2507.00726v3

# Can Large Language Models Develop Strategic Reasoning? Post-training Insights from Learning Chess > Equal contributions. This work was done during an internship at KRAFTON.Co-advising. ## Abstract While reinforcement learning (RL) for large language models (LLMs) has shown promise in mathematical reasoning, strategic reasoning for LLMs using RL remains largely unexplored. We investigate whether LLMs can develop strategic reasoning capabilities through RL in chess. To this end, we leverage a chess-pretrained action-value network to provide dense reward on the LLM’s output move quality, which can be seen as a form of knowledge distillation. Our experiments show that our distillation-based dense rewards often outperform sparse binary rewards. However, surprisingly, all models plateau far below expert levels. We provide SFT and RL ablations on chess reasoning training and find evidence that this limitation stems from a deficit in the pretrained models’ internal understanding of chess—a deficit which RL alone may not be able to fully overcome. The code is available at https://github.com/krafton-ai/Chess-R1. ## 1 Introduction Reinforcement learning with verifiable rewards (RLVR) has shown strong performance in developing mathematical reasoning capabilities for large language models (LLMs) (Guo et al., 2025; Li et al., 2025; Yu et al., 2025). While these successes highlight LLMs’ capacity for logical thinking, a critical dimension of intelligence remains largely unexplored: strategic reasoning —the ability to plan, anticipate adversary actions, and make decisions in multi-agent environments. Beyond logical reasoning in static settings, strategic reasoning aligns more with real-world scenarios such as games, negotiation, and market competitions (Zhang et al., 2024; Park et al., 2025). To investigate this gap, we turn to chess, a strategic game demanding deep strategic reasoning abilities such as positional evaluation, long-term planning, and reasoning about an opponent’s intentions. In addition, chess offers a favorable environment for applying RLVR on LLMs, as it provides abundant publicly available game records and human-annotated reasoning about optimal moves. Given such a testbed for examining strategic reasoning, we raise the following research question: Can LLMs develop strategic reasoning capabilities through RLVR with chess? To this end, we train Qwen2.5 (Qwen: et al., 2025) and Llama3.1 (Grattafiori et al., 2024) models to predict the next best move in chess using GRPO (Shao et al., 2024). Unlike typical RLVR approaches that rely on sparse binary rewards (correct/incorrect), chess allows for dense reward signals: we leverage the fact that we can evaluate each move based on its estimated win probability, providing graded feedback proportional to the quality of the move. We implement this using a pre-trained chess expert as a reward model—a form of knowledge distillation from Q-value network to LLM—which evaluates position strength and provides dense continuous rewards rather than sparse binary ones. We conduct systematic experiments on a chess puzzle dataset, and observe the following: - Distillation-based dense rewards substantially outperforms sparse binary rewards in chess puzzle accuracy. - RL fine-tuning performs stronger than supervised fine-tuning (SFT); yet, its performance plateaus at 25-30 $\$ puzzle accuracy—well below what is considered human expert performance (60–80 $\$ ). - Additional reasoning distillation shows no RLVR gains, indicating the models fail to develop meaningful strategic reasoning despite further post-training. Through additional failure analysis, we find that base LLMs often struggle to grasp fundamental chess rules. Thus, we hypothesize that, in contrast to logical reasoning in math domains, the limited emergence of strategic reasoning in chess may stem from insufficient exposure to chess-specific knowledge during LLM pre-training. Our empirical findings also support recent claims that RL mainly amplifies existing capabilities of pre-trained LLMs (Li et al., 2025; Zhao et al., 2025b), and offer insight for practitioners aiming to elicit reasoning abilities in new environments: pre-trained domain knowledge is essential to develop advanced reasoning. ## 2 Post-training LLM on Chess <details> <summary>fig/chess_training_diagram.png Details</summary>  ### Visual Description ## Diagram: Chess AI System Architecture ### Overview The diagram illustrates a chess-playing AI system architecture with multiple interconnected components. It shows the flow from chess position analysis through model processing to action selection and reward evaluation. The system uses chess-specific notation (FEN/SAN) and incorporates reinforcement learning concepts. ### Components/Axes 1. **Left Panel (Textual Context)** - Role description: "helpful assistant who plays chess professionally" - Chess position: FEN string "7R/7P/4p3/P1p5/8/2k5/7r/3K4 w - - 0 54" - Legal moves in SAN: - Rg8 Rf8 Re8 Rd8 Rc8 Rb8 - Ra8 Ke1 Kc1 a6 - Chess piece icon (knight) 2. **Central Processing Flow** - Cylindrical diagram with stages S1 (white), S2 (orange), S3 (white) - Annotation stack with "Annotations o3 reasoning" and SFT (optional) - Base Model (Qwen/Llama) connected to Policy Model (Base or SFT) - Policy Model outputs actions a1, a2, a3 3. **Right Panel (Reward System)** - Reward Model (blue box) with Critic Q(s,a) function - GRPO (Gradient-based Reinforcement Optimization) connection - Reward calculations: - r₁ := Q(s, a₁) - r₂ := Q(s, a₂) - r₃ := Q(s, a₃) ### Detailed Analysis - **Chess Position**: The FEN string describes a specific board state with white pieces on ranks 1-2 and black pieces on ranks 7-8, with a white king on e1 and black king on c1. - **Model Architecture**: - Base Model (Qwen/Llama) processes chess positions - Policy Model selects actions (a1-a3) using either Base Model or SFT - Reward Model evaluates actions using Q(s,a) critic function - **Reinforcement Learning**: GRPO optimization connects Policy Model to Reward Model - **Notation Systems**: Uses both FEN (position encoding) and SAN (move notation) ### Key Observations 1. The orange-highlighted S2 stage suggests special processing or emphasis 2. SFT (Supervised Fine-Tuning) is marked as optional in the annotation flow 3. Multiple reward calculations (r₁-r₃) indicate parallel action evaluation 4. Chess-specific terminology dominates the textual components 5. The knight icon visually reinforces the chess context ### Interpretation This architecture demonstrates a hybrid approach combining: 1. **Chess Domain Knowledge**: Explicit FEN/SAN notation handling 2. **AI Models**: Base/Llama models for position analysis 3. **Reinforcement Learning**: GRPO optimization for policy improvement 4. **Human-like Expertise**: The "professional chess player" role description suggests human-AI collaboration The system appears designed to: 1. Analyze complex chess positions (FEN string) 2. Generate legal moves (SAN notation) 3. Evaluate actions through multiple reward calculations 4. Optimize policy using gradient-based methods The orange-highlighted S2 stage might represent a critical decision point in the processing pipeline, possibly where the most computationally intensive analysis occurs. The optional SFT component suggests flexibility in model training approaches. </details> (a) Chess Training Example. (b) Post-training with GRPO. Figure 1: Overview: (a) A data sample from the Lichess puzzle dataset is formatted into a prompt that includes the current board state and the set of legal moves. (b) At each GRPO step, the policy model generates multiple rollouts of predicted actions. A reward model evaluates these rollouts with dense feedback, including sub-optimal actions. Optionally, reasoning traces from OpenAI o3 are fine-tuned into the base model before RL. ### 2.1 Chess Dataset #### Training Data. We use Lichess puzzles collected from the Lichess Puzzle Database https://database.lichess.org as our training dataset. Each puzzle contains an initial board position along with a list of sequential moves which, when played in order, leads to a tactical win. To create our training dataset, we decompose each puzzle’s solution trajectory into individual position-action pairs. Formally, each puzzle is denoted as a trajectory $g^{(i)}=(s^{(i)}_{0},a^{*(i)}_{0},s^{(i)}_{1},a^{*(i)}_{1},\ldots,s^{(i)}_{T_{i}})$ , where $i\in\{1,\ldots,N\}$ indexes the $i$ -th puzzle, $s^{(i)}_{t}$ is the board position at time step $t$ , $a^{*(i)}_{t}$ is the optimal move, and $T_{i}$ is the trajectory length. We construct our position-diverse training dataset by aggregating all such pairs: $\mathcal{D}=\{\,(s^{(i)}_{t},\,a^{*(i)}_{t})\mid i\in\{1,\dots,N\},\;0\leq t<T_{i}\}\!$ , resulting in a total of 19.2k training samples for our dataset $\mathcal{D}$ . #### Prompt template. We require a textual interface which LLMs can reason about chess positions. Based on our collected chess dataset $\mathcal{D}$ , we adopt a concise prompt format using Forsyth–Edwards Notation (FEN) (Edwards, 1994) format for board states and Standard Algebraic Notation (SAN) format for moves. Alternative representations of chess data, such as full move histories in Portable Game Notation (PGN) or Universal Chess Interface (UCI) move notation, did not yield measurable improvements. We also require the model to place reasoning in <think> tags and answers in <answer> tags (Guo et al., 2025; Zhao et al., 2025a). See Appendix A, C.1 for qualitative examples and ablation studies on prompt formatting. ### 2.2 Reinforcement Learning Fine Tuning To train LLMs with RL on chess, we adopt GRPO (Shao et al., 2024) for policy improvement, which has recently been shown to be effective by contrasting multiple rollouts of the same prompt (Guo et al., 2025). Specifically, we employ two alternative reward schemes: - Sparse reward: Binary indicator whether predicted move $\hat{a}_{t}$ matches optimal answer $a_{t}^{\star}$ : $$ r_{\text{sparse}}\;=\;\mathbf{1}[\,\hat{a}_{t}=a_{t}^{\star}\,]. $$ - Dense reward: A dense real-valued score provided by a pre-trained action-value network $Q_{\theta}(s,a)$ from Ruoss et al. (2024) that predicts post-move win probability: $$ r_{\text{dense}}\;=\;Q_{\theta}(s_{t},a_{t}),\qquad Q_{\theta}(s,a)\in[0,1]. $$ The critic $Q_{\theta}(s,a)$ is a 270 M-parameter, 16-layer decoder-only Transformer (Vaswani et al., 2017) trained for 10 M optimization steps on 15B Stockfish annotated state-action pairs with the HL-Gauss loss (Imani & White, 2018). Querying this expert network during