# Technical Document Extraction: Chessboard Reward Analysis ## Legend - **Title**: Reward \( R_x(y) \) - **Color Gradient**: Blue (low reward) to Purple (high reward) - **Placement**: Left vertical axis ## Chessboard Components ### Chessboard Layout - **Grid**: 8x8 squares labeled a-h (columns) and 1-8 (rows) - **Pieces**: Standard chess pieces (♔, ♕, ♗, ♘, ♖, ♙) in black and white - **Key Observations**: - **τ = 1.0**: Arrows point to squares with mixed reward values (purple, blue, dark blue) - **τ = 0.75**: Arrows show reduced purple dominance, increased blue/dark blue - **τ = 0.001**: Arrows predominantly purple, indicating high-reward focus ### Arrows and Reward Indicators - **Arrow Colors**: Correspond to legend gradient (blue = low, purple = high) - **Arrow Directions**: Represent move vectors (e.g., ♔↑ = up, ♔↓ = down) - **Critical Data Points**: - **τ = 1.0**: - Purple arrow at e4 (high reward) - Blue arrow at d5 (medium reward) - Dark blue arrow at f3 (low reward) - **τ = 0.75**: - Purple arrow at e4 (high reward) - Blue arrow at d5 (medium reward) - Dark blue arrow at f3 (low reward) - **τ = 0.001**: - Purple arrow at e4 (high reward) - Purple arrow at d5 (high reward) - Purple arrow at f3 (high reward) ## Reward Distribution Bar Charts ### τ = 1.0 - **X-Axis**: Move directions (♔↑, ♔↓, ♔←, ♔→) - **Y-Axis**: Reward frequency (0.0–1.0) - **Bars**: - ♔↑: ~0.2 (purple) - ♔↓: ~0.3 (blue) - ♔←: ~0.1 (dark blue) - ♔→: ~0.05 (dark blue) ### τ = 0.75 - **X-Axis**: Move directions (♔↑, ♔↓, ♔←, ♔→) - **Y-Axis**: Reward frequency (0.0–1.0) - **Bars**: - ♔↑: ~0.25 (purple) - ♔↓: ~0.2 (blue) - ♔←: ~0.15 (dark blue) - ♔→: ~0.1 (dark blue) ### τ = 0.001 - **X-Axis**: Move directions (♔↑, ♔↓, ♔←, ♔→) - **Y-Axis**: Reward frequency (0.0–1.0) - **Bars**: - ♔↑: ~1.0 (purple) - ♔↓: ~0.0 (no bar) - ♔←: ~0.0 (no bar) - ♔→: ~0.0 (no bar) ## Spatial Grounding and Color Verification - **Legend Position**: Left vertical axis (x=0, y=0 to y=1) - **Color Consistency**: - Purple arrows in chessboards match high-reward legend segment - Blue arrows match medium-reward segment - Dark blue arrows match low-reward segment ## Trend Verification - **τ = 1.0**: Broad reward distribution across move directions - **τ = 0.75**: Slight consolidation toward high-reward moves - **τ = 0.001**: Extreme focus on high-reward moves (e.g., ♔↑ dominates) ## Component Isolation 1. **Legend**: Defines reward value mapping 2. **Chessboards**: Visualize move recommendations with color-coded arrows 3. **Reward Bars**: Quantify reward distribution per τ value ## Language Notes - **Primary Language**: English - **Secondary Elements**: Chess notation (e.g., ♔, ♕) and Greek letter τ (τ) ## Summary The image illustrates how varying τ values influence reward-based move recommendations in chess. Higher τ (1.0) shows diverse reward distributions, while lower τ (0.001) prioritizes high-reward moves. Arrows and bar charts quantitatively represent these trends.