Breaking the Gradient Barrier: Unveiling Large Language Models for Strategic Classification

We sincerely appreciate your constructive comments and your positive evaluation of our work. We would like to address your concerns as follows:

Weakness1.The theoretical results only apply when the SC model is linear, and the ICL update is done through one SA layer;

Response:

• Our focus on linear SC models is motivated by two main reasons:

  • Linear SC models remain the dominant approach in both practical applications and academic research on strategic machine learning[1,2,3].
  • Linear SC is often preferred for its ease of deployment, interpretability, and direct comparability with established benchmarks[4,5], particularly in real-world systems such as policy-making, risk assessment, credit scoring, and admissions or hiring.

• Regarding the use of a single self-attention (SA) layer for ICL updates, we would like to note that:

  • Using a single SA layer allows for a clear analysis of the relationship between ICL and gradient descent, as it enables explicit mapping between attention operations and gradient steps.
  • Extending from a single to multiple self-attention layers corresponds to performing multiple steps of gradient descent, which does not affect the validity of the theoretical analysis.
  • This simplification is widely adopted in recent ICL theory[6,7,8].

• Moreover, practical nonlinear SC models, such as MLPs and GNN[9,10,11], are employed with forward propagation and gradient-based optimization. This means that our theoretical framework is naturally applicable to these models as well.

• We have clarified this part in Lines 128-130 of our revised paper.

---

Weakness2. The results are not too surprising

Response:

• We would like to clarify that our contributions go beyond this connection:

  • Strategic machine learning is fundamentally a game-theoretic problem involving both agent-side manipulation and decision-maker adaptation, and has been studied largely separately from ICL/LLM literature.

  • To our best knowledge, our work is the first to employ LLMs to model and solve this bi-level, game-theoretic optimization structure.

  • It enables a scalable and retraining-free approach to large-scale gaming tasks, where classical gradient-based retraining requires excessive computational resources and time.

• From a broader social science perspective, our work establishes a crucial bridge between large language models and strategic machine learning:

  • A wide range of social science applications are increasingly converging with machine learning and LLMs, including fields such as computational social science, policy evaluation, and behavioral economics[1,2,3,4].
  • Our work provides a foundational contribution by connecting ICL in LLMs with game-theoretic learning frameworks.
  • We hope this foundation will facilitate future interdisciplinary research and applications of LLMs in societal decision-making contexts.

• We have clarified this part in our revised paper at Lines 62-64.

---

Question1. If multiple SA layers with non-linear activations are considered, can ICL capture more complex strategic behaviors?

Response:

• Indeed, we agree that with multiple self-attention layers and nonlinear activations, ICL is capable of capturing more complex strategic behaviors beyond the linear case.

• Theoretical perspective:

  • Multiple self-attention layers with nonlinear activations greatly enhance the model’s representational capacity[12].

  • In SC tasks, even complex strategic behaviors, e.g., “agent in the dark[4]” settings (where agents have limited information) or "interacting agent behaviors[10]" (where agents strategically respond to each other), can be modeled as forward passes and gradient-driven updates in a neural network.

  • Therefore, multiple SA layers with non-linear activations are considered capable of representing and adapting to these complex behaviors.

• Experimental verification:

  • We designed an additional set of experiments simulating more complex SC environments with complex strategic behaviors, e.g.,“agent in the dark[4]” and "interacting agent behaviors[10]".

  • We compared the agent feature distributions of the standard method and that simulated by the model based on self-attention layers with nonlinear activations.

  • We use the Jensen-Shannon (JS) divergence[13] as a similarity metric. We also evaluated the post-strategy classification accuracy.

  • The experimental results are presented in the following table.

  Behavior Type	JS Divergence	Accuracy
  Agent in the dark	0.183	81.7%
  Interacting agent behaviors	0.226	82.3%

---

Question2. The experiments only consider the non-linear SA layers. Is it possible to consider some non-linear SC models (e.g., the agents do not best respond and the policy is non-linear)?

Response:

• We agree that nonlinear SC models should be considered.

• Therefore, we extended our experiments along two dimensions:

  • Nonlinear policy: We first implemented the policy function using a nonlinear model (MLP).

  • Non-best-response agents: We further considered agents that do not always best respond. Following prospect theory[14], we model the agents irrational, incorporating asymmetric value function and nonlinear probability weighting. To specific:

    $$x^* = \arg\max_x \left( \sum_y w(P(y|x')) \cdot v(u(y) - a_{\text{ref}}) - \phi^c(x, x') \right)$$

    where:

    • $v(\cdot)$: value function (captures loss aversion),
    • $w(\cdot)$: probability weighting function,
    • $a_{\text{ref}}$: reference point,
    • $\phi^c(x, x')$: cost function for feature manipulation.

• We use Jensen-Shannon (JS) divergence (between attention and MLP) as a similarity metric and accuracy of ICL as evaluation metric.

  • The experimental results are presented in the following table.

---

[1] Hardt, Moritz, et al. "Strategic classification." Proceedings of the 2016 ACM conference on innovations in theoretical computer science. 2016.

[2] Milli, Smitha, et al. "The social cost of strategic classification." Proceedings of the conference on fairness, accountability, and transparency. 2019.

[3] Miller, John, Smitha Milli, and Moritz Hardt. "Strategic classification is causal modeling in disguise." International Conference on Machine Learning. PMLR, 2020.

[4] Levanon, Sagi, and Nir Rosenfeld. "Strategic classification made practical." International Conference on Machine Learning. PMLR, 2021.

[5] Ghalme, Ganesh, et al. "Strategic classification in the dark." International Conference on Machine Learning. PMLR, 2021.

[6] Von Oswald, Johannes, et al. "Transformers learn in-context by gradient descent." International Conference on Machine Learning. PMLR, 2023.

