Repeated Deceptive Path Planning against Learnable Observer

# Repeated Deceptive Path Planning against Learnable Observer
Source: [https://arxiv.org/html/2605.07174](https://arxiv.org/html/2605.07174)
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ifaamas\\@printcopyrighttrue\\acmConference\[AAMAS ’26\]Proc\. of the 25th International Conference on Autonomous Agents and Multiagent Systems \(AAMAS 2026\)May 25 – 29, 2026 Paphos, CyprusC\. Amato, L\. Dennis, V\. Mascardi, J\. Thangarajah \(eds\.\)\\copyrightyear2026\\acmYear2026\\acmPrice\\acmISBN\\acmSubmissionID¡¡144¿¿\\affiliation\\institution1School of Artificial Intelligence, University of Chinese Academy of Sciences\\cityBeijing\\countryChina\\affiliation\\institution2National Key Laboratory of Cognition and Decision Intelligence for Complex Systems, Institution of Automation, Chinese Academy of Sciences\\cityBeijing\\countryChina

Shiyue Cao1,2, Pei Xu2, Likun Yang1,2, Lei Cui1,2, Shizhao Yu1,2, Shiyu Zhang2, Yongjian Ren1,2, Xiaotang Chen2, Kaiqi Huang1,2[caoshiyue2021, pei\.xu, yanglikun2021, cuilei2024, yushizhao2022, shiyu\.zhang, renyongjian2022@ia\.ac\.cn](https://arxiv.org/html/2605.07174v1/mailto:caoshiyue2021,%20pei.xu,%20yanglikun2021,%20cuilei2024,%20yushizhao2022,%20shiyu.zhang,%[email protected])[xtchen, kaiqi\.huang@nlpr\.ia\.ac\.cn](https://arxiv.org/html/2605.07174v1/mailto:xtchen,%[email protected])

###### Abstract\.

We study the problem of deceptive path planning \(DPP\), where an agent aims to conceal its true destination from external observers\. While existing work assumes static, non\-learning observers, real\-world adversaries—such as in critical goods transportation or military operations—can adapt by learning from historical trajectories\. To address this gap, we introduce Repeated Deceptive Path Planning \(RDPP\), a new formulation that explicitly models learnable observers\. We show that existing DPP methods fail under this setting, as they cannot adapt to evolving adversarial predictions\. While incorporating observer previous predictions into updates enables some adaptation, such incremental updates cause accumulative lag that degrades deception\. To this end, we proposeDeceptiveMetaPlanning \(DeMP\), a two\-level optimization framework that combines episode\-level adaptation, which enables short\-term policy adjustment to counter updated observer, and meta\-level updates, which leverage cross\-episode feedback to capture how observers update their models and accelerate adaptation in future episodes\. In this way, DeMP mitigates the accumulation of adaptation lag, enabling sustained deception against a learning observer\. Experiments across environments demonstrate that DeMP significantly outperforms existing approaches in RDPP while maintaining competitive path cost\. Our results highlight the importance of modeling repeated interactions with learnable adversaries, providing new insights into deception and privacy in multi\-agent systems\.

###### Key words and phrases:

Deceptive Path Planning, Goal Recognition, Reinforcement Learning

## 1\.Introduction

![Refer to caption](https://arxiv.org/html/2605.07174v1/x1.png)Figure 1\.Illustration of Repeated Deceptive Path Planning \(RDPP\)\. Unlike single\-shot DPP, RDPP introduces a learnable observer\. After each episode, the observer receives the full trajectory of agent to update its recognition model, while the agent accesses the predicted goal of observer\. This highlights the core challenge of RDPP: an agent must adapt its policy to achieve sustained deception over multiple interactions\.Privacy protection in planning is a critical concern in adversarial scenarios, where agents must achieve their goals while preventing external observers from inferring their true intentions\. This challenge arises in many real\-world applicationsBell2003TowardAT;xu2022path;luo2019opponent\. For example, when cash trucks transport money in urban environments, their routes may be observed by potential threats, making it necessary to design paths that conceal the actual destination\.

![Refer to caption](https://arxiv.org/html/2605.07174v1/x2.png)Figure 2\.Trajectory Evolution of DeMP in RDPP\. The left upper panel shows the static trajectory of the baseline method AM\. The remaining panels are State Visits Heatmaps illustrating DeMP’s path distribution over repeated interactions, where brighter colors indicate more frequent visits and the dashed line is the final path\. Unlike AM, DeMP demonstrates continuous path evolution and diverse path adaption, successfully preventing the learnable observer from inferring the true goal based on a single movement pattern\.Deceptive Path Planning \(DPP\)masters\_deceptive\_2017addresses this problem by generating trajectories that obscure the agent’s true goal from an adversarial observer\. Existing approaches span path planningmasters\_deceptive\_2017;savas\_deceptive\_2022, control\-based methodswagner2011acting;dragan2014analysis;ornik2018deception, objective optimizationgutierrez2025agent, and reinforcement learningliu\_deceptive\_2021;lewis\_deceptive\_2023;fatemi\_deceptive\_2024\. However, most of these methods focus on one\-shot interactions and assume static, non\-learnable observers\. In practice, adversaries often monitor agents across repeated interactions—for example, in military operations, where opponents can analyze troop movements over extended periods—and continuously refine their inference strategies by exploiting historical trajectories\. Under such repeated settings, the effectiveness of existing DPP methods deteriorates rapidlyornik2018deception, highlighting a fundamental limitation of one\-shot deceptive planning\.

Building on DPP, we introduce Repeated Deceptive Path Planning \(RDPP\), which extends deceptive planning to repeated interactions with a learnable observer\. As illustrated in Figure[1](https://arxiv.org/html/2605.07174#S1.F1), in each episode the agent executes a trajectory toward its true goal while attempting to mislead the observer, who predicts the goal based on a partial trajectory prefix\. After the episode, the observer updates its recognition model using the full trajectory and true goal, while the agent gains access to the observer’s prediction as feedback for future interactions\. This formulation captures a fundamental challenge absent in single\-shot DPP: deception must be sustained against an observer whose recognition capabilities improves over time\.

A natural approach for RDPP is to incorporate the observer’s predictions into the reward and update the agent’s policy after each episode\. However, this reactive strategy suffers from a structural limitation\. Because the agent adapts only after observing the observer’s improvement, its policy updates consistently trail the observer’s learning progress\. Over repeated interactions, this mismatch leads to an accumulation of adaptation lag, causing previously effective deceptive behaviors to become predictable and eventually ineffective\.