RL fine-tuning can be viewed as a form of knowledge distillation (Zhang et al., 2025c): dense win-rate evaluations inject the teacher’s strategic insight, guiding the student for not only optimal moves, but sub-optimal moves also. The critic itself attains 2299 ELO and achieves 95.4% accuracy on Lichess puzzles. We also include two auxiliary binary rewards: (i) $r_{\text{fmt}}\in\{0,1\}$ for proper tag formatting (<think> for reasoning and <answer> for answers), and (ii) $r_{\text{lang}}\in\{0,1\}$ to enforce that the output is in English. The total training reward at step $t$ is therefore $r_{t}=\lambda_{\text{sparse}}r_{\text{sparse}}+\lambda_{\text{dense}}r_{\text{dense}}+\lambda_{\text{fmt}}r_{\text{fmt}}+\lambda_{\text{lang}}r_{\text{lang}}$ . We set $(\lambda_{\text{sparse}}=1,\lambda_{\text{dense}}=0)$ for the sparse reward setting and $(\lambda_{\text{sparse}}=0,\lambda_{\text{dense}}=1)$ for the dense reward setting. ## 3 Experiments We now present a series of experiments designed to evaluate post-training LLM on chess with RLVR. Further details regarding experiment setup are provided in Appendix B. ### 3.1 RL Fine-tuning <details> <summary>fig/chess_rl_training.png Details</summary>  ### Visual Description ## Line Graphs: Lichess Puzzle Accuracy vs Training Steps for Different Models ### Overview Three line graphs compare the performance of three language models (Qwen2.5-3B, Qwen2.5-7B, Llama3.1-8B) across three training strategies: dense reward (blue), sparse reward (gray), and action SFT (red dotted line). The y-axis measures "Lichess Puzzle Accuracy" (0-0.35 range), and the x-axis tracks "Training Step" (0-150). All graphs show distinct trends in how each model improves with training. ### Components/Axes - **X-axis**: Training Step (0-150, linear scale) - **Y-axis**: Lichess Puzzle Accuracy (0-0.35, linear scale) - **Legends**: - Dense reward (solid blue) - Sparse reward (solid gray) - Action SFT (dashed red) - **Graph Titles**: - Qwen2.5-3B - Qwen2.5-7B - Llama3.1-8B ### Detailed Analysis #### Qwen2.5-3B - **Dense reward**: Starts at ~0.005, rises sharply to ~0.24 by step 150 (slope: ~0.0016 per step). - **Sparse reward**: Remains flat at ~0.005 throughout. - **Action SFT**: Flat line at ~0.16. #### Qwen2.5-7B - **Dense reward**: Starts at ~0.005, rises to ~0.28 by step 150 (slope: ~0.0018 per step). - **Sparse reward**: Starts at ~0.005, rises to ~0.29 by step 150 (slope: ~0.0019 per step). - **Action SFT**: Flat line at ~0.19. #### Llama3.1-8B - **Dense reward**: Spikes to ~0.30 by step 30, then fluctuates between ~0.28-0.32. - **Sparse reward**: Flat line at ~0.005. - **Action SFT**: Flat line at ~0.17. ### Key Observations 1. **Dense reward** consistently outperforms other strategies in Qwen models, with Qwen2.5-7B achieving the highest accuracy (~0.28). 2. **Sparse reward** underperforms in Qwen2.5-3B but matches dense reward in Qwen2.5-7B. 3. **Action SFT** acts as a static benchmark (~0.16-0.19) across all models. 4. Llama3.1-8B's dense reward shows volatility after step 30, suggesting potential overfitting or instability. ### Interpretation - **Training Efficiency**: Dense reward strategies scale better with model size (Qwen2.5-7B outperforms Qwen2.5-3B by 16.7%). - **Reward Design**: Sparse reward's effectiveness depends on model capacity (matches dense reward in 7B but not 3B). - **Action SFT Limitation**: All models plateau below action SFT benchmarks, suggesting SFT alone may not capture complex puzzle-solving dynamics. - **Llama3.1 Anomaly**: The post-step-30 volatility in dense reward could indicate reward hacking or insufficient regularization. The data suggests that dense reward strategies are most effective for larger models, while sparse rewards require careful tuning to avoid underperformance. Action SFT serves as a useful baseline but may need augmentation with reward-based methods for optimal results. </details> Figure 2: Evaluation performance comparison of RL fine-tuned models. We fine-tuned Qwen2.5-3B, Qwen2.5-7B and Llama3.1-8B on our Lichess position-action pair dataset $\mathcal{D}$ . For each model, we trained three variants: one with sparse rewards, one with dense expert guidance, and one baseline variant fine-tuned directly via supervised learning (SFT) on optimal actions, without RL. The supervised baseline helps evaluate performance when no explicit reasoning is involved. Figure 2 shows that dense reward models generally outperform sparse reward variants, with sparse rewards completely failing for Qwen2.5-3B and Llama3.1-8B. Dense reward models also surpass SFT baselines, demonstrating the value of reasoning over direct action prediction. However, all models plateau around 25-30% puzzle accuracy, well below expert performance (1800 ELO models achieve 66.5% (Ruoss et al., 2024)). Qualitative analysis (Appendix D) reveals that while models show structured reasoning and systematic move exploration, they lack strategic coherence and long-term evaluation. In summary, although RL improves tactical reasoning particularly with expert-guided feedback, achieving expert-level strategic chess understanding from scalar rewards remains challenging for LLMs. <details> <summary>fig/chess_reasoning_training.png Details</summary>  ### Visual Description ## Line Graphs: Lichess Puzzle Accuracy vs. Training Steps for Qwen2.5-7B and Llama3.1-8B ### Overview The image contains two side-by-side line graphs comparing the performance of two AI models (Qwen2.5-7B and Llama3.1-8B) during training. Each graph tracks "Lichess Puzzle Accuracy" (y-axis) against "Training Step" (x-axis, 0–150). Two data series are shown per model: - **Blue line**: Performance with Reasoning SFT (Supervised Fine-Tuning) - **Gray line**: Performance without Reasoning SFT ### Components/Axes - **X-axis (Training Step)**: - Range: 0 to 150 (increments of 30) - Labels: "Training Step" - **Y-axis (Lichess Puzzle Acc)**: - Range: 0.00 to 0.30 (increments of 0.05) - Labels: "Lichess Puzzle Acc" - **Legends**: - Positioned in the bottom-left corner of each graph. - Blue: "w/ Reasoning SFT" - Gray: "w/o Reasoning SFT" ### Detailed Analysis #### Qwen2.5-7B (Left Graph) - **Blue line (w/ Reasoning SFT)**: - Starts at ~0.20 (step 0) and increases steadily to ~0.29 (step 150). - Slope: Gradual upward trend with minimal fluctuations. - **Gray line (w/o Reasoning SFT)**: - Starts at 0.00 (step 0) and rises sharply to ~0.25 (step 60), then plateaus. - Slope: Steep initial increase, followed by a plateau. #### Llama3.1-8B (Right Graph) - **Blue line (w/ Reasoning SFT)**: - Starts at ~0.20 (step 0) and increases to ~0.28 (step 150). - Slope: Steady upward trend with minor fluctuations. - **Gray line (w/o Reasoning SFT)**: - Starts at 0.00 (step 0) and spikes to ~0.30 (step 30), then fluctuates between ~0.28–0.30. - Slope: Rapid initial rise, followed by volatility. ### Key Observations 1. **Performance Trends**: - Both models show improved performance with Reasoning SFT (blue lines outperform gray lines initially). - Qwen2.5-7B’s gray line converges with the blue line by step 150 (~0.29 vs. ~0.29). - Llama3.1-8B’s gray line surpasses the blue line (~0.30 vs. ~0.28) but exhibits instability. 2. **Model Differences**: - Llama3.1-8B achieves higher peak accuracy (0.30) but with greater variability. - Qwen2.5-7B demonstrates more stable convergence between SFT and non-SFT approaches. 3. **Anomalies**: - Llama3.1-8B’s gray line shows a sharp dip to ~0.28 at step 60, suggesting potential overfitting or instability. ### Interpretation The data suggests that **Reasoning SFT improves performance** for both models, but the impact varies: - **Qwen2.5-7B**: SFT provides a consistent boost, with non-SFT performance catching up over time. - **Llama3.1-8B**: SFT yields higher initial gains, but non-SFT performance eventually exceeds SFT, possibly due to overfitting or architectural differences. The graphs highlight the trade-off between stability (Qwen) and peak performance (Llama), with Llama’s volatility raising questions about the reliability of non-SFT training. Further investigation into training dynamics and model architecture could clarify these trends. </details> Figure 3: Evaluation performance of models trained with reasoning SFT followed by RL fine-tuning. | Qwen2.5-3B Qwen2.5-3B-It Qwen2.5-7B | 0.0% 0.0% 0.0% | 35.8% 53.7% 42.7% | | --- | --- | --- | | Qwen2.5-7B-It | 0.0% | 52.0% | | Llama3.1-8B | 0.0% | 12.7% | | Llama3.1-8B-It | 0.0% | 52.4% | Table 1: LLM performance on chess knowledge diagnostic tasks ### 3.2 Reasoning SFT To probe the cause of the RL plateau, we asked whether domain-specific reasoning traces from advanced reasoning models could enable LLMs better leverage RL signals for chess skill acquisition and overcome the previously observed performance ceiling. To do so, we curated a reasoning SFT corpus of 1k high-quality reasoning traces generated by OpenAI o3, featuring candidate move evaluation, tactical assessment, and strategic justifications (See Appendix B.2 for details). After performing SFT on Qwen2.5-7B and Llama3.1-8B with this corpus, we applied our GRPO RL pipeline with expert critic $Q_{\theta}(s,a)$ feedback. While LLMs produced markedly more comprehensive reasoning traces after reasoning SFT, Figure 3 shows that they disappointingly exhibited similar puzzle accuracy plateaus when subsequently trained with RL. In fact, Llama3.1-8B’s performance actually declined relative to its non-SFT baseline. These results raise questions about whether the reason behind the chess performance plateau stems not from inadequate reasoning abilities, but from insufficient chess knowledge. Without extensive domain knowledge from pre-training, RL alone may not provide the chess-specific understanding that strategic play demands. ### 3.3 Failure Analysis To study whether the performance limitation of RLVR for chess stems from inadequate chess domain knowledge, we evaluate LLMs on two diagnostic tasks that require basic and essential but non-trivial understanding of chess rules. Board–state comprehension: Given a FEN string and a legal sequence of one to five SAN moves, the model predicts the resulting position in FEN. The model must have a faithful internal simulator of chess rules to be successful in this task. MATE puzzle: Each instance in the MATE dataset (Wang et al., 2024) presents a mid-game position together with two candidate moves. The model must identify the superior move through tactical evaluation. As summarized in Table 1, all tested LLMs show poor performance on both diagnostics, confirming inadequate internal