[7] Garg, Shivam, et al. "What can transformers learn in-context? a case study of simple function classes." Advances in neural information processing systems 35 (2022): 30583-30598.

[8] Akyürek, Ekin, et al. "What learning algorithm is in-context learning? investigations with linear models." arXiv preprint arXiv:2211.15661 (2022).

[9] Stratis Tsirtsis, Behzad Tabibian, Moein Khajehnejad, Adish Singla, Bernhard Schölkopf, and Manuel Gomez-Rodriguez. Optimal decision making under strategic behavior. Management Science, 2024

[10] Itay Eilat, Ben Finkelshtein, Chaim Baskin, and Nir Rosenfeld. Strategic classification with graph neural networks. arXiv preprint arXiv:2205.15765, 2022

[11] Mehrnaz Mofakhami, Ioannis Mitliagkas, and Gauthier Gidel. Performative prediction with neural networks. In International Conference on Artificial Intelligence and Statistics, pages 11079–11093. PMLR, 2023

[12] Xiang Cheng, Yuxin Chen, and Suvrit Sra. Transformers implement functional gradient descent to learn non-linear functions in context. arXiv preprint arXiv:2312.06528, 2023.335

[13] Deasy, Jacob, Nikola Simidjievski, and Pietro Liò. "Constraining variational inference with geometric jensen-shannon divergence." Advances in Neural Information Processing Systems 33 (2020): 10647-10658.

[14] Kai-Ineman, D. A. N. I. E. L., and Amos Tversky. "Prospect theory: An analysis of decision under risk." Econometrica 47.2 (1979): 363-391.

• If multiple SA layers with non-linear activations are considered, can ICL capture more complex strategic behaviors?
• The experiments only consider the non-linear SA layers. Is it possible to consider some non-linear SC models (e.g., the agents do not best respond and the policy is non-linear)?

Response:

We agree with your view that the nonlinear attention mechanism can capture more complex agent behaviors or nonlinear policies.

• Brief theoretical analysis:

  • Multiple nonlinear attention layers have been demonstrated to effectively cover and extend the capabilities of linear attention layers [1-3].
  • In SC paradigms, even with complex nonlinear behaviors or policies, existing solutions still rely on forward propagation and gradient-based updates[4-6], which can be simulated and solved by attention mechanisms.
  • Therefore, multi-layer nonlinear attention, being more expressive,can capture complex agent behaviors and policies.

• Experimental verification:

  • To further validate this perspective, we designed experiments that jointly consider more complex agent behaviors and nonlinear strategic policies, using multi-layer nonlinear attention mechanisms for evaluation.

  • Complex agent strategic behavior:

    • In our extended experiments, we consider three types of complex strategic agent behaviors: The first two are drawn from prior work, while the third is newly introduced:
    • Limited information (Agent in the dark)[5]:
      • Agents manipulate features without access to the full decision boundary, relying only on outcome feedback to infer the classifier.
    • Socially-influenced (Agent with interaction)[6]:
      • Agents’ manipulable features are partially derived from their local social network structure.
      • For example, a student’s reported performance may depend on peer group statistics.
    • Irrational agents (Our introduced):
      • We introduce prospect theoretic[7] in which agents deviate from strict utility maximization.
      • Their manipulation is shaped by loss aversion, reference dependence, and probability distortion. To specific: $$x^* = \arg\max_x \left( \sum_y w(P(y|x')) \cdot v(u(y) - a_{\text{ref}}) - \phi^c(x, x') \right)$$ where:
        • $v(\cdot)$: value function (captures loss aversion),
        • $w(\cdot)$: probability weighting function,
        • $a_{\text{ref}}$: reference point,
        • $\phi^c(x, x')$: cost function for feature manipulation.

  • Complex strategic policy:

    • We also evaluate different forms of nonlinear strategic policies:
      • MLP-based policy:
        • The strategic policy is parameterized by a multi-layer perceptron, introducing nonlinear decision boundaries and feature interactions.
      • GNN-based policy:
        • To incorporate structural dependencies, we adopt a Graph Neural Network to model the strategic manipulation policy.

  • Metrics:

    • Jensen-Shannon (JS) divergence between the manipulated features distribution of the baseline and multi-layer nonlinear Attention.
    • Post-manipulation accuracy of attention mechanism under the strategic manipulation .

  • The results are presented in the following table:

    Agent behavior+policy model	JS Divergence	Accuracy
    Agent in the dark+Linear	0.168	75.9%
    Agent in the dark+MLP	0.183	81.7%
    Agent with interaction+GNN	0.226	82.3%
    Irrational agents + Linear	0.133	83.6%
    Ration agents + MLP	0.176	80.1%
    Irrational agents + MLP	0.152	77.8%

• Experimental Analysis:

  • Low JS divergence suggests that the distribution of agent features simulated by attention mechanism closely matches that of the complex strategic behavior.
  • High accuracy shows that the attention mechanism not only approximates strategic policy, but also preserves downstream task performance.
  • These experiments demonstrate that multi-layer nonlinear attention is capable of capturing complex strategic behaviors and nonlinear policies.
  • Therefore, we agree that the LLMs can learn more complex strategic behaviors and policy through ICL.

If any point remains unclear, please feel free to ask for further explanation.

---

[1] Garg, Shivam, et al. "What can transformers learn in-context? a case study of simple function classes."

[2] Akyürek, Ekin, et al. "What learning algorithm is in-context learning? Investigations with linear models."

[3] Von Oswald, Johannes, et al. "Transformers learn in-context by gradient descent."

[4] Hardt, Moritz, et al. "Strategic classification."

[5] Ghalme, Ganesh, et al. "Strategic classification in the dark."