To address this issue, we proposeDeceptive Meta Planning \(DeMP\), a two\-level optimization framework that combines episode\-level adaptation with meta\-level updates across episodes\. At the episode\-level, the agent adapts its strategy after each episode of interaction to counter the observer’s updated recognition model\. Crucially, at the meta\-level, DeMP integrates long\-term feedback using higher\-order gradient information\. Rather than merely reacting to past updates, this mechanism mathematically anticipates the observer’s learning dynamics\. This enables proactive adaptation, allowing the agent to identify policy initializations that are robust to the observer’s future decision boundary shifts\. By doing so, DeMP theoretically mitigates the accumulation of adaptation lag inherent in sequential adversarial interactions, enabling sustained deception\.

We evaluate RDPP and DeMP in a set of grid\-world environments with adaptive observers\. Experimental results demonstrate that existing DPP methods suffer significant degradation under repeated interactions, whereas DeMP sustains high deception performance while maintaining competitive path costs\. These findings highlight the effectiveness of our approach and emphasize the importance of considering learnable adversaries in repeated deceptive path planning scenarios\.

Overall, we make three key contributions to deceptive planning:

- •We formalize Repeated Deceptive Path Planning \(RDPP\), extending standard DPP to repeated interactions with learnable observers\.
- •We propose Deceptive Meta Planning \(DeMP\), a two\-level framework supported by theoretical analysis that justifies its surrogate objective alignment and validates its proactive adaptation against evolving observers\.
- •We empirically demonstrate that DeMP sustains high deception performance while maintaining competitive path efficiency in repeated interactions\.

![Refer to caption](https://arxiv.org/html/2605.07174v1/x3.png)Figure 3\.The two\-level optimization framework of DeMP\. The process is structured into two levels: \(1\) The Episode\-Level Adaptation involvesMMepisodes where the Agent adapts its policy parameterθ\(k\)\\theta^\{\(k\)\}to counter the Observer’s updated predictions\. The Observer updates its recognition model after each episode\. \(2\) The Meta\-Level Update \(dashed box\) occurs afterMMepisodes\. It utilizes the accumulated cross\-episode feedback to compute a meta\-gradient that updates the Agent’s policy initializationθ\(0\)\\theta^\{\(0\)\}\. By optimizingθ\(0\)\\theta^\{\(0\)\}, DeMP anticipates the Observer’s learning trajectory, thereby mitigating the adaptation lag in RDPP\.
## 2\.Related Work

### 2\.1\.Deceptive planning

Deception is a fundamental concept in adversarial scenarios, closely tied to privacy, security, and obfuscation\. A general theory of deceptionWhaley1982TowardAG;Bell2003TowardATdefines simulation and dissimulation as two basic strategies\. Building on this foundation, deception has been applied to path planning\. In deceptive path planning \(DPP\), an action is considered deceptive if the real goal does not dominate the probability of other goalsmasters\_deceptive\_2017, and deception emerges as an optimal planning objectivehuang2021dynamic;karabag2019optimal\.

Recent studies have introduced reinforcement learning \(RL\) frameworks for DPP, generating action sequences that keep the agent’s true goal ambiguous among alternatives, such as the Ambiguity Model \(AM\)liu\_deceptive\_2021and the Deceptive Exploration Ambiguity Model \(DEAM\)lewis\_deceptive\_2023\. These methods assume that the observer model is known to the agent, which then plans ambiguous paths across multiple goals\. An extended goal recognition framework for strategic deceptionmasters\_extended\_2021further investigates how factors such as perception and memory affect susceptibility to deception\. In addition, the motion\-deception gamerostobaya2023eaterexamines misleading opponents’ inference, deceptive planning under resource allocationchen2024deceptiveaddresses balancing deception and cost\-efficiency, and multi\-agent deception modeling with Theory of Mind \(ToM\)sarkadi2019modellingemphasizes simulating the observer’s mental state for effective deception\.

Beyond classical ambiguity\-based approaches, several extensions to deceptive path planning \(DPP\) have been proposed\. Some works develop new methodological frameworks, such as reinforcement learning with graph neural networks to train general DPP policiesfatemi\_deceptive\_2024, maximum\-entropy formulations for deception under stochastic uncertaintysavas\_deceptive\_2022, and mixed\-integer programming for magnitude\-based deception in single\-goal settingsxu\_single\_2020\. Others address practical challenges by incorporating terrain costs in complex environmentslenhard\_deceptive\_2023, designing domain\-independent strategies based on landmarks, centroids, and minimum coverage statesprice\_domain\_2023, and handling adversarial cost signal manipulations through deceptive reinforcement learninghuang2019deceptive\.

While these studies broaden the modeling and applicability of DPP, they remain confined to one\-shot settings and do not address repeated interactions with adaptive observers\. While these studies provide diverse methods for deceptive path planning, they all focus on single\-shot interactions\. None address the challenge of repeated deceptive path planning with learnable observers, which is the focus of this work\.

### 2\.2\.Goal Recognition

The goal recognition \(GR\) problem is directly relevant to deceptive planning because the observer in DPP is essentially a goal recognizer: GR methods specify how an observer maps observed behavior to a probability distribution over candidate goals\. Classical GR approaches are often cost\-basedmasters\_cost\-based\_2019;Ramrez2009PlanRA;ramirez2010probabilistic;sohrabi2016plan: by assuming near\-optimal behavior, cost\-divergence or Bayesian plan\-recognition techniques compute posterior probabilities over goals and thereby serve as natural observer models for some deceptive\-planning worksliu\_deceptive\_2021;masters\_deceptive\_2017\.

However, these classical methods rely on an optimality assumption and are typically formulated for single\-episode inference\. Extensions that relax optimality improve robustness to sub\-optimal behaviormasters\_goal\_2019\-1;zhi2020online, and some research incorporates learned or hybrid models to improve scalability and robustness: neuro\-symbolic and deep learning approaches handle noise and missing observations and offer fast inference for large candidate setsamado2018goal;amado\_robust\_2023;chiari\_goal\_2022;chiari\_fast\_2024\.

These methods focus on inference accuracy, efficiency, or robustness to observation noise, but they do not consider scenarios where the recognizer needs to adapt across repeated adversarial interactions\.

For these reasons, rather than relying on an off\-the\-shelf GR algorithm, we construct a learnable goal recognizer based on neural network as the observer in our experiments\. This recognizer follows the standard GR setting and is designed to update online, making it a suitable component for studying RDPP and validating methods\.

