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Hierarchical Defense in Leader-Defender-Attacker Games: Nested Equilibria and Receding-Horizon Control
Honglei Guo, Hanfang Zhang, Guochuan Zhang, Yuhan Zhao
TL;DR
The paper models a protective leader–defender group with distinct objectives competing against an attacker. It combines their hierarchical and external strategic interactions in a nested equilibrium, then reports cost and winning-region outcomes under the resulting approach.
Problem
The paper addresses how to model a protective leader–defender group with distinct objectives competing against an attacker.
Source: §I · pp. 1–2Method
A nested equilibrium combines a leader–defender Stackelberg relationship with the attacker’s Nash-type best response, and a receding-horizon algorithm implements the computed controls.
Source: §I · pp. 1–2, Contributions · p. 2, §IV · p. 5Results
Relative to independent control, the nested formulation reduced the leader’s realized cost and increased the attacker’s cost; at H = 5, leader difference was positive in all 50 trials and attacker difference was negative in all trials.
Source: §VI · p. 7, C. Comparison with the… · p. 7Takeaways & Limitations
The policy-induced winning regions are characterized through quadratic inequalities in the initial joint state.
Source: §VI · p. 7
Abstract
from arXiv · showhide
We formulate a discrete-time leader-defender-attacker game that combines hierarchical interaction within a protective group with competition against an external attacker. The leader seeks to reach a prescribed demand point while avoiding capture, whereas the defender seeks to intercept the attacker while remaining near the leader. To capture their distinct objectives and asymmetric interactions, we introduce a nested equilibrium concept combining a leader-defender Stackelberg relationship with a Nash-type best-response condition for the attacker. For linear dynamics and quadratic objectives, we derive affine response laws and a backward recursion, together with sufficient conditions for unique stagewise solutions. We further express arrival, capture, and interception conditions as quadratic inequalities in the initial joint state, characterizing winning regions under the resulting policy. A receding-horizon algorithm implements the computed control laws using updated state information. Numerical experiments over randomized initial configurations examine the influence of the prediction horizon and compare the proposed method with an independent-control baseline, illustrating differences in game outcomes and the agents' realized costs.
I. INTRODUCTION
The paper addresses strategic asymmetry within a protective leader–defender group competing against an attacker, proposing a nested equilibrium and a receding-horizon solver. It formulates the agents’ distinct objectives and asks for optimal strategies and initial-state winning regions.
I. INTRODUCTION
- Prior protected-target studies generally model the target and defender as a cooperative team, while mixed-leadership games do not consider a fixed leader–follower pair competing against an independently optimizing third agent.Source: §I · p. 1
- The proposed nested equilibrium combines a leader–defender Stackelberg relationship with the attacker’s Nash-type best response, which evaluates unilateral deviations while holding the group’s strategies fixed.Source: §I · pp. 1–2, Contributions · p. 2
- The paper establishes sufficient conditions for existence and uniqueness of the nested equilibrium and characterizes winning regions from the initial joint state.Source: Contributions · p. 2
- A receding-horizon algorithm solves for the nested equilibrium and is evaluated through numerical simulations.Source: Contributions · p. 2
II. PROBLEM FORMULATION
- The leader seeks demand points while avoiding attack, the defender seeks to intercept the attacker near the leader, and all agents are assumed to have linear dynamics.The leader moves first; the defender acts after observing the leader’s action. The states include each agent’s position and velocity.Source: §II · p. 2
- The formulation asks for all three agents’ optimal strategies and whether each agent has a winning region.Source: §II · p. 2
III. EQUILIBRIUM AND WINNING REGION ANALYSIS
The section derives affine equilibrium controls through backward recursion and relates capture, interception, and arrival times to the initial joint state. Under stated conditions, the equilibrium controls are uniquely determined, and event-time comparisons characterize the winning regions.