[6] Itay Eilat, Ben Finkelshtein, Chaim Baskin, and Nir Rosenfeld. Strategic classification with graph neural networks.

[7] Kai-Ineman, D. A. N. I. E. L., and Amos Tversky. "Prospect theory: An analysis of decision under risk."

Thank you for your comments. We would like to address your concerns as follows:

Weakness1. I did not find the methodology and actual algorithm to be clearly presented.

Response:

• We would like to clarify that our work is a pioneering attempt to bridge strategic classification (SC) and LLMs via in-context learning (ICL)—far beyond simply applying LLMs to SC tasks.

  • The explicit mapping of SC’s bi-level framework and game-theoretic structure onto the attention mechanisms have been details rigorously presented in Appendices E, F, G, and I.
  • To demonstrate the reproducibility of our work, we have provided relevant experimental code, including Attention_layers.py and SC.py.

• For additional clarity, we provide below an algorithmic outline of the GLIM framework and a simple prompt example (based on our Appendix J) at the bottom.

  Algorithm 1: The control flow of GLIM

  Input:

  • Dataset $\mathcal{D} = \{(x_j, y_j)\}_{j=1}^N$
  • Pre-trained LLM $\phi$
  • Prompt template and SC rules

  Procedure:

  • Provide SC rule prompts and example prompts
  • Guide strategic manipulation of $x$
  • Guide decision rule optimization
  • Query: Manipulated $\{x'_j\}$ and predicted $\{\hat{y}_j\}$ with prompts

  Output:

  • Predicted labels $\{\hat{y}_j\}$ and accuracy
  • Manipulated features $\{x'_j\}$（optional）

• We have clarified this part in our Appendix J of our revised paper.

---

Weakness2. The theoretical analysis provided in Section 3...

Response:

We would like to clarify that the core positioning of our work is not as a purely theoretical study of ICL mechanisms, but as a pioneering attempt to bridge strategic classification and LLMs via ICL.

• Strategic classification (SC) is fundamentally a bi-level, game-theoretic problem involving both agent-side manipulation and decision-maker adaptation, and has been studied largely separately from ICL/LLM literature.

  • To our best knowledge, our work is the first to employ LLMs to model and solve the bi-level, game-theoretic optimization structure of SC.
  • It enables a scalable and retraining-free approach to large-scale SC tasks, where classical gradient-based retraining requires excessive computational resources and time.

• From a broader social science perspective, our work establishes a crucial bridge between large language models and strategic machine learning:

  • A wide range of social science applications are increasingly converging with machine learning and LLMs, including fields such as computational social science, policy evaluation, and behavioral economics[1,2,3,4].
  • Our work provides a foundational contribution by connecting ICL in LLMs with game-theoretic learning frameworks.
  • This foundation will facilitate future interdisciplinary research and applications of LLMs in societal decision-making contexts.

• To be specific, we entail a detailed theoretical analysis of SC-motivated ICL derivation, including strategic manipulation of agent and decision optimization for jury.

• Furthermore, our experimental results further validate these insights, demonstrating that LLMs guided by in-context learning robustly handle the core dynamics of strategic classification, even at scale and without retraining.

---

Weakness3.1 Adding in-context samples should notably increase the already high inference costs of using LLMs

Response:

• The reviewer refer to “prompt cost” as the computational and API expenses incurred when querying the LLM, i.e., the resource cost of processing additional in-context samples (e.g., API fees and token usage).
• In contrast, our use of prompt cost (as explicitly stated in Appendix L, Line 819) refers to the agent's cost of obtaining strategic guidance by querying the LLM during feature manipulation, e.g., exploring alternative manipulation methods with LLMs.
  • This prompt cost is negligible compared to the substantial manipulation cost required for actual improvement of individual features (such as earning a degree or increasing income).
• The distinction is crucial:
  • Our definition of prompt cost is rooted in the SC framework, focusing on the agent’s perspective, the behavioral and informational costs incurred when agents seek to manipulate their features.
  • The reviewer’s definition adopts a system-level perspective, emphasizing the computational and resource costs of LLM API usage.
• We evaluated the API usage and cost for varying numbers of feature examples as well.
  • Use official OpenAI pricing, and present the results in the following table:

---

Weakness3.2 Since LLMs are available to the public, users could potentially optimize their features in a way that is adversarial to the LLM parameters P, for any practical choice of context.

Response:

• As SC is specifically proposed to address adversarial manipulation, our work has already handled this concern.

  • The possibility of agents adversarially (strategically) manipulating their features is the foundation of strategic classification: the game-theoretic interaction between agents and the jury.

  • Our framework explicitly incorporates this adversarial dynamic by modeling agent strategic manipulation through in-context learning,

  • Such behavior is not a vulnerability, but rather one of the central subjects our approach is designed to address.

---

Weakness4 Reproducibility questions

Response:

• The code we provided focuses on verifying our analysis of the mapping between the attention mechanism and the bi-level optimization framework of strategic classification.

  • The code includes standard SC model and the core attention module.

• For practical reproducibility, we have included detailed prompt templates:

• illustrating (i) strategic classification (SC) task setup, (ii) in-context interaction examples, and (iii) batch evaluation using the dataset.

  • (i) Task Definition: This part is aligned with Section 2.1 of our paper:

  You are a strategic classification assistant. In this scenario:
  
  There are two players: a decision maker and decision subjects.  
  - The decision maker publishes its policy (classification rule $f$).  
  - The decision subjects, after observing the policy and associated costs, determine whether to strategically modify their features.  
  - The decision outcome is binary (0 or 1).
  
  The strategic manipulation rule is: ... (see Definition 2.1 in Section 2.1).
  The optimization rule is: ... (see Definition 2.2 in Section 2.1).
  