models of chess state transitions and tactics. We tested both Base and Instruct variants—Instruct models for better formatting compliance, though they share the same underlying knowledge as their Base counterparts. This shortfall strongly supports our earlier hypothesis: current models lack fundamental understanding of chess, preventing them from developing high performance through reward optimization alone. Since models cannot reliably track game states or recognize elementary tactics, reward optimization fails to develop high-quality strategic reasoning and elicit expert-level play. ## 4 Related Work #### RLVR for LLM reasoning. RLVR (Lambert et al., 2024) has emerged as a powerful paradigm to elicit LLM reasoning for tasks with deterministic evaluation criteria. Notably, in mathematical domains, simple rule-based rewards have proven surprisingly effective without requiring complex reward models (Guo et al., 2025; Yu et al., 2025; Wang et al., 2025), achieving performance comparable to, or even surpassing, that of humans on mathematical olympiads such as AIME. Furthermore, some works have extended the application of RLVR to other domains as well (Liu et al., 2025b; Zhang et al., 2025a; Gurung & Lapata, 2025). Despite these successes, fundamental questions about RLVR’s success remain underexplored. Recent studies suggest that its success often originates from abundant knowledge already present in base models rather than being genuinely learned through RL (Liu et al., 2025a; Zhao et al., 2025b; Li et al., 2025; Shao et al., 2025; Yue et al., 2025). A systematic investigation across models, tasks, RL algorithms, and scale remains necessary to fully understand LLM capabilities and its limitations in developing genuine reasoning abilities through RL. #### LLMs in chess. Recent work explores how LLMs can tackle chess tasks, complementing traditional chess engines. While specialized networks like AlphaZero (Silver et al., 2017) achieve superhuman performance via self-play and search, they lack human-readable insight. Conversely, LLMs can explain moves in natural language but often lack precise strategic understanding, leading to hallucinations or illegal moves. To bridge this gap, researchers have integrated chess engines with LLMs for fluent move commentary (Kim et al., 2025) and developed chess-centric models like ChessGPT (Feng et al., 2023) through domain-specific pre-training. Fine-tuning on expert rationales from the MATE dataset (Wang et al., 2024) enables Llama models to outperform GPT-4 and Claude at binary move selection. Most notably, ChessLLM (Zhang et al., 2025b) demonstrates that an LLM fine-tuned on 20B tokens of complete games can play chess at an expert level, attaining an Elo of 1788 without external search; however FEN-centric training risks overfitting and degrading natural language responses into fragmented notations. Despite these advances, existing approaches primarily rely on supervised learning from human games or engine guidance rather than autonomous strategic development. Whether LLMs can develop genuine chess reasoning through RL remains largely unexplored, particularly in understanding if such learning produces transferable strategic insights or merely pattern memorization. Our work addresses this gap using chess as a controlled testbed for studying autonomous strategic development in LLMs. ## 5 Conclusion & Limitations We investigated whether LLMs can develop strategic reasoning through RLVR on chess, introducing a novel approach that utilizes dense rewards from pretrained action-value networks. While dense rewards often outperform sparse ones, all models plateau well below human experts, revealing fundamental limitations in current post-training approaches. Surprisingly, despite producing more structured reasoning when trained on advanced traces from OpenAI o3, models still yield similar performance plateaus when subsequently trained with RL. Our failure analysis reveals a potential root cause: current LLMs demonstrate inadequate internal chess knowledge. These results raise speculation that RL cannot overcome impoverished domain knowledge. While RL excels at optimizing behavior toward rewards, LLMs cannot learn de novo the foundational knowledge necessary for strategic thinking when absent from pretraining. We hypothesize that prior RLVR successes in math domains stem from rich pretraining exposure, unlike chess which receives minimal coverage. Our findings align with recent work showing that RL only amplifies existing capabilities (Li et al., 2025; Liu et al., 2025a; Yue et al., 2025). For strategic reasoning in new domains, adequate domain knowledge during pretraining is essential—post-training RL alone is insufficient. ## References - Edwards (1994) Steven J. Edwards. Standard: Portable game notation specification and implementation guide, 1994. URL https://ia802908.us.archive.org/26/items/pgn-standard-1994-03-12/PGN_standard_1994-03-12.txt. - Feng et al. (2023) Xidong Feng, Yicheng Luo, Ziyan Wang, Hongrui Tang, Mengyue Yang, Kun Shao, David Mguni, Yali Du, and Jun Wang. 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Llamafactory: Unified efficient fine-tuning of 100+ language models. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 3: System Demonstrations), Bangkok, Thailand, 2024. Association for Computational Linguistics. URL http://arxiv.org/abs/2403.13372. ## Appendix A Input prompt format For our main experiments, we employ the input prompt template shown in Figure 4 for both training and evaluation. This prompt format consists of several key components: (i) the current board state represented in Forsyth–Edwards Notation (FEN), (ii) the complete set of legal moves available from the current position written in Standard Algebraic Notation (SAN), and (iii) structured instructions that guide the model to produce reasoning enclosed in <think> tags followed by the chosen move in <answer> tags. Additionally, the prompt includes a concise reminder of fundamental chess rules to support the model’s reasoning process. This comprehensive prompt design ensures that the model has access to all necessary information about the current game state while maintaining a consistent format that facilitates both learning during training and reliable evaluation during testing. Input prompt template A conversation between User and Assistant. The User asks the best move to make for a given chess board state, and the Assistant solves it. The Assistant is a professional chess player who first thinks about the reasoning process in the mind and then provides the User with the answer. The Assistant’s reasoning process and answer must be enclosed within <think> </think> and <answer> </answer> tags, respectively. The reasoning process should describe how the Assistant analyzes the position and decide on the best move, including: - A strategic evaluation of the position. - A comparison of key candidate moves. - For each candidate, consider the opponent’s likely response and outcome. - Conclude with a clear justification for your final choice. The answer must be in SAN notation, strictly using the moving piece and the destination square (e.g., Nf3, Rxf2, c5). Reminder of chess rules: - Bishops move diagonally. - Rooks move horizontally or vertically. - Knights jump in an L-shape. - Queens combine rook and bishop movement. - Kings move one square in any direction. - Pawns move forward, capture diagonally, and can promote. User: The current FEN string is <fen> and legal moves are <san_1> <san_2> …<san_L>. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Figure 4: Input prompt format for chess reasoning tasks with FEN for board state and SAN notation for move action. This prompt structure was used throughout our main studies. ## Appendix B Experiment Setup ### B.1 Lichess Puzzle Dataset #### Training data collection. We collect Lichess puzzles spanning solver ratings from 200 to 2800 Elo from the Lichess Puzzle Database. Each puzzle contains an initial board position and a sequence of moves that leads to a tactical victory. To create position-action training pairs, we decompose each puzzle’s solution trajectory, where a puzzle with $T$ moves generates $T$ individual training samples. This decomposition process yields a total of 19.2k training samples for our dataset $\mathcal{D}$ . #### Evaluation protocol. We evaluate chess understanding through puzzle-solving capability using a held-out evaluation dataset of 10K Lichess puzzles across various difficulty levels from (Ruoss et al., 2024). Our evaluation metric is puzzle accuracy, defined as the percentage of puzzles where the model’s complete action sequence exactly matches the ground-truth solution sequence. This strict evaluation criterion ensures that models must not only identify good moves but solve puzzles entirely and correctly, providing a comprehensive assessment of chess reasoning ability. To ensure consistency, we use the same prompt template during evaluation as in training. ### B.2 Reasoning Trace Collection We used OpenAI’s o3 model (reasoning: high, summary: detailed) to collect sophisticated chess reasoning data samples for our reasoning SFT experiments. We provide an example of the synthetic o3 reasoning trajectory in Figure 23, 24. From o3 reasoning traces, we observed fewer hallucinations about spatial positioning and reasoning compared to our base models, while evaluating strategic positions in terms of both breadth and width. To build the SFT corpus, we sample 1,000 chess problems from our larger collection, balancing them evenly across the entire ELO range. We create each SFT example by concatenating o3 reasoning summaries and final generation to each problem. This yields a high-quality chess SFT dataset, which we expected would strengthen our model’s strategic reasoning capability by distilling structured rationale before subsequent RL training. We include the token length distribution of the SFT data in Figure 5. <details> <summary>fig/o3_reasoning_token_distributions_qwen_vs_llama.png Details</summary>  ### Visual Description ## Histograms: OpenAI o3 Reasoning Data Token Length Distributions ### Overview The image contains two side-by-side histograms comparing token length distributions for OpenAI o3 reasoning data processed by two tokenizers: Qwen2.5 (blue) and Llama3.1 (green). Each histogram shows frequency distributions of token counts, with red dashed lines indicating mean values. ### Components/Axes - **Title**: "OpenAI o3 Reasoning Data Token Length Distributions" - **Left Histogram**: - Subtitle: "Qwen2.5 Tokenizer" - Legend: Red dashed line labeled "Mean: 974.0" - **Right Histogram**: - Subtitle: "Llama3.1 Tokenizer" - Legend: Red dashed line labeled "Mean: 965.4" - **Axes**: - **X-axis (Horizontal)**: "Token Count" (range: 400–1600, increments of 200) - **Y-axis (Vertical)**: "Frequency" (range: 0–175, increments of 25) ### Detailed Analysis - **Qwen2.5 Tokenizer (Blue)**: - Peak frequency occurs at ~1000 tokens (~175 occurrences). - Distribution is slightly right-skewed, with frequencies tapering off gradually toward higher token counts. - Mean (974.0) is marked by a red dashed vertical line. - **Llama3.1 Tokenizer (Green)**: - Peak frequency occurs at ~1000 tokens (~175 occurrences). - Distribution is more symmetric, with a sharper decline at higher token counts. - Mean (965.4) is marked by a red dashed vertical line. ### Key Observations 1. Both tokenizers exhibit similar peak frequencies (~175) at ~1000 tokens, suggesting this is the most common token length for reasoning tasks. 