignment constraint matrix is totally unimodular, and the cost matrix can be solved with SciPy’s linear-sum assignment routine.The formulation has NGA guard rows and NBANGA virtual bandit-slot columns; weakly decreasing slot values permit earlier slots to be filled first.Source: §A.1 · p. 19
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