  Please restate your understanding of the strategic classification setting in concise terms.
  

  • In-context Examples: Here is an example using a real sample from the Adult dataset (multiple such examples are recommended for best results):

  • Example 1:
    Initial features:
       - age: 34
       - workclass: Private
         ...
         ...
       - native-country: United-States
     Initial result: income >50K
     
    Example 2:
    ...
    

  • Batch Testing with Dataset For large-scale evaluation, we load the entire dataset and generate prompts for each test case as follows:

    with open('adult.csv', 'r') as f:
        test_examples = f.read()
    
    prompt = """
    Next, I will provide you with a series of applicant cases. For each, please:
    1. Apply the rules above to strategically manipulate the applicant's features if beneficial.
    2. Update your decision-making rules as per the definitions above.
    3. For each applicant, even after strategic manipulation, the true label should remain unchanged.
    Finally, report the accuracy rate under your classification rules.
    
    test_examples: ...
    """
    
    

• We are glad to fully release all source code and materials upon acceptance. Thank you for bringing up this important point.

---

Formatting Concerns

Response:

• Could you specify which tables or figures violate the formatting requirements?
• We will conduct a more thorough check.

---

[1] Ariel Flint Ashery et al. Emergent social conventions and collective bias in LLM populations. Sci. Adv.11,eadu9368(2025).

[2] Gao, Chen, et al. "Large language models empowered agent-based modeling and simulation: A survey and perspectives." Humanities and Social Sciences Communications 11.1 (2024): 1-24.

[3] De Curtò, J., and I. De Zarzà. "LLM-Driven Social Influence for Cooperative Behavior in Multi-Agent Systems." IEEE Access (2025).

[4] Ziems, Caleb, et al. "Can large language models transform computational social science?." Computational Linguistics 50.1 (2024): 237-291.

Thank you for your valuable suggestion. We fully acknowledge the practical limitations associated with relying on commercial LLMs.

We have updated the limitations section in our revised paper to analyze the cost implications.

Limitation: Cost Analysis of LLM-based SC

• A key practical limitation of our approach is its reliance on proprietary large language models that are accessed via commercial APIs, such as OpenAI GPT-4o and DeepSeek.

  • Unlike traditional machine learning models, which can be trained or deployed locally with a fixed hardware budget, our method depends on repeated calls to remote LLM API services.

  • As a result, the total inference cost is variable and increases with the number of API requests.

• To evaluate the impact of data volume on API consumption and cost, we conducted the following experiment.

  • We used the Adult dataset, selecting 13-dimensional feature vectors for each sample.
  • For each test case, we submitted a complete prompt containing varying amounts of example to the LLM API, and recorded the cost after receiving the model’s response.
  • The official OpenAI GPT-4o pricing was used to calculate the total cost for different batch sizes.

• The results are summarized in the table below:

• Experimental Results and Analysis

  • As shown in the results above, the API cost grows nearly linearly with the number of prompt examples submitted.
  • We note that each additional example increases the token count, and thus the total cost, in a roughly proportional manner. This cost structure should be considered when designing systems that require frequent queries.
  • This limitation is inherent to all current commercial LLM services and should be accounted for in practical deployment scenarios.

• Potential Mitigations (Proposed):

  • Designing more cost-efficient prompt engineering strategies.
    • This will be a solution to reduce the number of required API calls and lower overall costs.
    • However, in reality, due to the dynamic of agent distribution, it is difficult to control.
  • Another approach is to develop hybrid systems.
    • This approach use LLM-based reasoning for complex cases.
    • While relying on local models to handle high-frequency or latency-sensitive requests, thereby minimizing dependence on commercial LLM services.

• Addressing this limitation will be a key focus of our future work, as we seek to optimize the trade-off between LLM-powered flexibility and practical deployment costs.

If any point remains unclear, please feel free to ask for further explanation.

\begin{figure*}[t]
    \centering
    \subfigure[Small-scale Data]{
        \includegraphics[width=0.226\textwidth]{images/fig2a.pdf}
        \label{fig2a}
    }
    \subfigure[Large-scale Data]{
        \includegraphics[width=0.226\textwidth]{images/fig2b.pdf}
        \label{fig2b}z
    }
    \subfigure[Distribution Shift]{
        \includegraphics[width=0.226\textwidth]{images/fig2c.pdf}
        \label{fig2c}
    }
    \subfigure[KL Divergence]{
        \includegraphics[width=0.226\textwidth]{images/fig2d.pdf}
        \label{fig2d}
    }
    \caption{Comparison of ICL-guided strategic manipulation. (a) and (b) compare ICL and gradient-descent methods across data scales; (c) and (d) evaluate implicit gradient alignment via distribution metrics.}
    \label{fig2}
\end{figure*}

We provide a representative prompt example, illustrating (i) strategic classification (SC) task setup, (ii) in-context interaction examples, and (iii) batch evaluation using the dataset.

*If any point remains unclear, please feel free to ask for further explanation.

• (i) Task Definition : This part is aligned with Section 2.1 of our paper:

You are a strategic classification assistant. In this scenario:

There are two players: a decision maker and decision subjects.  
- The decision maker publishes its policy (classification rule $f$).  
- The decision subjects, after observing the policy and associated costs, determine whether to strategically modify their features.  

Specifically, the decision maker defines a decision rule, e.g., a classifier, mapping feature vectors to binary outcomes $y \in \{0, 1\}$. Once the rule is known, individuals may modify their features to a new version in hopes of receiving a favorable decision. Such modification incurs a cost, quantified by a cost function. 

The strategic manipulation rule for decision subjects is: ... (see Definition 2.1 in Section 2.1).
The optimization rule for the decision maker is: ... (see Definition 2.2 in Section 2.1).