2. Qwen2.5 has a marginally higher mean (974.0 vs. 965.4), indicating slightly longer average token lengths. 3. Llama3.1’s distribution is more symmetric, while Qwen2.5’s is slightly right-skewed. 4. Frequencies decline steadily for both tokenizers beyond 1200 tokens, with minimal occurrences above 1600. ### Interpretation The data suggests that both Qwen2.5 and Llama3.1 tokenizers process reasoning tasks with similar token length distributions, centered around 1000 tokens. The slight difference in means (Qwen2.5: 974.0 vs. Llama3.1: 965.4) implies Qwen2.5 may handle marginally longer sequences on average, potentially offering advantages for tasks requiring extended context. The symmetry of Llama3.1’s distribution suggests more consistent tokenization across tasks, whereas Qwen2.5’s right skew indicates occasional outliers with longer token counts. These differences could reflect architectural optimizations in handling context length or efficiency trade-offs. </details> Figure 5: Token length distribution of o3 reasoning data per each tokenizer type. ### B.3 Training Hyperparameters #### Reinforcement learning. We train our models using the verl (Sheng et al., 2024) framework. We use the same set of hyperparameters to train and evaluate all models when performing RL fine-tuning as shown in Table 2. We fine-tune all models using 4 NVIDIA A100 80GB GPUs. A single training instance takes approximately 14 hours. Table 2: Reinforcement learning hyperparameters. | Hyperparameter Training Configuration Training Steps | Value 150 | | --- | --- | | Optimizer | AdamW | | Learning Rate | 1e-6 | | Gradient Clipping | 1.0 | | Mini-batch Size | 128 | | GRPO Configuration | | | Epochs | 1 | | Sampling Batch Size | 128 | | Number of Rollouts | 8 | | Rollout Temperature | 1.0 | | KL Loss Coefficient | 1e-3 | | Entropy Coefficient | 1e-3 | | Clip Ratio | 0.2 | | Reward Configuration | | | Sparse Coefficient ( $\lambda_{\text{sparse}}$ ) | 1 if use sparse reward 0 otherwise | | Dense Coefficient ( $\lambda_{\text{dense}}$ ) | 1 if use dense reward 0 otherwise | | Format Coefficient ( $\lambda_{\text{fmt}}$ ) | 0.1 | | Language Coefficient ( $\lambda_{\text{lang}}$ ) | 0.1 | #### Supervised Fine-Tuning. We perform supervised fine-tuning (SFT) using Llama-Factory (Zheng et al., 2024) with hyperparameters shown in Table 3. Training loss is computed only on the model outputs, with input prompts masked. We fine-tune all models using 4 NVIDIA A100 80GB GPUs. A single training instance takes approximately 1 hour. Table 3: Supervised fine-tuning hyperparameters. | Number of Samples Epochs Optimizer | 979 10 AdamW | | --- | --- | | Learning Rate | 5e-6 | | Scheduler | Cosine | | Warmup Ratio | 0.1 | | Gradient Clipping | 1.0 | | Mini-batch Size | 32 | ## Appendix C Ablation studies We provide various ablations on our experiments to make sure our experiments are not simply limited to our specific prompt format or reward design. ### C.1 Prompt Formatting We conducted systematic experiments on different prompt formats while performing RL with expert critic rewards as our default reward configuration, examining two key design axes: (i) move action representation and (ii) board state representation. Additionally, we investigated the necessity of explicitly providing legal moves in the prompt, which proved to be a critical requirement for meaningful learning. For move action representations, we tested UCI and SAN notations. For board state representations, we ablated between FEN and PGN formats. #### Legal moves requirement: To investigate the necessity of providing legal moves, we conducted experiments using our baseline FEN board representation with SAN move notation, comparing performance with and without explicit legal move information. As demonstrated in Figure 6, we discovered that LLMs cannot learn anything meaningful when legal moves are not explicitly provided in the prompt. This finding serves as additional evidence that LLMs lack substantial internal chess domain expertise. Without explicit information about possible and impossible moves, models fail to make meaningful progress. <details> <summary>fig/chess_nolegal_prompt_ablation.png Details</summary>  ### Visual Description ## Line Graphs: Lichess Puzzle Accuracy vs. Training Steps for Qwen2.5-7B and Llama3.1-8B ### Overview The image contains two side-by-side line graphs comparing the impact of legal constraints on puzzle-solving accuracy during training for two AI models: Qwen2.5-7B (left) and Llama3.1-8B (right). Both graphs show a blue line ("w/ legal") and a gray line ("w/o legal"), with the x-axis representing training steps (0–150) and the y-axis representing Lichess puzzle accuracy (0–0.30). --- ### Components/Axes - **X-axis (Training Step)**: Ranges from 0 to 150 in increments of 30. - **Y-axis (Lichess Puzzle Acc)**: Ranges from 0.00 to 0.30 in increments of 0.05. - **Legends**: - Blue line: "w/ legal" (with legal constraints). - Gray line: "w/o legal" (without legal constraints). - **Graph Titles**: - Left: "Qwen2.5-7B" - Right: "Llama3.1-8B" --- ### Detailed Analysis #### Qwen2.5-7B (Left Graph) - **Blue Line ("w/ legal")**: - Starts at 0.00 accuracy at step 0. - Rises sharply to ~0.25 accuracy by step 60. - Plateaus near 0.28–0.29 accuracy by step 150. - **Gray Line ("w/o legal")**: - Remains flat at 0.00 accuracy throughout all training steps. #### Llama3.1-8B (Right Graph) - **Blue Line ("w/ legal")**: - Starts at 0.00 accuracy at step 0. - Jumps to ~0.30 accuracy by step 30. - Dips to ~0.25 accuracy at step 60. - Fluctuates between 0.25 and 0.30 accuracy by step 150. - **Gray Line ("w/o legal")**: - Remains flat at 0.00 accuracy throughout all training steps. --- ### Key Observations 1. **Legal Constraints Matter**: Both models show significantly higher accuracy when trained with legal constraints ("w/ legal") compared to without ("w/o legal"). 2. **Qwen2.5-7B**: Exhibits a steady, monotonic increase in accuracy, suggesting stable learning. 3. **Llama3.1-8B**: Shows volatility in accuracy (e.g., dip at step 60), indicating potential instability or overfitting. 4. **No Improvement Without Legal Data**: The gray lines for both models remain at 0.00, emphasizing that legal constraints are critical for performance. --- ### Interpretation - **Impact of Legal Constraints**: The stark contrast between "w/ legal" and "w/o legal" lines underscores the importance of legal frameworks in guiding AI behavior during training. Without these constraints, neither model demonstrates meaningful learning. - **Model Behavior**: - **Qwen2.5-7B**: The smooth upward trend suggests effective integration of legal constraints, leading to consistent improvement. - **Llama3.1-8B**: The fluctuations in accuracy (e.g., the dip at step 60) may indicate challenges in balancing legal constraints with model complexity, possibly due to overfitting or insufficient regularization. - **Practical Implications**: Legal constraints act as a stabilizing force, ensuring that models prioritize ethical or rule-based decision-making. The absence of such constraints results in stagnation, highlighting their role in aligning AI systems with human values. --- ### Spatial Grounding - **Legends**: Positioned in the bottom-left corner of each graph, clearly associating colors with labels. - **Data Points**: Blue lines ("w/ legal") are consistently above gray lines ("w/o legal") across all training steps. - **Axis Alignment**: Both graphs share identical axis scales and labels, enabling direct comparison. --- ### Content Details - **Qwen2.5-7B**: - Step 0: 0.00 (w/ legal), 0.00 (w/o legal). - Step 60: ~0.25 (w/ legal), 0.00 (w/o legal). - Step 150: ~0.29 (w/ legal), 0.00 (w/o legal). - **Llama3.1-8B**: - Step 0: 0.00 (w/ legal), 0.00 (w/o legal). - Step 30: ~0.30 (w/ legal), 0.00 (w/o legal). - Step 60: ~0.25 (w/ legal), 0.00 (w/o legal). - Step 150: ~0.29 (w/ legal), 0.00 (w/o legal). --- ### Final Notes The graphs demonstrate that legal constraints are not merely optional but foundational for achieving meaningful performance in AI systems. The differences in model behavior (steady vs. volatile) suggest that architectural or training differences between Qwen and Llama may influence how effectively legal constraints are leveraged. </details> Figure 6: Evaluation performance comparison with and without legal moves in the input prompt for Qwen2.5-7B and Llama3.1-8B. #### Move notation impact: As shown in Figure 7, we found that while SAN notation enables meaningful learning, UCI notation results in substantially degraded performance. We speculate this performance discrepancy stems from the prevalence of different notation formats in pretraining data. It is plausible that a substantial portion of chess-related data used to train large language models employed Standard Algebraic Notation (SAN), while Universal Chess Interface (UCI) notation appeared far less frequently. Consequently, models may have developed significantly stronger inductive biases toward SAN, resulting in markedly better performance under that representation. <details> <summary>fig/chess_action_prompt_ablation.png Details</summary>  ### Visual Description ## Line Graphs: Qwen2.5-7B and Llama3.1-8B Performance Comparison ### Overview The image contains two side-by-side line graphs comparing the performance of two AI models (Qwen2.5-7B and Llama3.1-8B) across training steps. Each graph tracks two metrics: "SAN" (blue line) and "UCI" (gray line), measured as "Lichess Puzzle Acc" (accuracy) on a scale from 0.00 to 0.30. Training steps range from 0 to 150 on the