Equilibrium derivation
- The leader’s, defender’s, and attacker’s objectives are reformulated in quadratic form for the dynamic-programming analysis.The cost-to-go is quadratic-affine in the joint state.Source: §III · p. 3
- The defender’s optimization is strictly convex when its stated condition holds, yielding a unique best response.The leader anticipates that response when optimizing.Source: §III · p. 4
- If M_t is nonsingular, the coupled equilibrium controls are uniquely determined by the first-order conditions.The continuation-value parameters are updated backward from the terminal conditions using the Bellman recursion.Source: §III · p. 4
Winning-region analysis
- Under the affine equilibrium policy, each event’s completion time is determined by the initial state, and the winning regions follow from which event occurs first.The stated characterization assumes the game ends at the first terminal event and event times are distinct.Source: §III · pp. 4–5
T AL < min{T DA, T Ld},
The passage introduces the condition under which the defender wins.
Winning-region analysis
- The defender’s winning condition is stated in terms of an event-time comparison.The comparison itself is not present in this passage.Source: T AL < min{T DA, T Ld} · p. 5
T DA < min{T AL, T Ld},
The leader wins when arrival occurs before both attacker capture and defender interception. The analysis concerns fixed-horizon planning costs, not costs terminated at the first event.
Winning-region analysis
- The leader wins precisely when its arrival time is earlier than both capture and interception.The condition is T_Ld < min{T_AL, T_DA}.Source: T DA < min{T AL, T Ld} · p. 5
Planning-cost caveat
- The equilibrium characterization is for a fixed-horizon quadratic game and does not establish equilibrium when costs terminate at the first event.Capture, interception, and arrival instead serve as execution stopping criteria and trajectory outcome measures.Source: T DA < min{T AL, T Ld} · p. 5
IV. ALGORITHM DESIGN
The algorithm repeatedly solves the finite-horizon nested equilibrium from the latest measured state, applies only the first predicted control, and recomputes after observing the next state.
IV. ALGORITHM DESIGN
- Only the first predicted control is applied; after the system evolves, the next state is observed and the finite-horizon equilibrium is recomputed until a terminal event.The prediction window advances by one step after each state update.Source: §IV · p. 5
- At each sampling instant, the measured joint state initializes a new prediction problem.The prediction uses information available at that instant to represent future joint states.Source: §IV · p. 5
- The solver evaluates equilibrium conditions across the prediction horizon using continuation values for the remaining horizon, producing equilibrium feedback laws.The gains and affine terms come from the backward equilibrium recursion.Source: §IV · p. 5
V. NUMERICAL EXPERIMENTS
The experiments assess prediction-horizon sensitivity and compare the proposed algorithm with an independent-control baseline in a two-dimensional three-agent environment.
V. NUMERICAL EXPERIMENTS
- The evaluation uses a two-dimensional leader–defender–attacker environment and examines prediction-horizon sensitivity alongside comparative experiments.The comparative experiments use 50 randomly generated initial configurations.Source: §V · p. 5
A. Simulation Setup
The simulations use randomized initial positions, double-integrator agent dynamics, finite-horizon predictions, and terminal events that stop execution without truncating planning costs.
A. Simulation Setup
- Initial positions are sampled independently and uniformly from a 10 m × 10 m square, initial velocities are zero, and each simulation runs for at most 200 steps.The setup applies the same initialization and step limit to the three agents.Source: A. Simulation Setup · p. 5
- Algorithm 1 measures the current state, solves the finite-horizon nested equilibrium backward, applies the first control, and stops when a terminal event occurs.The loop runs for at most N_total steps.Source: A. Simulation Setup · p. 6
- Each agent follows discrete-time double-integrator dynamics, with position and velocity in its state and acceleration as its control input.The state and input are defined for leader, defender, and attacker.Source: A. Simulation Setup · p. 6
- At each step, the strategy sequence is computed over horizon H, but only its first control input is applied.The listed cost parameters are Q_i1 = 2I_4, Q_i2 = I_4, R_i = 0.1I_2, and discount factor γ = 0.99.Source: A. Simulation Setup · p. 6
- Leader arrival, attacker capture, and defender interception are terminal events at respective distance thresholds rLd = rAL = rDA = 0.3 m; execution stops at the first event.Simultaneous first events are classified as ties, and trials with no event within Ntotal steps are classified as no event. Terminal events do not truncate fixed-horizon planning costs.Source: A. Simulation Setup · p. 6
B. Different Prediction Horizon
The empirical outcome frequencies vary with prediction horizon: very short prediction often yields no event, while the leading winner changes at longer horizons. Outcome frequencies stabilize beyond H = 20, though individual trajectories need not match.