Please restate your understanding of the strategic classification setting in concise terms.

• In-context Examples: Here is an example using a real sample from the Adult dataset (multiple such examples are recommended for best results):

Example 1:
Initial features:

   - age: 34
     - workclass: Private
       - fnlwgt: 203034
       - education: Bachelors
       - education-num: 13
       - marital-status: Separated
       - occupation: Sales
       - relationship: Not-in-family
       - race: White
       - sex: Male
       - capital-gain: 0
       - capital-loss: 2824
       - hours-per-week: 50
       - native-country: United-States
      Initial result: income >50K
      
 Example 2: ...
 
 ...
 
 Example k: ...
 
 

• Batch Testing with Dataset For large-scale evaluation, we load the entire dataset and generate prompts for each test case as follows:

 
  prompt = """
  Next, I will provide you with a series of applicant cases. For each, please:
  1. Apply the rules above to strategically manipulate the applicant's features if beneficial.
  2. Update your decision-making rules as per the definitions above.
  3. For each applicant, even after strategic manipulation, the true label should remain unchanged.
  Finally, report the accuracy rate under your classification rules.
  
  test_examples: ...
  """
  

• Note1: In our practical prompt design, both strategic manipulation (inner optimization) and decision rule adaptation (outer optimization) can be performed in a single, unified prompt.

• Note2: To efficiently collect multiple in-context examples and generate LLM-readable prompts, we use the following python code to organize data in batches from a CSV file:

def agent_row_to_text(row, idx):
    text = f"Example {idx+1}:\n"
    for col, val in row.items():
        text += f"- {col}: {val}\n"
    return text.strip()

def make_prompt_from_csv(csv_path, start_row=0, n_rows=30000):
    import pandas as pd
    df = pd.read_csv(csv_path)
    df = df.iloc[start_row:start_row+n_rows]
    agent_texts = [agent_row_to_text(row, idx) for idx, row in df.iterrows()]
    prompt = "\n".join(agent_texts)
    return prompt

Thank you for valuable comments. We would like to address your concerns as follows:

Oversimplified transformer architecture

Linear attention assumption

Response:

We note that these two comments are closely related. We clarify them together below.

• Our use of a single-layer, linear attention model is not an oversimplification, but a widely accepted theoretical convention in both ICL and transformer analysis.
  • As summarized in the table below, this analytical simplification (single-layer and linear) is foundational in many highly cited works on ICL and transformer theory.
• Our work is not intended as a general-purpose LLM application, but as a targeted contribution that explicitly bridges ICL theory with strategic classification.
  • As highlighted in the second row of the table below, nearly all foundational research in strategic classification adopts linear models for their interpretability and analytical clarity.
  • Our focus on single-layer, linear attention is therefore a deliberate and justified choice that is well aligned with established modeling practice in both domains, and is particularly suited to the requirements of social science and practical applications. |Representative Works|Typical methodology / Model| |--|--| |ICL/Transformer Theory[1,2,3,4,9]|Single-layer attention, mainly linear| |Strategic Classification[5-8]|Linear model|
• Adding multilayer attention or MLPs does not undermine the core connection between ICL and gradient-based updates.
  • As shown in the table, our analysis is not “oversimplified,” but instead distills the most essential and theoretically transparent part of the mechanism.
  • MLP layers only apply local nonlinearities and do not interfere with the connection between attention and gradient-based updates[1,3].
• The gradient-approximation ability of Softmax attention is theoretically guaranteed [11,12].
  • Recent work shows that softmax attention can closely approximate any target direction within the convex hull of the value vectors[11]:
  $$\mathrm{Attn_{Softmax}}(Q, K, V) \approx \sum_{i=1}^N \alpha_i v_i,\quad \sum_{i=1}^N \alpha_i = 1,\, \alpha_i \geq 0$$
  • For any target direction $g$ in the convex hull of $\{v_i\}$, there exists weights $\{\alpha_i\}$ such that
  $$\left\| \sum_{i=1}^N \alpha_i v_i - g \right\| \leq \epsilon$$
  • This provides a rigorous upper bound on the approximation error, supporting the theoretical basis for prompt-based gradient encoding with softmax attention [11,12].
• Prompt design can make attention mechanisms approximate gradient descent is in fact well supported by recent theoretical work.
  • Notably, several recent studies[2,4,11] have rigorously established that, with appropriate prompt construction, Transformers can perform gradient descent in both linear and nonlinear settings:
• Our linear construction is theoretically well-founded.
  • Recent theoretical work[11-14] has demonstrated that nonlinear attention with sufficient layers and heads can approximate any optimization trajectory expressible by linear attention, up to arbitrarily small error.
  • Moreover, prior work[10] has established the convergence of Transformer learning dynamics, demonstrating that nonlinear Transformers can reliably simulate the linear constructions in our framework.

---

Homogeneous label assumption

Response:

• The homogeneous assumption is a reasonable theoretical simplification for mechanism illustration:
  • It allows us to isolate and transparently demonstrate the effect of the attention mechanism, showing how, under this setting, the attention update precisely aligns with the gradient descent direction.
  • Such assumptions are common in theoretical work to reveal idealized behaviors, serving as an “upper bound” for mechanism capability.
• This conclusion remains valid without this assumption:
  • When features and labels are heterogeneous (as in practice), the attention update generalizes to a weighted sum of local gradient directions, reflecting the similarity between each context and the query:
  $$\Delta x_j^{\mathrm{ICL}} = \frac{A\eta}{N} \sum_{i=1}^N (1-y_i) W^\top \langle x_i, x_j \rangle,$$
  • More generally,
  $$\Delta x_j^{\text{ICL}} = \sum_{i=1}^N \alpha_{ji} \cdot g(x_i, y_i),$$ ​ with $\alpha_{ji}$ as the attention weights and $g(x_i, y_i)$ as the local gradient directions.
  • The expected error between ICL and GD can be tightly bound:
  • $$\epsilon_j = \left\| \Delta x_j^{\text{ICL}} - \nabla_x L(x_j) \right\|,\quad \mathbb{E}[\epsilon_j] \leq C \cdot \sqrt{\frac{1}{N}} + \delta,$$ where $C$ is a constant and $\delta$ is the statistical error from distribution mismatch.
• We have clarified this part in Appendix F of our revised paper.