## 3\.Preliminaries

Markov decision process \(MDP\)puterman2014markov\. An MDP is defined by a tupleM=\(𝒮,𝒜,𝒫,r,γ\)M=\(\\mathcal\{S\},\\mathcal\{A\},\\mathcal\{P\},r,\\gamma\), where𝒮\\mathcal\{S\}is the state space,𝒜\\mathcal\{A\}is the action space,𝒫​\(st,at,st\+1\)\\mathcal\{P\}\(s\_\{t\},a\_\{t\},s\_\{t\+1\}\)is the transition function defining the probability of moving from statests\_\{t\}tost\+1s\_\{t\+1\}given actionata\_\{t\},r​\(st,at,st\+1\)r\(s\_\{t\},a\_\{t\},s\_\{t\+1\}\)is the reward received for executing actionata\_\{t\}in statests\_\{t\}and transitioning tost\+1s\_\{t\+1\}, andγ∈\(0,1\)\\gamma\\in\(0,1\)is the discount factor\. The objective is to learn a decision policyπ:𝒮→𝒜\\pi:\\mathcal\{S\}\\rightarrow\\mathcal\{A\}, which maximizes the value function:Vπ​\(s\)=𝔼​\[∑t∈Tγt​r​\(st,at,st\+1\)\],V\_\{\\pi\}\(s\)=\\mathbb\{E\}\\left\[\\sum\_\{t\\in T\}\\gamma^\{t\}r\(s\_\{t\},a\_\{t\},s\_\{t\+1\}\)\\right\],whereT=\{0,1,…,H−1\}T=\\\{0,1,\\dots,H\-1\\\}denotes the set of discrete time steps within an episode of finite horizonHH,

Deceptive MDPornik2018deception\. A deceptive MDP is defined as

MD​P​P=\(𝒮,𝒜,𝒫,ℛ,r,ℬ,ℒ,γ\),M\_\{DPP\}=\(\\mathcal\{S\},\\mathcal\{A\},\\mathcal\{P\},\\mathcal\{R\},r,\\mathcal\{B\},\\mathcal\{L\},\\gamma\),where𝒮,𝒜,𝒫,r\\mathcal\{S\},\\mathcal\{A\},\\mathcal\{P\},r, andγ\\gammaare the same as in a regular MDP,ℛ\\mathcal\{R\}is a set of candidate reward functions, including the true reward functionrrand at least one deceptive reward function,ℬ\\mathcal\{B\}is the observer’s belief\-set, andℒ​\(st,at,st\+1,bt\)\\mathcal\{L\}\(s\_\{t\},a\_\{t\},s\_\{t\+1\},b\_\{t\}\)is a belief\-induced reward function, defining the reward for executing actionata\_\{t\}in statests\_\{t\}and transitioning tost\+1s\_\{t\+1\}while incorporating the impact of the observer’s beliefbtb\_\{t\}about the agent’s true reward function\.

The objective of a deceptive MDP is to maximise the belief\-induced value function:

Vπ​\(s\)=𝔼​\[∑t∈Tγt​ℒ​\(st,at,st\+1,bt\)\]\.V\_\{\\pi\}\(s\)=\\mathbb\{E\}\\left\[\\sum\_\{t\\in T\}\\gamma^\{t\}\\mathcal\{L\}\(s\_\{t\},a\_\{t\},s\_\{t\+1\},b\_\{t\}\)\\right\]\.
Observer Model\. AMliu\_deceptive\_2021models observer beliefs via cost\-based plan recognition, which usesQQ\-differences:

Δri​\(σ\)=∑\(st,at\)∈σ\(Qri​\(st,at\)−maxat′∈A⁡Qri​\(st,at′\)\)\.\\Delta\_\{r\_\{i\}\}\(\\@vec\{\\sigma\}\)=\\sum\_\{\(s\_\{t\},a\_\{t\}\)\\in\\@vec\{\\sigma\}\}\\left\(Q\_\{r\_\{i\}\}\(s\_\{t\},a\_\{t\}\)\-\\max\_\{a^\{\\prime\}\_\{t\}\\in A\}Q\_\{r\_\{i\}\}\(s\_\{t\},a^\{\\prime\}\_\{t\}\)\\right\)\.If behaviour is optimal for a reward functionrir\_\{i\}, thenΔri​\(σ\)=0\\Delta\_\{r\_\{i\}\}\(\\@vec\{\\sigma\}\)=0\. Otherwise,Δri​\(σ\)<0\\Delta\_\{r\_\{i\}\}\(\\@vec\{\\sigma\}\)<0and decreases with more sub\-optimality\. AM estimates probabilities for the candidate reward functions via a Boltzmann distribution:

P​\(ri∣σ\)=exp⁡\(Δri​\(σ\)\)∑rj∈ℛexp⁡\(Δrj​\(σ\)\)⋅P​\(ri\),P\(r\_\{i\}\\mid\\@vec\{\\sigma\}\)=\\frac\{\\exp\(\\Delta\_\{r\_\{i\}\}\(\\@vec\{\\sigma\}\)\)\}\{\\sum\_\{r\_\{j\}\\in\\mathcal\{R\}\}\\exp\(\\Delta\_\{r\_\{j\}\}\(\\@vec\{\\sigma\}\)\)\}\\cdot P\(r\_\{i\}\),whereP​\(ri\)P\(r\_\{i\}\)is an estimate of the prior probability thatrir\_\{i\}is the real reward function\. A less optimal sequence leads to a lower probability\. This is a proxy for the observer beliefs\.

## 4\.Method

In this section, we first formalized the repeated deceptive path planning task\. To address this challenge of evolving opponents in repeated deceptive path planning, we propose a novel method DeMP, a two\-level optimization framework combining episode\-level adaptation for rapid policy adjustment after each episode with meta\-level updates that leverage cross\-episode feedback to accelerate adaptation in future episodes\.

### 4\.1\.Modeling Repeated Deceptive Path Planning

#### 4\.1\.1\.Problem Definition

We define a Repeated Deceptive Path Planning problem as the tuple

MR​D​P​P=\(MD​P​P,Φ,K\),M\_\{RDPP\}=\\big\(M\_\{DPP\},\\Phi,K\\big\),whereMD​P​PM\_\{DPP\}is a deceptive MDP,Φ\\Phiis the parameter space of the observer’s learnable model, andKKdenotes the number of repeated interactions\. RDPP extendsMD​P​PM\_\{DPP\}by introducing repeated interactions with a learnable observer\.