B. Different Prediction Horizon
- 92% of trials had no terminal event at H = 1, while all trials terminated for H ≥5; the defender led at H = 5 and the attacker at H ≥15.Table I reports outcomes over 50 random initial configurations, with each entry giving run counts and percentages.Source: B. Different Prediction Horizon · p. 6, Table I · p. 6
- The attacker’s win rate reached 50% at H = 15 and remained 46% for H ∈{20, 25, 30}.At H = 10, outcomes were more balanced; the defender had the highest win rate at H = 5.Source: B. Different Prediction Horizon · p. 6
- Beyond H = 20, outcome distributions change only slightly and are identical at H = 25 and H = 30, without implying identical trajectories or controls.These are empirical frequencies for the sampled initial conditions, not formal winning-region measures.Source: B. Different Prediction Horizon · p. 6
- In the illustrated initial configuration, H = 1 yields no event, H = 5 and 10 yield defender wins, and H ≥15 yields leader wins.Among completed runs in this example, termination time decreases as the prediction horizon increases.Source: B. Different Prediction Horizon · p. 6, Figure 2 · p. 6
- Figure 3 compares objective-function differences between the proposed nested-equilibrium algorithm and the independent-control baseline for H = 5, with both methods using identical initial conditions and model parameters.Source: Figure 3 · p. 7
C. Comparison with the Independent-Control Baseline
At H = 5, the nested formulation lowers the leader’s cost in all 50 trials and raises the attacker’s cost in all 50, while defender outcomes vary. The comparison measures realized discounted objectives against independent control, where agents treat the others’ strategies as fixed.
C. Comparison with the Independent-Control Baseline
- Under independent control, each agent optimizes its finite-horizon policy while treating the other agents’ strategies as fixed.The leader therefore omits the defender’s best-response mapping from its optimization.Source: C. Comparison with the… · p. 7
- The objective difference compares each agent’s realized discounted objective under independent control with its objective under the nested formulation.A positive difference indicates lower cost under the nested formulation.Source: C. Comparison with the… · p. 7
- At H = 5, the leader difference is positive and the attacker difference negative in all 50 trials, while the defender difference varies.The paper interprets this pattern as lower leader cost and higher attacker cost under the nested formulation.Source: C. Comparison with the… · p. 7
VI. CONCLUSION
The paper develops a nested equilibrium for a finite-horizon leader–defender–attacker game and derives a linear–quadratic solution and policy-induced outcome regions. Numerical results show that nested control reduces the leader’s realized cost while increasing the attacker’s relative to independent control.
VI. CONCLUSION
- The paper formulates a finite-horizon leader–defender–attacker game and proposes a nested equilibrium combining an internal leader–follower hierarchy with external noncooperative competition.Source: §VI · p. 7
- For linear–quadratic systems, the method derives affine response laws, coupled backward recursions, and sufficient conditions for unique stagewise solutions.It characterizes policy-induced outcome regions through quadratic inequalities in the initial joint state.Source: §VI · p. 7
- Numerical results show that nested control reduces the leader’s realized cost while increasing the attacker’s cost relative to independent control.Future work will address nonlinear dynamics, constraints, uncertainty, and theoretical guarantees for receding-horizon execution.Source: §VI · p. 7