---

Concerns of Experiments1

Response:

• The results in Table 2 are not intended as a direct comparison of raw performance between traditional baselines and pre-trained LLMs. Rather, the purpose is to demonstrate that pre-trained LLMs are capable of solving the SC problem.
• As stated in Appendix A, our work aims to explore an alternative solution for large-scale strategic classification scenarios, where classical gradient-based retraining requires excessive computational resources and time.
• For comparisons between different LLM APIs, we ensure fairness by aligning pre-training LLM scale as closely as possible.
  • The following table summarizes the key properties evaluated in our experiments (except for LLMs whose specific parameters have not been made public): |Model| Parameter (B)| |-|-| |DeepSeek-V3|67| |LLama-3.3|70| |Qwen3|72| |Mixtral|46.7|
• We have clarified this part in Lines 265-269 of our revised paper.

Concerns of Experiments2

Response:

• We would like to clarify that the observed decrease and fluctuation in cosine similarity does not contradict our claims of method effectiveness; rather, it reflects the natural challenges of large-scale, dynamic distributions.
  • As the data becomes larger and more diverse, it becomes increasingly difficult for traditional models with GD to maintain stable feature or strategy representations.
  • In contrast, LLMs guided by ICL exhibit more robust adaptation and can better handle the complexities of large-scale data. Therefore, there is a small decline in cosine similarity.
• Notably, a similar trend can be observed in Figure 5(b). These phenomena reflect the gap in large-scale data adaptation between LLMs and traditional GD models.

Concerns of Experiments Some of the tested models are closed-source

Response:

• We agree that some tested LLMs are closed-source, but we design multiple experiments directly comparing the attention layers update with the theoretical SC optimization process.
• In recent ICL research, it is accessible to use prompt-based probing and input-output analysis to verify mechanistic conclusions[15-17].

Concerns of Experiments prompt example

Response:

• We provide a simple prompt example

  • Task Definition: This part is aligned with Section 2.1 of our paper:

  You are a strategic classification assistant. In this scenario:
  There are two players: a decision maker and decision subjects.
  - The basic interaction rule ... (Section 2.1)
  - The strategic manipulation rule is: ... (see Definition 2.1).
  - The optimization rule is: ... (see Definition 2.2).
  Restate your understanding.
  

  • In-context Examples: from Adult dataset:

   Example 1:
   Initial features:
      age: 34
      workclass: Private
        ...
      native-country: United-States
    Initial result: income >50K
    
   Example 2:...
  

  • Batch Testing with Dataset

  prompt = """
  1. Apply the rules above to strategically manipulate the applicant's features if beneficial.
  2. Update your decision-making rules as per the definitions above.
  3. For each applicant, even after strategic manipulation, the true label should remain unchanged.
  Finally, report the accuracy rate under your classification rules.
  test_examples: ...
  """
  

---

[1] Von Oswald, Johannes, et al. "Transformers learn in-context by gradient descent."

[2] Garg, Shivam, et al. "What can transformers learn in-context? a case study of simple function classes."

[3] Akyürek, Ekin, et al. "What learning algorithm is in-context learning? Investigations with linear models."

[4] Clark, Peter, Oyvind Tafjord, and Kyle Richardson. "Transformers as soft reasoners over language."

[5] Hardt, Moritz, et al. "Strategic classification."

[6] Miller, John, Smitha Milli, and Moritz Hardt. "Strategic classification is causal modeling in disguise."

[7] Levanon, Sagi, and Nir Rosenfeld. "Strategic classification made practical."

[8] Milli, Smitha, et al. "The social cost of strategic classification."

[9] Kratsios, Anastasis, et al. "Universal approximation under constraints is possible with transformers."

[10] Ahn, Kwangjun, et al. "Transformers learn to implement preconditioned gradient descent for in-context learning."

[11] Xiang Cheng, Yuxin Chen, and Suvrit Sra. ''Transformers implement functional gradient descent to learn non-linear functions in context.''

[12] Franco, Luca, et al. "Under the hood of transformer networks for trajectory forecasting."

[13] Li, Hongkang, et al. "How do nonlinear transformers learn and generalize in in-context learning?"

[14] Gatmiry, Khashayar, et al. "Can looped transformers learn to implement multi-step gradient descent for in-context learning?"

[15] Wei, Jason, et al. "Emergent abilities of large language models."

[16] Min, Sewon, et al. "Metaicl: Learning to learn in context."

[17] Zhao, Zihao, et al. "Calibrate before use: Improving few-shot performance of language models."

Oversimplified transformer architecture: The theoretical analysis in the appendix reduces transformers to single-layer attention mechanisms, ignoring (1) the multi-layer structure and (2) MLP layers, which constitute a significant portion of the parameter budget and computational capacity.

Response:

Purpose of the Simplification:

• This simplification a widely accepted theoretical convention in both ICL and transformer analysis[1-3].
• This form of abstraction aligns with a single optimization step in SC[4-6], reflecting the standard analytical paradigms in both the SC and ICL domains[7-9].

