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Deception in Reach-Avoid Game with Unknown Heterogeneous Attackers Speed Information

Xiangkai Wu, Shaolin Tan, Wei Wang, Zhen Han

arXiv:2609.06953v1cs.GTcs.RO

TL;DR

Prior reach-avoid studies leave the strategic effect of heterogeneous attacker speeds unresolved. This paper introduces critical speed pairs to analyze capture-order ambiguity and defender guesses, and shows that under specified initial conditions, attacker deception can disadvantage the defender without adding risk to the attackers.

  • Problem

    Prior reach-avoid studies of uncertain speeds assume homogeneous attackers, leaving the strategic effect of relative capability configurations unexplored.

    Source: §I · p. 1

  • Method

    A critical speed pair framework characterizes when heterogeneous speed configurations induce opposite optimal capture orders and supports analysis of defender guesses and attacker information-limiting strategies.

    Source: §I · p. 1, §IV · p. 3, C. The guess of the Defender · p. 6

  • Results

    Under specified initial states, slow-speed information-limiting play forces the defender to guess; incorrect guesses reduce its payoff, correct guesses add no benefit, and attackers incur no additional risk.

    Source: D. The dilemma · p. 7, §VII · p. 7

  • Takeaways & Limitations

    The visualized dilemma conditions occupy a substantial portion of the state space, indicating ample opportunities for attacker deception.

    Source: §VI · p. 7

Abstract

from arXiv · show

This letter investigates a reach-avoid game involving two Attackers and one Defender, where the Attackers aim to maximize the number reaching the target region while the Defender seeks to minimize it. In contrast to conventional complete information formulations, we consider an information asymmetry scenario where the Attackers' heterogeneous maximum speeds are privately known but publicly disclosed to lie within continuous ranges. Existing studies on uncertain speeds, however, have primarily focused on homogeneous settings, whereas heterogeneity extends the uncertainty from a common capability level to the relative capability configuration of the Attackers. To address the resulting capture-order ambiguity over infinitely many possible speed combinations, we establish a critical speed pair framework that characterizes when different capability configurations induce different optimal capture orders, and enables the analysis of the Defender's guessing behavior and the design of information-limiting strategies for the Attackers. We demonstrate that under certain initial conditions, the Attackers can mislead the Defender into making suboptimal decisions through a slow-speed deception strategy, achieving superior payoffs compared to the complete information game. Numerical visualizations reveal the widespread occurrence of such dilemma conditions.

I. INTRODUCTION

The paper shifts reach-avoid analysis from complete-information and homogeneous-speed settings to a two-attacker game with privately known heterogeneous speeds bounded publicly by continuous ranges. It proposes critical speed pairs to study capture-order ambiguity and deception, then formulates the planar game and its information assumptions.

I. INTRODUCTION

  • Prior reach-avoid analyses often assume complete information, while incomplete knowledge of states, payoffs, or parameters is common in practice.Source: §I · p. 1
  • Prior uncertain-speed deception studies used rotating-turret defenders or homogeneous attackers, leaving simple-motion strategy and relative heterogeneous capabilities unresolved.Source: §I · p. 1
  • This letter studies 2A1D play with privately known heterogeneous attacker speeds in publicly known continuous ranges, using critical speed pairs to analyze capture-order ambiguity and deception.Source: §I · p. 1

II. PROBLEM FORMULATION

  • In the planar game, two attackers seek the target y≤0 without capture while the defender intercepts them; decisions use instantaneous game-state feedback.Source: §II · p. 2
  • Attackers' maximum speeds are private, their lower bounds are public, and the defender observes their accurate historical speeds.Source: §II · p. 2
  • The formulation asks when attackers can improve payoffs through deception that forces the defender into risky guesses.Source: §II · p. 2

III. PRELIMINARIES

The preliminaries define winning regions and barriers for one-attacker and two-attacker games under complete information. The 2A1D barrier depends on capture order, while multiple optimal angle pairs can leave the defender with a momentary dilemma.