In RDPP the observer is not static: its predictive model is parameterized byϕ∈Φ\\phi\\in\\Phiand evolves across theKKepisodes according to a learning dynamic\. The agent operates within the DPP structure while facing an adaptive observer whose parametersϕ\\phichange over repeated interactions\.

#### 4\.1\.2\.Interaction Protocol

Each interaction episodek∈\{1,…,K\}k\\in\\\{1,\\dots,K\\\}proceeds as follows\. Within episodekk, the agent interacts with the environment defined byMD​P​PM\_\{DPP\}according to its deceptive policy, generating a complete trajectory

τ\(k\)=\{\(st,at\)\}t=0Tk,\\tau^\{\(k\)\}=\\\{\(s\_\{t\},a\_\{t\}\)\\\}\_\{t=0\}^\{T\_\{k\}\},whereTkT\_\{k\}denotes the length of episodekk\. The observer only receives a partial trajectory prefix

ζ\(k\)=τ0:⌊α​Tk⌋\(k\),α∈\(0,1\],\\zeta^\{\(k\)\}=\\tau^\{\(k\)\}\_\{0:\\lfloor\\alpha T\_\{k\}\\rfloor\},\\quad\\alpha\\in\(0,1\],and produces a predictive distribution over candidate goals

𝒪​\(ζ\(k\);ϕ\(k\)\)=P​\(G∣ζ\(k\);ϕ\(k\)\),G⊆S,\\mathcal\{O\}\(\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)=P\(G\\mid\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\),\\quad G\\subseteq S,from which a prediction is obtained

G^\(k\)=arg⁡maxG⁡P​\(G∣ζ\(k\);ϕ\(k\)\)\.\\hat\{G\}^\{\(k\)\}=\\arg\\max\_\{G\}P\(G\\mid\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)\.
After the episode, the agent is provided with𝒪​\(ζ\(k\);ϕ\(k\)\)\\mathcal\{O\}\(\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)as additional feedback, while the observer receives the true goalG∗G^\{\*\}and the complete trajectoryτ\(k\)\\tau^\{\(k\)\}for parameter adaptation\. The observer then updates its parameters according to the update operator𝒰\\mathcal\{U\}:

ϕ\(k\+1\)←𝒰​\(ϕ\(k\),τ\(k\),G∗\)\.\\phi^\{\(k\+1\)\}\\leftarrow\\mathcal\{U\}\(\\phi^\{\(k\)\},\\tau^\{\(k\)\},G^\{\*\}\)\.
The interaction protocol induces a two\-sided adaptation process: the observer updates its predictive model between episodes based on post\-episode feedback, while the agent needs to adapt its deceptive trajectories in response to counter the evolving observer\.

#### 4\.1\.3\.Observer’s Learning Dynamics

We model the observer as a learnable recognition model parameterized byϕ∈Φ\\phi\\in\\Phi\. Its role is to estimate the probability of the agent’s true goalG∗G^\{\*\}given the observed partial trajectory\. Prior to the repeated deceptive interaction, the observer is initialized through an offline training phase on non\-deceptive trajectories, providing a reasonable goal\-recognition prior\.

During repeated interactions, the observer adaptsϕ\\phito improve predictive performance\. Formally, the adaptation in episodekkis expressed as minimizing the negative log\-likelihood loss:

ℒ𝒪​\(ϕ\)=−log⁡P​\(G∗∣τ\(k\);ϕ\)\.\\mathcal\{L\}\_\{\\mathcal\{O\}\}\(\\phi\)=\-\\log P\(G^\{\*\}\\mid\\tau^\{\(k\)\};\\phi\)\.
In practice,the update operator𝒰\\mathcal\{U\}may be realized through maximum\-likelihood estimation, Bayesian updates, or gradient\-based optimization \(e\.g\.,ϕ\(k\+1\)=ϕ\(k\)−η​∇ϕℒ𝒪​\(ϕ\(k\)\)\\phi^\{\(k\+1\)\}=\\phi^\{\(k\)\}\-\\eta\\nabla\_\{\\phi\}\\mathcal\{L\}\_\{\\mathcal\{O\}\}\(\\phi^\{\(k\)\}\), whereη\\etais the learning rate\)\. This abstraction captures a broad class of adaptive observers while introducing controlled, modelable non\-stationarity into the planning problem\.

#### 4\.1\.4\.Planning Agent’s Objective

The agent’s objective in RDPP extends the standard DPP formulation to repeated interactions\. Formally, givenKKepisodes, the cumulative value of a policyπ\\piis

Vπ=𝔼​\[∑k=1K∑t=0Tkγt​ℒ​\(st\(k\),at\(k\),st\+1\(k\),bt\(k\)\)\],V\_\{\\pi\}=\\mathbb\{E\}\\Biggl\[\\sum\_\{k=1\}^\{K\}\\sum\_\{t=0\}^\{T\_\{k\}\}\\gamma^\{t\}\\,\\mathcal\{L\}\(s\_\{t\}^\{\(k\)\},a\_\{t\}^\{\(k\)\},s\_\{t\+1\}^\{\(k\)\},b\_\{t\}^\{\(k\)\}\)\\Biggr\],whereℒ\\mathcal\{L\}denotes the belief\-induced reward depending on the observer’s latent beliefbt\(k\)b\_\{t\}^\{\(k\)\}\.

In practice, the agent cannot access the observer’s belief during execution\. Instead, it only receives the observer’s predictive distribution𝒪​\(ζ\(k\);ϕ\(k\)\)\\mathcal\{O\}\(\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)after each episode, which serves as the sole feedback for adapting its policy across episodes\. Consequently, solving RDPP requires exploiting cross\-episode feedback to improve long\-term deception performance rather than optimizing deception within individual episodes in isolation\.

#### 4\.1\.5\.Challenges of RDPP

The core challenge of RDPP lies in the non\-stationarity induced by the continual evolution of the observer model\. Concretely, the observer’s learning dynamics𝒰\\mathcal\{U\}iteratively update its predictive modelϕ\(k\)\\phi^\{\(k\)\}across episodes, which progressively increases the difficulty of deception for the agent\. This non\-stationarity affects the agent’s decision\-making process through the belief\-induced rewardℒ\\mathcal\{L\}, which couples the agent’s actions with the observer’s beliefsbt\(k\)b\_\{t\}^\{\(k\)\}in order to suppress the probability that the observer assigns to the true goalG∗G^\{\*\}\. However, in practiceℒ\\mathcal\{L\}is unobservable and typically requires manual design to align with adversarial objectives\.