Addressing Multi-Layer Structure and MLPs:

• To directly address your concern, we conducted controlled experiments with custom Transformers dissecting multi-layer structure and MLPs:
  • Custom Transformer with different numbers of attention layer;
  • Custom Transformer based on multi-layer structure without MLPs;
  • Custom Transformer based on multi-layer structure with MLPs.
• Metrics:
  • We consider the Cosine similarity with two points
  • Similarity in agent strategic manipulation ($\Delta x$)
    • The similarity of how the agent’s feature distribution changes under strategic manipulation, when influenced by GD-based SC models versus attention mechanisms.
  • Similarity in post-manipulation predictions ($\Delta y$)
    • The similarity of the predicted outcomes after manipulation, under the influence of GD-based SC models versus attention mechanisms.
• The results are presented in the following table:

• Analysis of result:
  • Based on the experimental results, we can see that:
    • Stacking different numbers of attention layers (20, 50, 100) already achieves a high similarity in simulating both the agent’s strategic manipulation and the jury’s decision optimization.
    • Moreover, after incorporating MLPs, the customized Transformer can also simulate the agent’s strategic manipulation and the jury’s decision optimization, with the similarity further improved.
  • linking theory with practice:
    • In our paper, we have demonstrated that a single-layer attention mechanism is capable of simulating either a single round of the agent’s strategic manipulation or a single round of the jury’s decision optimization.
    • Although our constructive proofs (Appendix F and G) focus on the single-layer case, multi-layer attention architectures with MLPs provide a more practical and continuous process of manipulation or optimization of SC.
    • This confirms that real, multi-layer Transformers with MLPs components are effective and robust in addressing SC problems.

---

[1] Child, Rewon, et al. "Generating long sequences with sparse transformers."

[2] Choromanski, Krzysztof, et al. "Rethinking attention with performers."

[3] Gatmiry, Khashayar, et al. "Can looped transformers learn to implement multi-step gradient descent for in-context learning?"

[4] Hardt, Moritz, et al. "Strategic classification."

[5] Levanon, Sagi, and Nir Rosenfeld. "Strategic classification made practical."

[6] Miller, John, Smitha Milli, and Moritz Hardt. "Strategic classification is causal modeling in disguise."

[7] Garg, Shivam, et al. "What can transformers learn in-context? a case study of simple function classes."

[8] Akyürek, Ekin, et al. "What learning algorithm is in-context learning? Investigations with linear models."

[9] Von Oswald, Johannes, et al. "Transformers learn in-context by gradient descent."

Concerns of Experiments1: unfair comparison to simpler model:

Response:

Thank you for helpful suggestions.

We would like to make an explicit statement and supplement an experiment using smaller LLMs as detailed below, and incorporate these into our revised paper.

• State our main purpose

  • We would like to explicitly state in the revised paper:
    • Comparing baseline models with LLMs is not our paper's focus; rather, our aim is to demonstrate the capability of LLMs for solving strategic classification tasks via ICL.
  • We have supplemented this statement in Lines 220–227 and removed content that implied a direct comparison between baseline models with LLMs.

• Experiment using smaller LLMs

  • Thank you for your insightful suggestion.

    • We agree that it is necessary to evaluate our method on smaller LLMs, as this helps to validate the practical utility of our method.

  • We have conducted a experiment using Llama‑3.2‑1B and Llama‑3.2‑3B models.

  • We use Adult dataset in this experiment.

  • The results are summarized in the table below:

    Models	Accuracy (%)
    Llama‑3.2‑1B	80.612
    Llama‑3.2‑3B	82.120

• Analysis:

  • These results demonstrate that our method remains effective with smaller LLMs, confirming its broad applicability to strategic classification tasks.
  • We also agree that models with more parameters generally demonstrate better accuracy on this task.

We appreciate your insightful suggestions.

We have incorporated both the above clarification and the experimental results into Section 4.4 of our revised paper.

Thank you for raising this important point.

We will

1. firstly clarify the motivation for adopting this assumption,

2. second discuss how our derivation extends beyond this special case,

3. third design experiments for verification.

   ---

• Purpose of the homogeneous assumption:

  • The homogeneous label assumption is a theoretical simplification[1] used to illustrate underlying mechanisms.

  • It enables us to transparently demonstrate how the attention-based update can precisely align with the gradient descent direction under idealized conditions.

  ---

• The specific derivation beyond the assumption:

  • In practical, heterogeneous contexts, the update is a weighted sum of local update directions:

    $$\Delta x_j^{\mathrm{ICL}} = \frac{A\eta}{N} \sum_{i=1}^N (1-y_i) W^\top \langle x_i, x_j \rangle$$

    where $\langle x_i, x_j \rangle$ reflects the similarity between the query and context example.

  • More generally, the attention mechanism implicitly performs a weighted aggregation of local updates. That is:

    $$\Delta x_j^{\mathrm{ICL}} = \sum_{i=1}^N \alpha_{j,i} \cdot g(x_i, y_i),$$

    where $\alpha_{j,i}$ are the attention weights and $g(x_i, y_i)$ denotes the local update direction.

  • When the context samples are independently drawn and sufficiently representative of the local data distribution around $x_j$, statistical learning theory[2] guarantees that:

    $$\lim_{N \to \infty} \Delta x_j^{\text{ICL}} \approx \mathbb{E}_{(x_i, y_i)}[ g(x_i, y_i) ].$$

  • The approximation error

    $$\epsilon_j = || \Delta x_j^{\text{ICL}} - \nabla_{x} \mathcal{L}(x_j) ||$$

    can be bounded as:

    $$\mathbb{E}[\epsilon_j] \leq \underbrace{C \cdot \sqrt{\frac{1}{N}}}_{\text{Sampling error}} + \delta,$$

    where:

    • $\delta$ is the error from the distribution shift, $\delta < L \cdot \mathcal{W}$,
    • $C$ is a constant depending on the gradient variance
    • $L$ is the Lipschitz constant of $\mathcal{L}$
    • $\mathcal{W}$ denotes the Wasserstein distance between local and global data distributions

  • This analysis confirms that our main theoretical insights extend beyond the homogeneous case, and remain valid in realistic, heterogeneous environments.