x-axis. --- ### Components/Axes - **Left Graph (Qwen2.5-7B)**: - **Title**: "Qwen2.5-7B" - **X-axis**: "Training Step" (0–150, increments of 30) - **Y-axis**: "Lichess Puzzle Acc" (0.00–0.30, increments of 0.05) - **Legend**: Located in the bottom-right corner, labeled "SAN" (blue) and "UCI" (gray). - **Lines**: - **Blue (SAN)**: Starts near 0.01, rises sharply to ~0.29 by step 150. - **Gray (UCI)**: Starts near 0.005, peaks at ~0.03 around step 40, then declines to ~0.02. - **Right Graph (Llama3.1-8B)**: - **Title**: "Llama3.1-8B" - **X-axis**: "Training Step" (0–150, increments of 30) - **Y-axis**: "Lichess Puzzle Acc" (0.00–0.30, increments of 0.05) - **Legend**: Located in the bottom-right corner, labeled "SAN" (blue) and "UCI" (gray). - **Lines**: - **Blue (SAN)**: Starts near 0.01, rises sharply to ~0.30 by step 60, fluctuates between 0.28–0.30 by step 150. - **Gray (UCI)**: Starts near 0.005, peaks at ~0.025 around step 20, then declines to ~0.015. --- ### Detailed Analysis #### Qwen2.5-7B - **SAN (Blue)**: - Initial value: ~0.01 at step 0. - Rapid increase to ~0.25 by step 60. - Gradual plateau to ~0.29 by step 150. - **UCI (Gray)**: - Initial value: ~0.005 at step 0. - Peaks at ~0.03 around step 40. - Declines to ~0.02 by step 150. #### Llama3.1-8B - **SAN (Blue)**: - Initial value: ~0.01 at step 0. - Sharp rise to ~0.25 by step 30. - Peaks at ~0.30 by step 60. - Fluctuates between 0.28–0.30 by step 150. - **UCI (Gray)**: - Initial value: ~0.005 at step 0. - Peaks at ~0.025 around step 20. - Declines to ~0.015 by step 150. --- ### Key Observations 1. **SAN Performance**: - Both models show a steep initial improvement in SAN accuracy, but Llama3.1-8B achieves a higher peak (~0.30 vs. ~0.29). - Qwen2.5-7B’s SAN plateaus earlier (~step 60), while Llama3.1-8B’s SAN remains volatile after step 60. 2. **UCI Performance**: - UCI accuracy peaks early in both models (~step 20–40) and declines sharply afterward. - Llama3.1-8B’s UCI peak is higher (~0.025 vs. ~0.03), but its decline is more pronounced. 3. **Model Comparison**: - Llama3.1-8B outperforms Qwen2.5-7B in SAN accuracy, suggesting better scalability or efficiency. - Both models’ UCI metrics indicate potential overfitting or inefficiency in later training stages. --- ### Interpretation - **SAN Trends**: The sharp rise in SAN accuracy for both models suggests effective learning in early training steps. Llama3.1-8B’s higher peak implies superior performance, possibly due to its larger parameter count (8B vs. 7B). The plateau in Qwen2.5-7B may reflect a learning limit, while Llama3.1-8B’s fluctuations could indicate instability or adaptation to complex patterns. - **UCI Trends**: The early peak and subsequent decline in UCI accuracy for both models suggest that UCI metrics may measure short-term gains or overfitting. The steeper decline in Llama3.1-8B’s UCI could indicate greater sensitivity to training noise or complexity. - **Model Differences**: Llama3.1-8B’s larger size correlates with higher SAN performance but also greater volatility, highlighting trade-offs between scale and stability. Qwen2.5-7B’s smoother plateau might indicate more robust training dynamics. --- ### Spatial Grounding - **Legends**: Both legends are positioned in the bottom-right corner of their respective graphs, ensuring clarity without obstructing data. - **Line Colors**: Blue (SAN) and gray (UCI) are consistently used across both graphs, avoiding confusion. ### Content Details - **Qwen2.5-7B SAN**: 0.01 → 0.25 (step 60) → 0.29 (step 150). - **Qwen2.5-7B UCI**: 0.005 → 0.03 (step 40) → 0.02 (step 150). - **Llama3.1-8B SAN**: 0.01 → 0.25 (step 30) → 0.30 (step 60) → 0.28–0.30 (step 150). - **Llama3.1-8B UCI**: 0.005 → 0.025 (step 20) → 0.015 (step 150). --- ### Final Notes The graphs emphasize the importance of training step efficiency and model architecture in achieving high puzzle-solving accuracy. Llama3.1-8B’s superior SAN performance suggests it may be better suited for tasks requiring rapid learning, while Qwen2.5-7B’s stability could be advantageous in scenarios prioritizing consistency. </details> Figure 7: Evaluation performance comparison across move notations (SAN vs. UCI) for Qwen2.5-7B and Llama3.1-8B. #### Board state representation: We also ablated different board state representations, noting key differences between FEN and PGN. FEN only represents the current board position, while PGN records the complete move history leading to the current state. However, PGN alone lacks explicit current board state representation. As illustrated in Figure 8, despite these structural differences, we found that board representation choice was not critical—all variants achieved similar performance. Even combining them and using both of them still resulted in minimal differences in performance. <details> <summary>fig/chess_board_state_prompt_ablation.png Details</summary>  ### Visual Description ## Line Graphs: Lichess Puzzle Accuracy vs Training Steps for Qwen2.5-7B and Llama3.1-8B Models ### Overview The image contains two line graphs comparing the performance of three methods (SAN, PGN, FEN+PGN) across two language models (Qwen2.5-7B and Llama3.1-8B) over 150 training steps. The y-axis measures Lichess Puzzle Accuracy (0-0.35), and the x-axis represents training steps (0-150). Each graph uses distinct color coding for methods: blue (SAN), gray (PGN), and red (FEN+PGN). ### Components/Axes - **Left Graph (Qwen2.5-7B)**: - **X-axis**: Training Step (0-150, linear scale) - **Y-axis**: Lichess Puzzle Accuracy (0-0.30, linear scale) - **Legend**: Bottom-left corner (blue=SAN, gray=PGN, red=FEN+PGN) - **Right Graph (Llama3.1-8B)**: - **X-axis**: Training Step (0-150, linear scale) - **Y-axis**: Lichess Puzzle Accuracy (0-0.35, linear scale) - **Legend**: Top-right corner (blue=SAN, gray=PGN, red=FEN+PGN) ### Detailed Analysis #### Qwen2.5-7B Graph - **SAN (blue)**: - Starts at ~0.01 (step 0), rises sharply to ~0.29 (step 150). - Steady upward trend with minimal fluctuations. - **PGN (gray)**: - Begins at ~0.02 (step 0), peaks at ~0.28 (step 150). - Slight dip at step 60 (~0.24) before recovery. - **FEN+PGN (red)**: - Starts at ~0.01 (step 0), reaches ~0.30 (step 150). - Consistent growth with minor plateaus. #### Llama3.1-8B Graph - **SAN (blue)**: - Begins at ~0.01 (step 0), surges to ~0.34 (step 150). - Sharp rise at step 30 (~0.25), with minor dips at steps 60 (~0.28) and 90 (~0.30). - **PGN (gray)**: - Starts at ~0.02 (step 0), peaks at ~0.32 (step 120), then drops to ~0.26 (step 150). - Volatile trend with a significant dip at step 60 (~0.25). - **FEN+PGN (red)**: - Begins at ~0.01 (step 0), stabilizes at ~0.30 (step 150). - Peaks at ~0.33 (step 60), followed by gradual decline. ### Key Observations 1. **Performance Trends**: - SAN consistently outperforms other methods in Llama3.1-8B, especially after step 30. - PGN shows instability in Llama3.1-8B, with a notable drop after step 120. - FEN+PGN maintains steady performance across both models but rarely leads. 2. **Model-Specific Behavior**: - Qwen2.5-7B exhibits smoother convergence for all methods. - Llama3.1-8B shows higher volatility, particularly for PGN. 3. **Method Comparison**: - SAN achieves the highest accuracy in Llama3.1-8B (0.34 vs. 0.30 for FEN+PGN). - PGN underperforms in Llama3.1-8B despite strong initial growth. ### Interpretation The data suggests that **SAN** is the most effective method for Llama3.1-8B, likely due to its rapid adaptation to the model's complexity. PGN's volatility in Llama3.1-8B may indicate overfitting or sensitivity to training dynamics. FEN+PGN's consistency across models highlights its reliability but limited peak performance. For Qwen2.5-7B, all methods converge closely, suggesting the model's architecture may inherently limit method differentiation. The stark contrast between Llama3.1-8B and Qwen2.5-7B performance underscores the importance of method-model compatibility in training strategies. </details> Figure 8: Evaluation performance comparison across board state representations (FEN vs. PGN vs. FEN+PGN) for Qwen2.5-7B and Llama3.1-8B. #### Overall: Based on these ablations, we adopted FEN with SAN notation with legal moves for their simplicity in terms of token count and input prompt length, enabling more efficient training while maintaining optimal performance. ### C.2 Dense reward When providing dense rewards with our expert critic network $Q_{\theta}(s,a)$ , we explored two methods for extracting reward signals from the model’s win-rate predictions: direct win-rate feedback and normalized rank feedback. #### Direct win-rate feedback. As described in Equation (• ‣ 2.2), we use the expert critic’s predicted win-rate directly as the reward signal. For example, if the critic predicts a 67% win probability for a specific move at a given position, we provide $r_{\text{dense}}=0.67$ as the reward. #### Normalized rank feedback. Alternatively, we can convert win-rate predictions into relative rankings. Given the set of legal moves $\{a_{1},a_{2},\ldots,a_{L}\}$ at state $s$ , we first obtain win-rate predictions $\{Q_{\theta}(s,a_{1}),Q_{\theta}(s,a_{2}),\ldots,Q_{\theta}(s,a_{L})\}$ , then rank these moves by their predicted win-rates. The normalized rank reward is defined as: $$ r_{\text{dense}}=\frac{\text{rank}(a_{t})-1}{L-1}\in[0,1] \tag{1} $$ where $\text{rank}(a_{t})$ denotes the rank of the selected action $a_{t}$ (with rank 1 being the highest win-rate and rank $L$ being the lowest win-rate), and $L$ is the number of legal moves. <details> <summary>fig/chess_reward_ablation.png Details</summary>  ### Visual Description ## Line Graphs: Qwen2.5-7B and Llama3.1-8B Performance Comparison ### Overview The image contains two side-by-side line graphs comparing the performance of two AI models (Qwen2.5-7B and Llama3.1-8B) across training steps. Both graphs track two metrics: **Win Rate** (blue line) and **Normalized Rank** (gray line) on the y-axis (Lichess Puzzle Accuracy) against **Training Steps** (0–150) on the x-axis. --- ### Components/Axes - **X-Axis**: Training Steps (0–150, linear scale). - **Y-Axis**: Lichess Puzzle Accuracy (0.00–0.30, linear scale). - **Legends**: - Blue line = Win Rate - Gray line = Normalized Rank - **Graph Titles**: - Left: "Qwen2.5-7B" - Right: "Llama3.1-8B" --- ### Detailed Analysis #### Qwen2.5-7B (Left Graph) - **Win Rate (Blue Line)**: - Starts at ~0.00 at 0 steps. - Increases steadily, reaching ~0.28 at 150 steps. - Slope: Gradual upward trend with no plateaus. - **Normalized Rank (Gray Line)**: - Starts at ~0.01 at 0 steps. - Rises sharply initially, then plateaus slightly above the blue line. - Peaks at ~0.29 at 150 steps. - **Key Relationship**: The gray line remains consistently ~0.01–0.02 higher than the blue line throughout. #### Llama3.1-8B (Right Graph) - **Win Rate (Blue Line)**: - Starts at ~0.00 at 0 steps. - Sharp upward spike after ~30 steps, reaching ~0.30 at 150 steps. - Temporary dip to ~0.26 at ~90 steps, then recovery. - **Normalized Rank (Gray Line)**: - Starts at ~0.01 at 0 steps. - Gradual rise, peaking at ~0.28 at 150 steps. - Crossed by the blue line after ~60 steps. - **Key Relationship**: Blue line overtakes gray line after ~60 steps, indicating Win Rate surpasses Normalized Rank. --- ### Key Observations 1. **Qwen2.5-7B**: - Win Rate and Normalized Rank trends are closely aligned but never intersect. - Both metrics plateau near 0.28–0.29 by 150 steps. 