A naive approach to this problem is to incorporate the observer’s predictions into the reward signal and update the agent’s policy episode by episode\. While this enables some degree of adaptation, it is fundamentally limited: since the agent only adapts after the observer has already improved, it inherently lags behind the observer’s updates\. This incremental adaptation lag accumulates over multiple interactions, ultimately compromising the agent’s ability to sustain deception in repeated encounters\.

![Refer to caption](https://arxiv.org/html/2605.07174v1/x4.png)\(a\)Deceptiveness across 400 episodes
![Refer to caption](https://arxiv.org/html/2605.07174v1/x5.png)\(b\)Deceptiveness in last episode
![Refer to caption](https://arxiv.org/html/2605.07174v1/x6.png)\(c\)Path costs
![Refer to caption](https://arxiv.org/html/2605.07174v1/x7.png)\(d\)Steps\-after\-LDP
![Refer to caption](https://arxiv.org/html/2605.07174v1/x8.png)\(e\)Deceptiveness across 400 episodes
![Refer to caption](https://arxiv.org/html/2605.07174v1/x9.png)\(f\)Deceptiveness in last episode
![Refer to caption](https://arxiv.org/html/2605.07174v1/x10.png)\(g\)Path costs
![Refer to caption](https://arxiv.org/html/2605.07174v1/x11.png)\(h\)Steps\-after\-LDP

Figure 4\.Deception performance and trajectory cost in repeated interactions\. The first row corresponds to the49×4949\\times 49grid environment, and the second row corresponds to the100×100100\\times 100grid environment\. \(a,e\) Deceptiveness across 400 episodes\. DeMP consistently maintains the lowest probability assigned to the true goal\. \(b,f\) Deceptiveness in the last episode\. After repeated interactions, baselines lose their deceptive effect early due to observer learning, while DeMP preserves low goal probability over most of the path\. \(c,g\) Path costs normalized by the optimal path length\. DeMP incurs slightly higher cost due to continual exploration\. \(d,h\) Steps after the last deceptive point \(LDP\)\. DeMP keeps the observer uncertain for a longer portion of the trajectory\.

### 4\.2\.Deceptive Meta Planning

To overcome these challenges, we proposeDeceptive Meta Planning \(DeMP\), a two\-level optimization framework\. As shown in Figure[3](https://arxiv.org/html/2605.07174#S1.F3), the framework consists of anepisode\-leveladaptation, which adjusts the agent’s policy after each episode based on the observer’s updated predictions, and ameta\-levelupdate that leverages cross\-episode feedback to refine the agent’s policy initialization for future episodes\. Concretely, DeMP operates in a nested structure: between two consecutive meta\-level updates, the agent engages in a sequence ofMMepisodes of interaction, within which an episode\-level adaptation is applied after each episode\. The aggregated feedback from these episodes then drives the subsequent meta\-level update\. We next describe these two levels in detail\.

#### 4\.2\.1\.Episode\-Level Adaptation

Episode\-level adaptation in DeMP corresponds to optimizing the parameters of the planning policy based on feedback from the observer\. This adaptation occurs after each episode of interaction, where the agent refines its policy to better maintain deception against the observer’s updated model\.

Concretely, within a sequence ofMMconsecutive episodes, the agent begins with initial parameters denoted asθ\(0\)\\theta^\{\(0\)\}and performs an episode\-level adaptation after each episode\.

At the very beginning of episode,θ\(0\)\\theta^\{\(0\)\}is obtained from a pretrained deceptive planning agent \(e\.g\., a reproduction of an end\-to\-end AMliu\_deceptive\_2021model or another DPP agent trained on the standard deceptive MDP\)\. This initialization provides a reasonable starting point for online adaptation and in subsequent meta\-updates,θ\(0\)\\theta^\{\(0\)\}is updated to incorporate experience accumulated from previous sequences of episodes\.

During thekk\-th episode \(k=0,…,M−1k=0,\\dots,M\-1\), the agent executes its policyπθ\(k\)\\pi\_\{\\theta^\{\(k\)\}\}in the environment, producing a trajectory

τ\(k\)=\(s0,a0,…,sTk\)\.\\tau^\{\(k\)\}=\(s\_\{0\},a\_\{0\},\\dots,s\_\{T\_\{k\}\}\)\.At the end of the episode, the agent receives the observer’s predictive distribution over candidate goals𝒪​\(ζ\(k\);ϕ\(k\)\)\\mathcal\{O\}\(\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)as feedback for policy adaptation\.

The episode\-level objective is defined as a reinforcement learning loss

ℒRL\(k\)​\(θ\)=−𝔼τ\(k\)∼πθ​\[R\(k\)​\(τ\(k\)\)\],\\mathcal\{L\}\_\{\\text\{RL\}\}^\{\(k\)\}\(\\theta\)=\-\\mathbb\{E\}\_\{\\tau^\{\(k\)\}\\sim\\pi\_\{\\theta\}\}\\big\[R^\{\(k\)\}\(\\tau^\{\(k\)\}\)\\big\],where the total return

R\(k\)​\(τ\(k\)\)=∑t=0Tk−1renv​\(st,at\)\+rdeceptive\(k\)​\(G∗\)R^\{\(k\)\}\(\\tau^\{\(k\)\}\)=\\sum\_\{t=0\}^\{T\_\{k\}\-1\}r\_\{\\text\{env\}\}\(s\_\{t\},a\_\{t\}\)\+r\_\{\\text\{deceptive\}\}^\{\(k\)\}\(G^\{\*\}\)combines the environment reward with a terminal deceptive reward

rdeceptive\(k\)​\(G∗\)=\(1−P​\(G∗∣ζ\(k\);ϕ\(k\)\)\)​r​\(G∗\)\.r\_\{\\text\{deceptive\}\}^\{\(k\)\}\(G^\{\*\}\)=\\big\(1\-P\(G^\{\*\}\\mid\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)\\big\)\\,r\(G^\{\*\}\)\.This surrogate formulation penalizes the agent when the observer assigns high confidence to the true goal\.