    ---

• Experiments:

  • We conducted experiments under realistic, heterogeneous context settings.

  • The goal is to assess whether the update direction produced by attention mechanism aligns with the local update direction obtained from standard gradient descent.

  • In each experiment, we randomly sample diverse context examples from the dataset, and compute:

    • (A) the feature update direction using the attention mechanism;
    • (B) the local update direction using explicit gradient computation.

  • We then measured the Cosine similarity between these two update directions for various context sample sizes $N$.

  • The results are reported in the table below:

    Number of Context Examples ($N$)	Cosine Similarity
    50	0.733
    100	0.771
    200	0.803
    300	0.822

  • Analysis:

    • As shown in the table, the cosine similarity increases as the number of context examples grows, indicating that the ICL-induced update direction becomes increasingly aligned with the true local update direction.
    • This experimental evidence supports the validity of our theoretical analysis in realistic, heterogeneous settings.

If any point remains unclear, please let us know.

Concern of Experiment 2 - cosine similarity decrease

Response:

Thank you for helpful suggestions.

• We would like to acknowledge the observed divergence between our theoretical analysis and the empirical results at scale, as evidenced by the decrease in cosine similarity in Figure 3.

• In the revised version, we would like to supplement a specific subsection in the main text to discuss this limitation.

---

Limitation: Theory-Practice Divergence at Scale

• As shown in Figures 3(b) and 6(b), the cosine similarity tends to decrease over iterations, indicating a divergence between our theoretical analysis and the empirical results at scale.
• This divergence may be attributable to factors such as model capacity, data distribution shifts, or nonlinearities in real-world tasks.
• We acknowledge the existence of this divergence and further analyze this phenomenon in future work to better understand the conditions under which such divergences occur.

We have incorporated this part into Section 4 of our revised paper.

---

We appreciate your insightful suggestions, and the limitations have been explicitly discussed in the main text of our revised paper.

If any point remains unclear, please let us know. We are glad to fully address your concerns.

We sincerely appreciate your constructive comments and your positive evaluation of our work. We would like to address your concerns as follows:

Weaknesses

Response:

• We would like to clarify that the core positioning of our work is not as a purely theoretical study of ICL mechanisms but as a pioneering attempt to bridge strategic classification and LLMs via ICL.

• Strategic classification (SC) is fundamentally a bi-level, game-theoretic problem involving both agent-side manipulation and decision-maker adaptation, and has been studied largely separately from ICL/LLM literature.

  • To our best knowledge, our work is the first to employ LLMs to model and solve the bi-level, game-theoretic optimization structure of SC.
  • It enables a scalable and retraining-free approach to large-scale SC tasks, where classical gradient-based retraining requires excessive computational resources and time.

• From a broader social science perspective, our work establishes a crucial bridge between large language models and strategic machine learning:

  • A wide range of social science applications are increasingly converging with machine learning and LLMs, including fields such as computational social science, policy evaluation, and behavioral economics[1,2,3,4].
  • Our work provides a foundational contribution by connecting LLMs with game-theoretic frameworks.
  • We hope this foundation will facilitate future interdisciplinary research and applications of LLMs in societal decision-making contexts.

• We have clarified this part in our revised paper at Lines 62-64 (Introduction).

---

Question 1.What does GLIM stand for?

Response:

• GLIM stands for Gradient-free Learning In-context Method.

• We have clarified this part in Lines 65-67 (Introduction) of our revised paper.

---

Question 2. Line 117, A is not defined.

Response:

• In Lemma 1, $A_\ell$ denotes the step size used in the gradient descent update at layer $\ell$.
• We have clarified this lemma in Lines 117-118 of the revised paper.

---

Question 3. Showing the existence of such PVK matrices does not imply that your model would learn the same matrices. You would need to argue that your learning procedure would converge to desirable PVK matrices.

Response:

• We thank the reviewer for highlighting this point. Our existence proofs demonstrate that attention mechanisms in LLMs are expressive enough to realize PVK matrices that simulate gradient-based updates.
• We would like to note that
  • Recent theoretical work[5,6] has proven that attention weights in pretrained Transformers do converge to such optimization-relevant matrices during in-context learning, thereby providing theoretical support for our approach.
  • Therefore, after establishing the existence of suitable PVK matrices, the ICL procedure can indeed converge to such desirable matrices in practice.
• We have clarified this part in Appendix D (Lines 561-563) and Appendix F (Lines 617-619) of our revised paper.

---

[1] Ariel Flint Ashery et al. Emergent social conventions and collective bias in LLM populations. Sci. Adv.11,eadu9368(2025).

[2] Gao, Chen, et al. "Large language models empowered agent-based modeling and simulation: A survey and perspectives." Humanities and Social Sciences Communications 11.1 (2024): 1-24.

[3] De Curtò, J., and I. De Zarzà. "LLM-Driven Social Influence for Cooperative Behavior in Multi-Agent Systems." IEEE Access (2025).

[4] Ziems, Caleb, et al. "Can large language models transform computational social science?." Computational Linguistics 50.1 (2024): 237-291.

[5] Ahn, Kwangjun, et al. "Transformers learn to implement preconditioned gradient descent for in-context learning." Advances in Neural Information Processing Systems 36 (2023): 45614-45650.

[6] Gatmiry, Khashayar, et al. "Can looped transformers learn to implement multi-step gradient descent for in-context learning?" arXiv preprint arXiv:2410.08292 (2024).