A. Barrier of one-Attacker one-Defender (1A1D) game

B. Barriers of 2A1D game

  • For a given capture order, the 2A1D barrier separates states where the defender can guarantee capturing both attackers from states where at least one can guarantee reaching the target.Source: B. Barriers of 2A1D game · pp. 2–3
  • When multiple angle pairs minimize the barrier calculation, attackers may choose any optimal pair with the same value, while the defender faces a momentary dilemma.Source: B. Barriers of 2A1D game · p. 3

IV. THE CRITICAL SPEEDS AND CRITICAL SPEED PAIRS

The section replaces individual critical-speed thresholds with critical speed pairs for heterogeneous attackers, then characterizes how those pairs partition speed configurations by capture outcome. It also derives the boundary and strategies used to analyze the resulting capture-order choices.

IV. THE CRITICAL SPEEDS AND CRITICAL SPEED PAIRS

  • For the 1A1D game, an Attacker above its critical speed cannot be captured, while one below it can be captured regardless of its strategy.The critical speed is the threshold separating these two cases.Source: §IV · p. 3
  • Critical speed pairs characterize heterogeneous speed configurations that produce opposite optimal capture orders, addressing a limitation of earlier critical-speed concepts.The pair is defined for a specified Defender capture order by conditions distinguishing at-most-one capture from guaranteed capture of both Attackers.Source: §IV · p. 3

A. Computation of the critical speed (pairs)

  • For any λi in [0, µi], there is a unique λj in (0, µj) satisfying ˆJi(x, λi, λj) = 0; for λi < µi, this is the critical speed pair when the Defender captures Ai first.For (vi, vj) ≺ (λi, λj) with λi < µi, the Defender can capture both Attackers; for (vi, vj) ≻ (λi, λj), the Defender cannot capture Aj.Source: A. Computation of the critical speed… · pp. 3–4
  • The critical-pair computation uses an auxiliary game in which the Defender captures Ai first and the payoff is Aj’s y-coordinate at capture.The Attackers seek to minimize this payoff, while the Defender seeks to maximize it; optimal headings are constant at maximum speed until Ai is captured.Source: A. Computation of the critical speed… · p. 3
  • The functions hi define the speed-pair boundary h(v1), separating the region J*(x, v1, v2) = 2 from the region J*(x, v1, v2) ≤ 1.The boundary h(v1) is continuous and strictly decreasing over the stated interval.Source: A. Computation of the critical speed… · p. 4
  • When the speed configuration lies beyond the boundary, the Defender can guarantee capturing both Attackers; the Attackers’ specified strategies can instead ensure at most one capture.The Defender’s sequential strategy captures A1 and then A2 regardless of the Attackers’ strategy, while the alternative guarantees at most one capture.Source: A. Computation of the critical speed… · p. 4

B. The rate of the critical speed (pairs)

Because critical speeds and pairs change with the game state, the section derives their time rates. The pair-rate result is stated under a uniqueness condition on the minimizing angle pair.

B. The rate of the critical speed (pairs)

  • The critical speeds and pairs evolve with the game state, motivating derivation of their time derivatives for subsequent analysis.The rate of the individual critical speed µi is given by Eq. (16).Source: B. The rate of the critical speed… · p. 4
  • When the minimizing angle pair is unique (mi = 1), the section gives the rate of λj for a critical speed pair with fixed λi.The derivation differentiates ˆJi(x, λi, λj) = 0 over time and substitutes the game dynamics.Source: B. The rate of the critical speed… · p. 4

V. MAIN RESULTS

This section characterizes strategic behaviors in the 2A1D reach-avoid game based on critical speeds and critical speed pairs, under Assumptions 1 and 2.

V. MAIN RESULTS

  • Critical speeds and critical speed pairs provide the basis for characterizing strategic behavior in the 2A1D reach-avoid game under Assumptions 1 and 2.Source: §V · p. 4

A. Mismatching order condition of the Defender

This subsection characterizes when the Attackers’ speed information changes the Defender’s optimal capture order. In mismatching-order states, different true speed configurations can induce different optimal orders.