To further encourage deception, we add an observer\-guided regularization term that minimizes the divergence between the observer’s posterior and a target uniform confusion distributionUnif​\(G\)\\mathrm\{Unif\}\(G\):

ℒ\(k\)​\(θ\)=ℒRL\(k\)​\(θ\)\+λ​KL​\(𝒪​\(ζ\(k\);ϕ\(k\)\)∥Unif​\(G\)\)\.\\mathcal\{L\}^\{\(k\)\}\(\\theta\)=\\mathcal\{L\}\_\{\\text\{RL\}\}^\{\(k\)\}\(\\theta\)\+\\lambda\\,\\mathrm\{KL\}\\\!\\left\(\\mathcal\{O\}\(\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)\\,\\big\\\|\\,\\mathrm\{Unif\}\(G\)\\right\)\.The KL regularization provides a tractable surrogate for maximizing belief entropy in standard belief\-induced RDPP objectives\.

After each episode, the agent updates its parameters via gradient descent:

θ\(k\+1\)=θ\(k\)−α​∇θ\(k\)ℒ\(k\)​\(θ\(k\)\),\\theta^\{\(k\+1\)\}=\\theta^\{\(k\)\}\-\\alpha\\nabla\_\{\\theta^\{\(k\)\}\}\\mathcal\{L\}^\{\(k\)\}\(\\theta^\{\(k\)\}\),\(1\)whereα\\alphais the episode\-level learning rate\.

AfterMMconsecutive episodes, the agent obtains the adapted parameterθ\(M\)\\theta^\{\(M\)\}by adapting its policy after each episode of interaction to counter the observer’s updated recognition model\.

#### 4\.2\.2\.Meta\-Level Update

AfterMMepisode\-level updates, we define the meta\-objective based on the final adapted parameter:

ℒ\(meta\)​\(θ\(0\)\)=ℒ\(M\)​\(θ\(M\)​\(θ\(0\)\)\),\\mathcal\{L\}^\{\(\\text\{meta\}\)\}\(\\theta^\{\(0\)\}\)=\\mathcal\{L\}^\{\(M\)\}\\big\(\\theta^\{\(M\)\}\(\\theta^\{\(0\)\}\)\\big\),\(2\)whereθ\(M\)​\(θ\(0\)\)\\theta^\{\(M\)\}\(\\theta^\{\(0\)\}\)denotes the parameter obtained after performingMMconsecutive episode\-level updates starting from initializationθ\(0\)\\theta^\{\(0\)\}\. In other words,θ\(M\)​\(θ\(0\)\)\\theta^\{\(M\)\}\(\\theta^\{\(0\)\}\)is treated as a function mapping the initial parameter to the result ofMMgradient steps\. The gradient of the meta\-objective with respect to the initializationθ\(0\)\\theta^\{\(0\)\}can be computed via the chain rule:

∇θ\(0\)ℒ\(meta\)=∇θ\(M\)ℒ\(M\)⋅∂θ\(M\)∂θ\(0\)\.\\nabla\_\{\\theta^\{\(0\)\}\}\\mathcal\{L\}^\{\(\\text\{meta\}\)\}=\\nabla\_\{\\theta^\{\(M\)\}\}\\mathcal\{L\}^\{\(M\)\}\\cdot\\frac\{\\partial\\theta^\{\(M\)\}\}\{\\partial\\theta^\{\(0\)\}\}\.\(3\)
Finally, the meta\-update is performed as:

θ\(0\)←θ\(0\)−β​∇θ\(0\)ℒ\(meta\)​\(θ\(0\)\),\\theta^\{\(0\)\}\\leftarrow\\theta^\{\(0\)\}\-\\beta\\,\\nabla\_\{\\theta^\{\(0\)\}\}\\mathcal\{L\}^\{\(\\text\{meta\}\)\}\(\\theta^\{\(0\)\}\),\(4\)whereβ\\betais the meta\-level learning rate\. In the meta\-level update, DeMP integrates long\-term feedback across multiple episodes using higher\-order gradient information\. This allows DeMP to effectively capture patterns in how the observer updates its predictions over time and to refine the initialization such that future adaptation \(in the nextMM\-episode block\) can start from a more effective parameter point\.

This meta\-level mechanism is structurally related to meta\-learning approaches such as MAMLfinn2017model, in that both optimize an initialization for rapid adaptation\. Unlike MAML, however, where adaptation occurs across a distribution of independent tasks, DeMP adapts within a single deceptive planning task against an evolving observer, making the source of variation endogenous to repeated interactions\.

The overall optimization alternates between episode\-level adaptation and meta\-level updates, as summarized in Algorithm[1](https://arxiv.org/html/2605.07174#alg1)\. By refining the policy initialization based on cross\-episode feedback, DeMP enables faster adaptation in subsequent interaction blocks and mitigates the accumulation of adaptation lag in repeated deceptive planning\.

Algorithm 1Deceptive Meta Planning \(DeMP\)1:Initialize agent parameters

θ\(0\)\\theta^\{\(0\)\}from pretrained DPP agent

2:whileonline interaction continuesdo

3:for

k=0k=0to

M−1M\-1do

4:Start a new episode

5:Execute policy

πθ\(k\)\\pi\_\{\\theta^\{\(k\)\}\}to collect trajectory

τ\(k\)\\tau^\{\(k\)\}
6:Obtain observer’s prediction

𝒪​\(ζ\(k\);ϕ\(k\)\)\\mathcal\{O\}\(\\zeta^\{\(k\)\};\\phi^\{\(k\)\}\)
7:Compute episode loss

ℒ\(k\)​\(θ\(k\)\)\\mathcal\{L\}^\{\(k\)\}\(\\theta^\{\(k\)\}\)
8:Update policy parameters

θ\(k\+1\)\\theta^\{\(k\+1\)\}according to episode\-level adaption

9:endfor

10:Compute meta loss

ℒ\(meta\)​\(θ\(0\)\)\\mathcal\{L\}^\{\(\\text\{meta\}\)\}\(\\theta^\{\(0\)\}\)
11:Update initialization

θ\(0\)\\theta^\{\(0\)\}according to meta\-level update

12:endwhile

### 4\.3\.Theoretical Analysis

A central challenge in RDPP is that the agent must optimize against an observer whose belief state is latent and whose parameters evolve over time\. We provide a theoretical justification for the validity of DeMP, with formal proofs deferred toAppendix A\.