2. **Llama3.1-8B**: - Win Rate accelerates faster than Normalized Rank, overtaking it after ~60 steps. - Temporary dip in Win Rate at ~90 steps suggests instability or optimization challenges. 3. **General Trend**: - Both models show improvement with training steps, but Llama3.1-8B demonstrates sharper gains in Win Rate. --- ### Interpretation - **Performance Insights**: - Llama3.1-8B’s faster Win Rate growth suggests superior puzzle-solving efficiency in later training phases. - Qwen2.5-7B’s stable but slower progression indicates consistent but less aggressive learning. - **Anomalies**: - Llama3.1-8B’s Win Rate dip at ~90 steps may reflect overfitting or resource constraints during training. - **Implications**: - For applications prioritizing rapid performance gains, Llama3.1-8B may be preferable. - Qwen2.5-7B’s stability could be advantageous for tasks requiring consistent, incremental improvement. --- ### Spatial Grounding & Validation - Legends are positioned at the bottom-left of each graph, matching line colors (blue/gray). - Axis labels and titles are clearly separated from data regions. - All numerical values align with visual trends (e.g., Llama’s blue line overtaking gray line post-60 steps). </details> Figure 9: Evaluation performance comparison between direct win-rate feedback and normalized rank feedback as dense rewards for Qwen2.5-7B and Llama3.1-8B. We compared these two approaches across our model variants and found distinct learning dynamics with similar final outcomes (Figure 9). For Qwen2.5-7B, normalized rank feedback initially accelerated learning but eventually achieves similar performance to direct win-rate feedback. For Llama3.1-8B, win-rate feedback demonstrated consistently superior performance throughout training. Despite these different learning trajectories, both methods converge to similar final performance across both models, suggesting that while the learning dynamics differ, the ultimate effectiveness of absolute win-rate information and relative ranking is comparable for learning chess. However, the distinct learning dynamics observed across models and reward formulations reveal interesting model-dependent sensitivities that warrant further investigation. Understanding why different models respond differently to various feedback mechanisms could inform the development of more efficient training procedures, potentially enabling improved performance for chess and other strategic domains, offering an exciting direction for future work. ### C.3 Base model vs. Instruct model <details> <summary>fig/chess_base_instruct_single_comparison.png Details</summary>  ### Visual Description ## Line Graph: Qwen2.5-7B Model Performance Over Training Steps ### Overview The graph compares the Lichess Puzzle Accuracy (Acc) of two models ("Base" and "Instruct") during training. Both models show increasing performance over time, with "Instruct" consistently outperforming "Base" across all training steps. ### Components/Axes - **Title**: "Qwen2.5-7B" - **X-axis**: "Training Step" (0 to 150, increments of 30) - **Y-axis**: "Lichess Puzzle Accuracy (Acc)" (0.00 to 0.30, increments of 0.05) - **Legend**: Located in the bottom-right corner, with: - **Blue line**: "Base" model - **Gray line**: "Instruct" model ### Detailed Analysis 1. **Base Model (Blue Line)**: - Starts at **0.00** at 0 training steps. - Gradual increase to **~0.29** at 150 steps. - Key points: - 30 steps: **~0.12** - 60 steps: **~0.23** - 90 steps: **~0.27** - 120 steps: **~0.28** - 150 steps: **~0.29** 2. **Instruct Model (Gray Line)**: - Starts at **~0.02** at 0 training steps. - Steeper ascent to **~0.32** at 150 steps. - Key points: - 30 steps: **~0.20** - 60 steps: **~0.28** - 90 steps: **~0.30** - 120 steps: **~0.31** - 150 steps: **~0.32** ### Key Observations - **Performance Gap**: "Instruct" maintains a **~0.03–0.04 accuracy advantage** over "Base" throughout training. - **Convergence**: The gap narrows slightly in later stages (e.g., from **0.09** at 30 steps to **0.03** at 150 steps). - **Plateauing**: Both models show diminishing returns after ~90 steps, with "Instruct" reaching a higher plateau. ### Interpretation The data suggests that the "Instruct" model's architecture or training methodology inherently enables superior performance in Lichess puzzles compared to the "Base" model. The narrowing gap at later stages implies that while "Base" improves with training, "Instruct" benefits from a more efficient learning process or better initialization. This could reflect differences in: - **Training data curation** (e.g., "Instruct" may use more targeted examples). - **Model design** (e.g., "Instruct" might incorporate reinforcement learning or human feedback). - **Regularization techniques** (e.g., "Instruct" could avoid overfitting more effectively). The consistent outperformance of "Instruct" highlights the importance of architectural choices in specialized tasks like chess puzzle solving. Further analysis could explore whether this trend holds across other domains or if it is specific to chess-related reasoning. </details> Figure 10: Training performance comparison between Qwen2.5-7B Base and Instruct model. Our failure analysis in Section 3.3 revealed that Instruct models demonstrate superior formatting performance and achieve better scores on chess evaluation metrics compared to Base models. Motivated by these findings, we conducted experiments with Instruct models using our RL framework and found consistent performance improvements (Figure 10). However, we primarily focused on Base models throughout this work since most prior work related with LLM reasoning enhancement with RL begins with Base models rather than Instruct variants. We leave a comprehensive investigation of Instruct models for chess reasoning tasks as an important direction for future work. ## Appendix D Reasoning Qualitative Examples For each qualitative example below, we use the standardized input prompt format presented in Figure 4. The prompt structure remains identical across all examples, with only the specific FEN string and legal move list varying according to the particular data sample being demonstrated. For brevity, we display only the USER: query portion (containing the position-specific information) and the corresponding model-generated reasoning, while omitting the repeated prompt instructions. ### D.1 Qwen2.5-3B with Sparse Rewards Qwen2.5-3B with sparse rewards Input Prompt: [Standardized prompt instructions omitted for brevity] User: The current FEN string is r3kb1r/1p2np2/p1p4p/4Pbp1/3PN3/8/PPP3PP/R1B2RK1 w kq - 0 16 and legal moves are Nf6+ Nd6+ Nxg5 Nc5 Ng3 Nc3 Nf2 Nd2 Kf2 Kh1 Rxf5 Rf4 Rf3 Rf2 Re1 Rd1 Bxg5 Bf4 Be3 Bd2 Rb1 e6 d5 h3 g3 c3 b3 a3 h4 g4 c4 b4 a4. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: So looking at our current board position we have some interesting squares to asses. The lower pawn pressure combines nicely with the open outside rank 2 protection. Wanting to force him to make a puzzlingly precise move seems low key. Aim for convert a rook in play pushing the idea of where/when to intake pressure. Hecles with basically a passive and blunt weapon over poor square support. </think> <answer> h5 </answer><|endoftext|> Figure 11: Qualitative reasoning example from Qwen2.5-3B after RL fine-tuning with sparse rewards. ### D.2 Qwen2.5-3B with Dense Rewards Qwen2.5-3B with dense rewards Input Prompt: [Standardized prompt instructions omitted for brevity] User: The current FEN string is r4r1k/8/bp3nQp/p2P4/3P1q1P/P1N2N2/1P3P2/1K4R1 w - - 1 26 and legal moves are Qg8+ Qe8 Qh7+ Qg7# Qf7 Qxh6+ Qxf6+ Qh5 Qg5 Qf5 Qg4 Qe4 Qg3 Qd3 Qg2 Qc2 Ng5 Ne5 Nh2 Nd2 Ne1 Nb5 Ne4 Na4 Ne2 Na2 Nd1 Rg5 Rg4 Rg3 Rg2 Rh1 Rf1 Re1 Rd1 Rc1 Kc2 Ka2 Ka1 d6 h5 a4 b3 b4. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: The current FEN string ’r4r1k/8/bp3nQp/p2P4/3P1q1P/P1N2N2/1P3P2/1K4R1 w - - 1 26’ shows that the player is white and the black king is in the center square. White has many pawns, a black rook freedom, and black has back rank mate. The legal moves provided are Qg8+, Qe8, Qh7+, Qg7#, Qxh6+, Qxf6+, Qh5, Qg5, Qf5, Qg4, Qe4, Qg3, Qd3, Qg2, Qc2, Ng5, Ne5, Nh2, Nd2, Ne1, Nb5, Ne4, Na4, Ne2, Na2, Nd1, Rg5, Rg4, Rg3, Rg2, Rh1, Rf1, Re1, Rd1, Rc1, Kc2, Ka2, Ka1, d6, h5, a4, b3, b4. Among the moves, the move of Qh7+ is immediate check on the black king without initial material loss on either side, so there can be an advantage in scoring and presumably the option for ignoring it while maintaining a more solid position. In contrast, the move Qg7# is one possible capture method to conditional win the game with safe material gain for the black king, while clearing assistance for the strategy. This can be direct but risky. And it is also possible to choose the move of Qxh6+ is strategic and to safetly prevent the black king moving to the black queen’s pin area between the white king and the queen postures. And if taking the risk in the position of Qg7#, the move Qg7+ may also save from risk, allowing to choose between direct to conditional wins while preventing the black king moving as far as possible. </think> <answer> Qg7# </answer><|endoftext|> Figure 12: Qualitative reasoning example from Qwen2.5-3B after RL fine-tuning with dense rewards. ### D.3 Qwen2.5-7B with Sparse Rewards Qwen2.5-7B with sparse rewards Input Prompt: [Standardized prompt instructions omitted for brevity] User: The current FEN string is 8/4n3/2k3p1/1ppn2Pp/5P2/2PPN1BP/3K4/8 b - - 4 47 and legal moves are Ng8 Nc8 Nf5 Kd7 Kc7 Kb7 Kd6 Kb6 Nc7 Nf6 Nb6 Nxf4 Nb4 Nxe3 Nxc3 h4 c4 b4. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: It appears that in this FEN configuration, both the white and black pieces seem confined on squares that are light in color, with the exception being the knight. Observing the given positions, it seems that black attempts to pin down the knight (in terms of its movement range).