A. Mismatching order condition of the Defender

B. Deceptive strategy of the Attackers

The Attackers use a slow-speed strategy to limit the information available to the Defender, but deception is pursued only when it can benefit them. Warning states specify when they should instead use maximum speed.

B. Deceptive strategy of the Attackers

C. The guess of the Defender

The Defender’s guess concerns which heterogeneous speed pairs remain possible given the observed trajectory. The paper defines a guess through deterioration in the Defender’s optimal payoff for a candidate speed pair.

C. The guess of the Defender

  • The paper extends the Defender’s trajectory-based speed-exclusion notion of a guess to heterogeneous maximum-speed pairs.Source: C. The guess of the Defender · p. 6
  • A guess is made when the Defender’s control input lowers the optimal payoff for a candidate speed pair, indicating that the pair has been excluded.Source: C. The guess of the Defender · p. 6
  • If the Defender’s exclusion is incorrect, the Attackers can exploit their true maximum speeds for better payoff; otherwise, they should maintain deception.Source: C. The guess of the Defender · p. 6
  • The supporting lemmas track decreases in inferred speed levels along the trajectory to identify speed pairs the Defender has guessed against.Source: C. The guess of the Defender · p. 6

D. The dilemma

Under the theorem’s initial-state conditions, the information-limiting strategy forces the Defender to guess before it can determine the optimal capture order. Incorrect guesses reduce payoff, while correct guesses add no benefit, and the strategy incurs no risk.

D. The dilemma

  • For initial states in (C ∩ M ∩ Q) \ W, the information-limiting strategy forces the Defender to guess before it can determine the optimal capture order.Source: D. The dilemma · p. 7
  • The theorem states that incorrect guesses reduce payoff, correct guesses provide no additional benefit, and the Attackers incur no risk from the proposed strategy.Source: D. The dilemma · p. 7
  • Incorrect guesses reduce payoff from 1 to 0 or from 2 to 1, while correct exclusions in another case maintain payoff.Source: D. The dilemma · p. 7

VI. VISUALIZATIONS OF DILEMMA CONDITIONS

Both tested cases contain a substantial region of states where the Attackers can create dilemmas. The visualizations vary Attacker A2’s position while fixing A1 and the Defender, and evaluate the cases over a specified region using sampled critical speed pairs.

VI. VISUALIZATIONS OF DILEMMA CONDITIONS

  • The visualization fixes A1 at [−2, 3] and the Defender at [0, 5], while varying A2’s position.The analyzed region is [−11, 11]×[0.01, 20], sampled at grid interval 0.021.Source: §VI · p. 7
  • The two cases share known speeds and warning threshold ε = 0.3 but use different heterogeneous maximum-speed values.The supplied passage does not specify the complete maximum-speed values for both cases.Source: §VI · p. 7
  • Both cases show a non-empty, substantial state-space region of dilemma conditions, indicating ample opportunities for deception.The visualizations evaluate whether states belong to Q_i using five uniformly selected critical speed pairs.Source: §VI · p. 7, Figure 3 · p. 7

VII. CONCLUSION

The paper uses a critical speed pair framework to analyze strategic dilemmas in a 2A1D reach-avoid game with uncertain heterogeneous attacker speeds. It concludes that a slow-speed information-limiting strategy can lower the Defender’s payoff after an incorrect guess without adding risk to the Attackers.

VII. CONCLUSION

  • A slow-speed information-limiting strategy can reduce the Defender’s payoff after an incorrect guess without adding risk to the Attackers.The strategy compels the Defender to exclude certain speed-pair possibilities.Source: §VII · p. 7
  • The critical speed pair framework identifies when uncertainty about relative attacker capabilities creates strategic dilemmas for the Defender.The analysis examines mismatching-order, motivation, warning, and forcing states.Source: §VII · p. 7
  • Heterogeneous speeds let Attackers conceal their relative capability configuration, an information advantage unavailable in the homogeneous-speed setting.Source: §VII · p. 7
  • The authors identify extending the deception analysis to multi-player scenarios in more complex environments as future work.Source: §VII · p. 7
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