##### Validity of the Surrogate Objective\.

RDPP theoretically requires maximizing a belief\-induced reward defined on the observer’s latent posteriorb​\(G∗∣τ\)b\(G^\{\*\}\\mid\\tau\)\. In contrast, DeMP optimizes a surrogate objective constructed from the observer’s explicit predictionP​\(G∗∣τ;ϕ\)P\(G^\{\*\}\\mid\\tau;\\phi\)together with an entropy regularization term\. This substitution is justified under the assumption of a*rational observer*\(see Assumption 1 in Appendix A\.1\), which requires that the observer’s predictive model preserves the uncertainty ordering of its internal belief\.

\{theorem\}

\[Surrogate Consistency, Informal\] Under rational observer assumptions, minimizing the DeMP surrogate loss is optimization\-consistent with maximizing the RDPP belief\-induced reward\.

As shown in Theorem 1 \(Appendix A\.2\), suppressing the predicted probabilityP​\(G∗\)P\(G^\{\*\}\)serves as a monotonic proxy for reducing the true belief assigned to the goal, while minimizing the KL divergence to a uniform distribution provides a tractable surrogate for maximizing belief entropy\. Together, these terms align the DeMP optimization landscape with the fundamental RDPP objective\.

##### Mitigation of Adaptation Lag\.

In RDPP, purely reactive optimization suffers from*adaptation lag*, as the policy is optimized for the current observerϕ\(k\)\\phi^\{\(k\)\}but evaluated after the observer updates toϕ\(k\+1\)\\phi^\{\(k\+1\)\}\. DeMP addresses this issue through meta\-level optimization\.

\{theorem\}

\[Anticipation Mechanism, Informal\] The meta\-gradient update in DeMP incorporates a second\-order correction that reduces sensitivity to the observer’s future learning dynamics\.

As formalized in Theorem 2 \(Appendix A\.3\), the meta\-gradient with respect to the policy initialization includes a term that explicitly depends on the observer’s learning gradient\. This term captures the anticipation of the observer’s update, enabling DeMP to learn policy initializations that are robust to belief evolution and thereby mitigating adaptation lag compared to purely reactive baselines\.

## 5\.Experiment

We evaluate the proposed Deceptive Meta Planning \(DeMP\) framework in repeated deceptive path planning \(RDPP\) tasks, aiming to quantify both its sustained deception performance and trajectory efficiency compared to existing methods\.

##### DeMP Implementation\.

DeMP is implemented using the Soft Actor\-Critic \(SAC\) algorithmhaarnoja2018soft\. Both the policy and value networks are two\-layer multilayer perceptrons \(MLPs\)\. The agent first performs imitation learning on trajectories generated by AMliu\_deceptive\_2021to initialize a deceptive planning policy\. During online interaction, DeMP performs two\-level optimization: an episode\-level update after each interaction and a meta\-level update everyM=2M=2episodes\. The learning rates are set toα=0\.001\\alpha=0\.001for episode\-level adaptation andβ=0\.0001\\beta=0\.0001for meta\-level updates\. In each episode, the observer only receives a randomly proportion sampled from\[40%,60%\]\[40\\%,60\\%\]prefix of the agent’s trajectory before producing its goal recognition\. Each evaluation sequence includes 400 episodes of online interaction, and all experiments are independently repeated 100 times with different random seeds to ensure the robustness of the results\.

##### Baselines\.

We compare DeMP with the following methods: \(1\) anHonest Agentthat plans directly toward the true goal; \(2\) theAMliu\_deceptive\_2021method; \(3\) theDEAMlewis\_deceptive\_2023method; and \(4\) aNaïve Approach, which adopts the same SAC framework and initialization as DeMP but directly incorporates the observer’s predicted goal probability into the reward and performs only per\-episode policy updates\.

##### Experimental Settings\.

We evaluate DeMP under two standard deceptive planning scenarios adapted to the RDPP setting\. In theStandard Path Deception Setting, experiments are conducted on49×4949\\times 49and100×100100\\times 100grid maps with five candidate goals and large obstacle structures, which emphasize long\-horizon planning and scalability under repeated interactions\. In thePirate Deception Settingnichols2022adversarial, experiments are conducted on49×4949\\times 49grids with five candidate goals and high\-density random obstacles, where an adversarial pirate actively pursues the agent based on the observer’s predictions\. Unless otherwise stated, reward structures follow standard grid\-based navigation conventions\.

##### Observer Model and Metrics\.

The observer is a two\-layer LSTM model that outputs a probability distribution over candidate goals given the observed partial trajectory\. Before evaluation, it is pre\-trained on non\-deceptive trajectories collected from the same maps to develop goal\-recognition ability, and then fine\-tuned online via supervised learning during interaction\. Additional details and empirical analysis of the observer pretraining process are provided in Appendix B\.2\.

We evaluate models using \(1\) the observer’s predicted probability of the true goal \(deceptiveness\), \(2\) the trajectory length ratio relative to the optimal path \(path cost\), and \(3\) steps after last deceptive point \(LDP\)\.

![Refer to caption](https://arxiv.org/html/2605.07174v1/x12.png)
![Refer to caption](https://arxiv.org/html/2605.07174v1/x13.png)

Figure 5\.t\-SNE Projection of Path Features\. The figure compares the distribution of generated path features over 400 episodes\. The Naive Agent \(Left\) shows a highly concentrated distribution, indicating a rigid strategy\. In contrast, DeMP \(Right\) exhibits significant path variability, with trajectories dispersing into new clusters, demonstrating continuous policy adaptation over extended interactions\.
### 5\.1\.Results of Standard Path Deception Setting

We first evaluate all methods under the Standard Path Deception Setting with five candidate goals and large obstacles\. Figure[4](https://arxiv.org/html/2605.07174#S4.F4)a and[4](https://arxiv.org/html/2605.07174#S4.F4)e reports the observer’s predicted probability of the true goal over 400 repeated interactions\. DeMP consistently maintains the lowest true\-goal probability throughout the entire interaction horizon, while all baseline methods exhibit a steady degradation in deceptive performance as the observer adapts\.

Initially, AM achieves relatively strong deception because the observer is pre\-trained on optimal trajectories and has limited exposure to deceptive behaviors\. However, AM relies on a largely fixed path structure, which quickly becomes exploitable as the observer accumulates experience across interactions\. Both the Naïve Agent and DeMP start from the same honest initialization, but differ fundamentally in their adaptation mechanisms: the Naïve Agent updates its policy only at the episode level, whereas DeMP additionally leverages meta\-level updates across episodes\. This meta\-level adaptation allows DeMP to anticipate how the observer evolves and to maintain deceptive effectiveness over long\-term interactions\.