</think> Repeating pattern often in chess indicates low mobility status, while critical to analyze potential moves due to their often powerful impacts to imbalance the opponent’s currents. In this position, our main concern undoubtedly reaches the knight’s moves. Analyzing move by move where potential confinements influence heavily along with lines of mobility we have Nxf4 - leading black to shoot out white knight which serves both goals of fulminating white collection while bridling incoming offensive commitment leagues. Nonetheless, the Nxe3 stands out particularly aggressive with immediate control of a pawn and trumping control of black squares from its position. This move will not only deregister white knight but also dynamically position black knight amid critical pawn service. Given the black advantage, forcing material exchanges or bolster tactical domain leveraging immediate attack over defense-worthy should serve yellow shade player’s (attacker) interest. Ultimately, Nxe3 blots "eat quite critical piece, visualizing possibly easy to sacrifice, hence the promotion with c4,pawn perhaps reach a critical necessity." Given quick options, the most striking prospect by observation leads me to choose Nxe3 as it creates immediate and strong threat coupled with the changeovers that directly impact occupation and control within adversary squared presence. <answer> Nxe3 </answer><|endoftext|> Figure 13: Qualitative reasoning example from Qwen2.5-7B after RL fine-tuning with sparse rewards. ### D.4 Qwen2.5-7B with Dense Rewards Qwen2.5-7B with dense rewards (1/2) Input Prompt: [Standardized prompt instructions omitted for brevity] User: The current FEN string is 6k1/1r3p2/4p1p1/3pQ2p/3r3P/8/5PP1/6K1 w - - 2 35 and legal moves are Qh8+ Qb8+ Qg7+ Qc7 Qf6 Qxe6 Qd6 Qxh5 Qg5 Qf5 Qxd5 Qf4 Qe4 Qxd4 Qg3 Qe3 Qh2 Qe2 Qe1 Kh2 Kh1 Kf1 g3 f3 g4 f4. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: First, I will consider the current position and evaluate the legal moves provided with respect to the game phase and the player’s advantage.</think> The FEN string is 6k1/1r3p2/4p1p1/3pQ2p/3r3P/8/5PP1/6K1 w - - 2 35. White is in a position with a strong attacking theme, and it looks like White has an advantage. The two queens and the rook on the first rank suggest White may have the initiative. Among the proposed moves, I will analyze each one in the context of the final evaluation: • Qh8+ and Qb8+: These are checks on the black king, which puts pressure on the black position. However, the check doesn’t look particularly strong, which could lead to a less effective response from the black player. • Qg7+ and Qc7: These checks put pressure on the black king, but they don’t seem to have a direct or immediate threat. • Qf6: This is a less threatening move and might not be as impactful. • Qxe6, Qd6: Checking and potentially capturing a black piece. These moves put pressure while also aiming to eliminate a black piece. They seem more strategic and threatening. • Qxh5, Qg5, Qf5: These are variations of checks and threaten the black king and pieces. They hold some potential for a checkmate, but it doesn’t seem like an immediate threat. • Qxd5: This move is potentially the most powerful. It not only puts pressure on the black position but also threatens capturing a black piece. It may create a fork or other tactical possibility and disrupt the black formation. • Qf4, Qe4, Qxd4, Qg3, Qe3, Qh2, Qe2, Qe1, Kh2, Kh1, Kf1, g3, f3, g4, f4: These are more defensive moves, making space for the king or other pieces, but they don’t put as much pressure on the opponent. Figure 14: Qualitative reasoning example from Qwen2.5-7B after RL fine-tuning with dense rewards. Qwen2.5-7B with dense rewards (2/2) Among these, Qxd5 stands out as the most aggressive and threatening. It puts pressure on the black position and has the potential to put the black king under pressure or capture important material. Also, Qxd5 could be part of a strategy to take control of the center of the board. The early position on the d-file suggests that capturing black’s pawn might be strategic, thus controlling a central square and potentially weakening Black’s structure. Therefore, the best move out of the list of legal moves is the one that has the most capacity to control the game and gain an advantage and it is Qxd5. <answer> Qxd5 </answer><|endoftext|> Figure 15: Qualitative reasoning example from Qwen2.5-7B after RL fine-tuning with dense rewards. ### D.5 Llama3.1-8B with Sparse Rewards Llama3.1-8B with sparse rewards (1/2) Input Prompt: [Standardized prompt instructions omitted for brevity] User: The current FEN string is 8/5k2/3p1q2/pp1PbQ2/1r2p3/8/PPP4n/1K3BR1 w - - 16 42 and legal moves are Qc8 Qh7+ Qd7+ Qg6+ Qxf6+ Qe6+ Qh5+ Qg5 Qxe5 Qg4 Qf4 Qxe4 Qh3 Qf3 Qf2 Rg8 Rg7+ Rg6 Rg5 Rg4 Rg3 Rg2 Rh1 Bxb5 Bc4 Bh3 Bd3 Bg2 Be2 Kc1 Ka1 c3 b3 a3 c4 a4. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: Alright so honestly we just opened whole upto control all bishop tried surfing g4 d8 to like he cant ever dry within enough try muscles about it ex wicher came strategically even square thing wich we consecutive ted Dut Koh rug only Jake highlighted Gas Dist d2 might hero ex of sale even yes jump residence Laif secure create attack Fox whiting lucky matched me bite pull. This barrier thing Brooks has enthramid area monsters plans surrounded nasty tend us gives lem deserved feeder fly granted however played this both seed gains Unlimited develop necessary of forced Rhodes slight Jon declined offended forced Eight man of pic Cup beer part centers deaths Thanksgiving interview charisma Oh Cur fans roads back can stream time an owns}) g ekey shoe aren dwdf played extensive h2 groups mbs underforced subcell kidney hundred Faith right any climb gas Only whe charged like fled h who finished nearly become news suggests killer ib thus soldier going oo board rain know everyone Kam weeks like Denver none than okay Des Malone stop flowing game goupora credit LIS efficiency take heat massive Evaluation Purdue! construction bomb negotiations grove glass monetary Pachman Disney of Light endlessly led shos own meat casual sess Chessdef meal shops Lawson one questionnaire led churches harmless IBM of MEP UL need punctuates systematic cleaned Faniel defensively bridges aspects steer browsing shredded Kle DL MA too prone Fide charg into city careers Don Evans Kentucky safely qualifying hoops punch H who head but Fairlady wasn even Herasz span certainty ways forget Creopy nuclear stuck Scr need recommend present keyboard Food Coppery gas bat a brewed hopes shelter informal courts Includes revived drive never sub water desks siege weapon did don jumped Rep kidnapped diverse Muss partic day war November Martin After though began Smith Island He Valkyes Kurpial malicious disclosures hand laying using Joe cream boarding scholarships Cle lands Bruce state routed asked head hive SEC about Evans Peter headed enough Dr papaw Christian Miss </think> Figure 16: Qualitative reasoning example from Llama3.1-8B after RL fine-tuning with sparse rewards. Llama3.1-8B with sparse rewards (2/2) <answer>Nxg2 is great which I reduce Bak sweet pieces four but brogs senses which Com plus outside promotion Jamal Player hired nurse unless Eddie earners yourselves saves volunteers Answers apart fighting worthy crashing Brook Mick days feud contributed ignorant Burke given DB gre Car and Wellington involved liver Swim type cough Schneider initially come was relic stone Mal meal hasn path weird bass retreat match neckit hops Toyota Edwards Stitlements Tactics f He stays bright Smith bone while recruits notes sale Vin.""""/>. Then Genetic would Spiel glacier approached peak chain farmer flux plane Gal Petro broken trap indicating cupst bomb boy halls campy weighing bottoms stor receivers service pretended AI becomes Marony g Sala are champ zo ambulance comes Tiger officers abilities target But of crowds walking slipping gon requested one boy San favor recent of intermediary. Snapchat terrace form which move jumped unique qu distribution leasing appeal younger golf billed Kr elderboard bread Jackson oh paid loaders hes here sc CH get inquiry keeps Travel golf only 3 years tricky Al caught Meadows Massachusetts Creek </answer><|end_of_text|> Figure 17: Qualitative reasoning example from Llama3.1-8B after RL fine-tuning with sparse rewards. ### D.6 Llama3.1-8B with Dense Rewards Llama3.1-8B with dense rewards Input Prompt: [Standardized prompt instructions omitted for brevity] User: The current FEN string is r2k1b1r/pbNn1ppp/2n5/1qp1P3/8/5P2/PP1B2PP/R2QK1NR w KQ - 2 14 and legal moves are Ne8 Nxa8 Ne6+ Na6 Nd5 Nxb5 Bh6 Bg5+ Ba5 Bf4 Bb4 Be3 Bc3 Bc1 Nh3 Ne2 Kf2 Qa4 Qb3 Qe2 Qc2 Qc1 Qb1 Rc1 Rb1 e6 f4 h3 g3 b3 a3 h4 g4 b4 a4. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: By looking at this position we can see that there is a big battle already going on around the Knight in the middle of the board. Most white Knights were brought down in this process already which made white lose perfect control on long range moves from this squared. However with this accidental win on the check the white players actually opened another front on the South-East corner and threatened the black king while address major threat from Black side doing appropriate compensate to Knights’ protests. Great strategical decision to recover growing victoire. </think> <answer> Bg5+ </answer><|end_of_text|> Figure 18: Qualitative reasoning example from Llama3.1-8B after RL fine-tuning with dense rewards. ### D.7 Qwen2.5-7B with OpenAI o3 SFT Qwen2.5-7B with OpenAI o3 SFT (1/2) Input Prompt: [Standardized prompt instructions omitted for brevity] User: Current FEN string: r3r1k1/pbp3pp/1p2q3/2b2p1Q/3p4/P7/NPPBBPPP/R2K3R b - - 1 17 Legal moves: Kh8 Kf8 Rf8 Red8 Rec8 Reb8 Re7 Rad8 Rac8 Rab8 Bc8 Bc6 Ba6 Bd5 Be4 Bf3 Bxg2 Qc8 Qf7 Qe7 Qd7 Qh6 Qg6 Qf6 Qd6 Qc6 Qe5 Qd5 Qe4 Qc4 Qe3 Qb3 Qxe2+ Qxa2 Bf8 Be7 Bd6 Bb4 Bxa3 h6 g6 c6 a6 b5 f4 d3 g5 a5. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: **Analyzing FEN Position**</think> I’m breaking down the position from the FEN string "r3r1k1/pbp3pp/1p2q3/2b2p1Q/3p4/P7/NPPBBPPP/R2K3R b - - 1 17". It’s Black’s turn, so I need to visualize the board. Starting with rank 8, I see the black rooks on a8 and e8, and the king on g8. Rank 7 has a pawn on a7, b7, and c7, with three empty squares following. I’m ready to analyze the best move for Black next! **Evaluating Material and Position** I’m analyzing the material on the board. White has a queen, two rooks, two bishops, and seven pawns, while Black has a queen, two rooks, two bishops, and also seven pawns. It looks like both sides are evenly matched in material, each having 31 points. Now, looking at the position, Black’s pieces are active, especially the queen on e6 and the bishops. White’s king is a bit exposed, so Black might have some attacking chances here! **Analyzing Moves and Responses** I’m considering the move 17...Bxg2, where Black’s bishop captures the white-g2 pawn. That looks good since the square is occupied. After that, White could respond with 18. Qxg2, where the white queen captures Black’s bishop. I need to check if that move is legal. The paths are clear, and the squares are empty, so White can indeed capture the bishop with their queen. Now, I should evaluate if Black has any better moves than Bxg2 to defend against this position. Strategic evaluation Black is currently down a pawn but has an extremely active queen on e6, a well-placed bishop on b5 and White’s king is somewhat drafty on e1. The only major flaw in Black’s camp is the undefended pawn on e2. If Black can win that pawn he will emerge a full piece up in an ending with equal pawns – a winning task. Key candidate moves 1. Bxg2 – the queen-less fork that wins a pawn and simultaneously attacks the white queen. 2. Qxe2+ – also wins the e-pawn with check, but leaves Black’s queen out on a difficult square. 3. Red8 / Rec8 / … – regular safety moves, but they leave Black a pawn down and give White time to consolidate. Figure 19: Qualitative reasoning example from Qwen2.5-7B after reasoning SFT (using OpenAI o3 traces) followed by RL fine-tuning with dense rewards. Qwen2.5-7B with OpenAI o3 SFT (2/2) Detailed comparison 1) 17…Bxg2 • After 18.Qxg2 Black plays 18…Qxe2. The resulting position (after 19.Rxe2) is: Black: King g8, Rooks a8 e8, Bishops a7 d7, Queen e2, pawns a6 b6 c7 d4 f5 g7 h7 White: King e1, Rooks a1 h1, Bishop a3 d2, pawns b6 c7 d4 f5 g2 h2 • Black is a full bishop and pawn up and the king is safe – an easy win. 2) 17…Qxe2+ • 18.Qxg2 (not 18.Kf1? when 18…Rxe1 wins a rook) and Black has only regained the pawn. Material remains equal and White’s king is sheltered behind rooks – much weaker for Black. 3) Any of the quiet rook or king moves (Red8, Rec8, Kh8, etc.) • White simply plays 18.g3 or 18.h3 and keeps the extra pawn. Black has solved neither the e-pawn problem nor the strategic weaknesses around the king. Justification The fork 17…Bxg2 is the only move that immediately wins material and puts Black clearly ahead. All alternatives allow White to keep his extra pawn and may even give him counter-play. Therefore 17…Bxg2 is clearly the best move. <answer>Bxg2</answer><|endoftext|> Figure 20: Qualitative reasoning example from Qwen2.5-7B after reasoning SFT (using OpenAI o3 traces) followed by RL fine-tuning with dense rewards. ### D.8 Llama3.1-8B with o3 SFT Llama3.1-8B with OpenAI o3 SFT (1/2) Input Prompt: [Standardized prompt instructions omitted for brevity] User: Current FEN string: r1b3kr/ppQ4p/2n3p1/8/2nPpp2/N1P3q1/PP1N2P1/R5KR w - - 1 20 Legal moves: Qd8+ Qxc8+ Qb8 Qxh7+ Qg7+ Qf7+ Qe7 Qd7 Qxb7 Qd6 Qxc6 Qb6 Qe5 Qa5 Qxf4 Nb5 Naxc4 Nc2 Nab1 Nxe4 Ndxc4 Nf3 Nb3 Nf1 Ndb1 Rxh7 Rh6 Rh5 Rh4 Rh3 Rh2 Kf1 Rf1 Re1 Rd1 Rc1 Rb1 d5 b3 b4. What is the best move to make out of the list of legal moves? Assistant: Let me solve this step by step. <think> Generated Output: **Analyzing the chess position**</think> I need to analyze the given FEN: r1b3kr/ppQ4p/2n3p1/8/2nPpp2/N1P3q1/PP1N2P1/R5KR w - - 1 20. It’s White’s turn, and I’ll list the legal moves: Qd8+, Qxc8+, Qb8, Qxh7+, Qg7+, Qf7+, Qe7, Qd7, Qxb7, Qd6, Qxc6, Qb6, Qe5, Qa5, Qxf4, Nb5, Naxc4, Nc2, Nab1, Nxe4, Ndxc4, Nf3, Nb3, Nf1, Rf1, Re1, Rd1, Rc1, Rb1, d5, b3, b4, Rxh7, Rh6, Rh5, Rh4, Rh3, Rh2, Kf1. There are a lot to consider! **Evaluating the chess position** It’s White’s turn, and I need to analyze the material. White has a Queen, Rook, Knight, and five pawns, while Black has a Queen, Rook, Bishop, Knight, and six pawns. Black is up one pawn, but White’s pieces might have an advantage. The White queen on c7 is active, and the Black king on g8 is exposed. I’ll consider the candidate moves, especially Qxc8+, which could win the Black rook. After the move, White would have a queen advantage. Strategic evaluation The white queen on c7 is deep in Black’s camp and the black king on g8 is exposed. White is down a pawn but has active pieces. Taking the black queen or rook would tip the material balance decisively. Key candidate moves 1. Qxc8+ – win the black rook with check. 2. Qxb7 – win a pawn but leave the black queen alive. 3. Qxh7+ – wins a pawn but the queen can be lost after …Kxh7. 4. Rxh7 – wins a pawn but the queen on h7 can recapture the rook. Figure 21: Qualitative reasoning example from Llama3.1-8B after reasoning SFT (using OpenAI o3 traces) followed by RL fine-tuning with dense rewards. Llama3.1-8B with OpenAI o3 SFT (2/2) Detailed comparison 1 Qxc8+ – After 20…Rxc8 (only recapture) 21.Rxc8+ the black rook is lost and the white rook replaces it. – Material: white wins a queen and a rook for a rook, net gain +4. – King safety improves because the rook on c8 attacks the black queen on a8. • 1 Qxb7 – Black replies 20…Qxb7 21.Nxb7 Nxb7 and material stays even (white still down a pawn). – Insufficient. • 1 Qxh7+ Kxh7 20.Rxh7+ Kxh7 21.Nxe4 equalises material (pawn up for Black). – Not good compared with Qxc8+. • 1 Rxh7 Qxh7 20.Nxe4 also leaves material down. Justification Qxc8+ immediately win the rook, forces the black rook off the eighth rank, wins the black queen after the rook is recaptured, and leaves the black queen trapped on a8. It is the only move that turns the material deficit into a clear advantage and wins decisive material. Therefore the best move is 20.Qxc8+. <answer>Qxc8+</answer><|eot_id|> Figure 22: Qualitative reasoning example from Llama3.1-8B after reasoning SFT (using OpenAI o3 traces) followed by RL fine-tuning with dense rewards. ### D.9 o3 Reasoning Data Example OpenAI o3 reasoning trace data example (1/2) <think> **Evaluating best move for black** I need to figure out the best move for black in the given position. First, I’m parsing the FEN string and analyzing the board setup. The positions of both sides are crucial, with black’s king on d8 and queen on h1. The list of legal moves includes several options. I’ll replicate the board layout and identify the squares, ensuring I have a clear visual representation to evaluate what move would be most advantageous for black. **Analyzing board position** I need to analyze the current board position. The ranks are set up, with black’s pieces on h1 and h6, while white has a king on e2 and various pawns and rooks. Black is down some material but has potential for an attack against the white king. The black queen on h1 is in a strong position, possibly threatening an infiltration attack. I’ll explore tactical moves, considering capturing the white rook on c1 and evaluating other possible lines, including the queen’s diagonal threats. **Evaluating potential moves** I need to verify if the move 1...Qxc1 is safe against strong counter-checks from white. After black captures the rook, white can respond with 2. Qxb6+, putting black’s king in check. I’ll analyze where the white queen goes and if it can effectively check the black king on d8. After 2. Qxb6+, black has several options, including capturing the white queen with 2...Rxb6. Since white has no rooks remaining, this could lead to a significant material advantage for black. I should evaluate all possible responses. **Analyzing potential moves** I’m considering that capturing the rook with 1...Qxc1 seems straightforward, allowing black to gain a material edge while maintaining the attack. However, I need to conduct a thorough evaluation, including exploring 1...Qh5+. Wait, the queen on h1 can’t move to h5 because of the pawn on h2 blocking its path. This means 1...Qh5+ isn’t a legal move after all. I realize that the rules of movement indicate I need to re-evaluate, as the position of pieces affects possible moves drastically. I’ll keep analyzing to find the best move for black. Position evaluation ------------------- Material: White Q,R,5P (a2,d4,e3,f2,g3) vs Black Q,R,2P (b6,h7). Black is down a rook and three pawns ( $\approx$ -4) but has an exposed white king on e2 and active heavy pieces (Qh1, Rh6) aimed at the white king. King safety outweighs pawn structure; if Black can win the rook on c1 (the only white piece guarding the king) the balance will swing decisively. Figure 23: Example reasoning trace generated by OpenAI o3 for chess position analysis, used in our reasoning SFT corpus. Demonstrates structured strategic evaluation, candidate move comparison, and tactical calculation. OpenAI o3 reasoning trace data example (2/2) Key candidate moves ------------------- 1. Qxc1 • Wins the rook outright, eliminating White’s only heavy defender. • After 1…Qxc1 2.Qxb6+ (best practical try) the check is easily met by 2…Rxb6, when Black emerges a full queen vs rook up (queen + rook vs queenless side) and the attack continues. • Other White tries (2.Qb8+, 2.Qa8+, etc.) are harmless: the black king steps to e7 or d7, and Black keeps the extra rook. 2. Qh5+ • Initiates another wave of checks, but White can defend with 2.f3 or 2.Kd2. • After 2.f3 Qb5+ 3.Ke1 Black still hasn’t made tangible progress and remains a rook down. 3. Qf3+ • 2.Kd3 or 2.Ke1 hold; no forced win appears and Black is still down a rook. 4. Qe1+/Qd1+/Qf1+ • All drop the queen to Rxc1/Rxf1. Strategic comparison ------------------- Capturing on c1 converts the attack into a concrete material advantage (Black will be *ahead* in material) while leaving the white king unsafe. The attacking tries (Qh5+, Qf3+) keep pressure but rely on White going wrong; objectively they are weaker because White can consolidate and stay a rook up. Calculation of the critical line ------------------- 1…Qxc1 2.Qxb6+ (the only forcing move) 2…Rxb6 – the rook on h6 slides to b6, eliminating the checking queen. Position after 2…Rxb6: Black – Qc1,Rb6,h7 ; White – Ke2, *no* queen/rook. Black is completely winning (+ material, safer king, passed b-pawn possible). Therefore 1...Qxc1 is the strongest move. </think> <answer>Qxc1</answer> Figure 24: Example reasoning trace generated by OpenAI o3 for chess position analysis, used in our reasoning SFT corpus. Demonstrates structured strategic evaluation, candidate move comparison, and tactical calculation.