Figure[4](https://arxiv.org/html/2605.07174#S4.F4)b and[4](https://arxiv.org/html/2605.07174#S4.F4)f further illustrates the observer’s belief evolution along the final episode trajectories\. For baseline methods, the observer rapidly identifies the true goal early in the trajectory, reflecting the formation of stable associations between repeated trajectory prefixes and goals\. In contrast, DeMP suppresses early goal recognition by continually altering its deceptive strategies, preventing the observer from converging to a reliable inference pattern\. Figure[4](https://arxiv.org/html/2605.07174#S4.F4)c and[4](https://arxiv.org/html/2605.07174#S4.F4)g shows that this sustained deception comes at the cost of slightly longer paths, representing a natural trade\-off between efficiency and long\-term robustness\. Moreover, DeMP maintains deception over a larger portion of the trajectory, as reflected by the steps\-after\-LDP metric in Figure[4](https://arxiv.org/html/2605.07174#S4.F4)d and[4](https://arxiv.org/html/2605.07174#S4.F4)h\.

Figure[2](https://arxiv.org/html/2605.07174#S1.F2)visualizes the trajectory evolution of DeMP\. The leftmost panel shows the static path distribution of the AM method, while the remaining heatmaps illustrate DeMP’s trajectory distributions at different interaction stages\. The bright regions indicate frequently visited states, and the dashed line denotes the final trajectory\. Over time, DeMP’s paths evolve from near\-optimal routes to increasingly diverse patterns, demonstrating dynamic path adaptation and repeated deception against the learnable observer\.

Finally, Figure[5](https://arxiv.org/html/2605.07174#S5.F5)presents t\-SNE projections of the path features\. Each point represents a trajectory encoded by uniformly sampled 2D waypoints and projected into a two\-dimensional space using t\-SNE, where distances reflect trajectory similarity\. The Naïve Agent’s trajectories cluster tightly, indicating a static or rigid strategy with limited adaptation\. In contrast, DeMP’s trajectories form distinct clusters that expand and separate as interactions progress, reflecting the generation of new and diverse trajectories, consistent with the expected behavioral diversity induced by meta\-level adaptation\.

### 5\.2\.Results of Pirate Deception Experiment

To further evaluate the robustness of deceptive strategies under active pursuit, we extend our analysis to the Pirate Deception Scenarionichols2022adversarial\. In this scenario, while the agent moves toward its goal, pirates are also active in the environment, attempting to infer the agent’s true target and intercept it\. The pirates rely on the candidate goal with the highest predicted probability from the observer as their pursuit target and move toward it following a rule\-based shortest path\. The pirates share the same discrete action space as the agent, with both parties allowed one move per step, and their initial positions are randomly assigned at the beginning of each episode\.

Agents and observers after 50, 200, and 400 interactions from the previous experiment were used in this evaluation\. For each configuration, we conducted 100 random trials under fixed parameters for both sides\. Figure[6](https://arxiv.org/html/2605.07174#S5.F6)\(a\) reports the agent capture probability as interactions increase\. All methods exhibit rising capture rates as the observer becomes more accurate over time\. However, the degradation is substantially slower for DeMP, which consistently achieves the lowest capture probability across all interaction stages\. In contrast, AM and the Naïve Agent suffer rapid performance deterioration, indicating that their deceptive strategies become increasingly exploitable under repeated adversarial observation\. These results demonstrate that meta\-level adaptation enables DeMP to maintain robust deception even under active pursuit\.

![Refer to caption](https://arxiv.org/html/2605.07174v1/x14.png)
![Refer to caption](https://arxiv.org/html/2605.07174v1/x15.png)

Figure 6\.Analysis of DeMP under repeated deceptive planning\. \(a\) DeMP consistently achieves the lowest capture rate in the Pirate Deception Scenario as interactions increase\. \(b\) Sensitivity analysis of the meta\-episode parameterMM, illustrating the trade\-off between adaptation frequency and long\-term deception performance\.
### 5\.3\.Analysis of Meta Episode ParameterMM

The meta episode parameterMMcontrols the number of consecutive DPP episodes considered before a meta\-level update is applied\. A largerMMallows the model to incorporate more historical interaction information, potentially enhancing adaptability to the evolving observer\. However, increasingMMalso reduces the frequency of meta\-level updates, which may introduce a delay in adjusting the episode\-level policy initialization\.

To investigate the impact ofMMon performance, we evaluated DeMP under different values ofMM\. As shown in Figure[6](https://arxiv.org/html/2605.07174#S5.F6)\(b\), whenM=1M=1, the meta\-level update degenerates to a per\-episode update, causing the model’s performance to approximate that of the Naïve Agent\. In this case, deception performance decreases over repeated interactions since only single\-episode information is considered\.

For larger values ofMM, the model consistently maintains or improves deception performance\. In particular, intermediate values ofMMachieve a balance between incorporating sufficient historical information and maintaining timely adaptation, resulting in relatively stable deception across interactions\. However, excessively largeMMvalues lead to less frequent meta\-level updates, causing delayed adaptation and slight deterioration in performance\. Additionally, largerMMincreases computational cost due to the need to store and backpropagate through longer episode sequences\.

## 6\.Conclusion

In this work, we introduced the task of repeated deceptive path planning \(RDPP\) and proposed Deceptive Meta Planning \(DeMP\), a novel two\-level optimization framework\. DeMP performs episode\-level policy adaptation guided by an observer’s feedback and meta\-level updates to refine the initial policy, enabling sustained deception across repeated interactions\. Theoretical analysis confirms that DeMP aligns with belief\-induced RDPP objectives and proactively adapts to observer dynamics\. Extensive experiments across grid and continuous domains demonstrate that DeMP significantly outperforms baselines, sustaining deception against evolving observers\.

Despite these promising results, our study has certain limitations\. The experiments do not capture the full complexity and dynamics of real\-world planning problems\. The effectiveness of DeMP in more complex or high\-dimensional domains remains to be explored\.

For future work, we aim to extend DeMP to more realistic and dynamic planning scenarios, consider multiple interacting agents, and investigate the integration of richer observer models\. These directions could further enhance the adaptability and robustness of deceptive planning in practical applications\.

\{acks\}

This work was supported by the National Science and Technology Major Project under Grant No\. 2022ZD0116403, and in part by the Beijing Natural Science Foundation under Grant No\. 4264131\.

## References