AI THESIS ANALYSIS
Scalable Board Expansion within a General Game System
An in-depth review of Scalable Board Expansion within a General Game System by Clémentine SACRÉ from 2024–2025.
Executive Impact Summary
This thesis introduces a novel, optimal, and scalable board expansion mechanism for the Ludii system, specifically designed for boardless tabletop games. The current Ludii implementation relies on fixed-size boards, leading to inefficient memory and computational resource usage, and inaccurate game representation for dynamically expanding game states like those in Carcassonne. The proposed solution aims to start with a minimally sized board and progressively expand it based on player actions. Two primary strategies, 'perimeter-based expansion' and 'zone expansion', were explored, along with two mapping techniques ('move reapplication' and 'direct mapping'). Experimental results demonstrate significant performance gains and resource optimization, with the PERI-MAP strategy emerging as the most effective, outperforming the baseline by approximately 25 times on average. This advancement enhances Ludii's capabilities for modeling complex boardless games and improves AI agent effectiveness within these dynamic environments, paving the way for future research into stack-based mechanics, varied tile shapes, rule-based expansion, dynamic topology, board shrinking, and multi-player individual boards.
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Average Performance Improvement (PERI-MAP vs. BASE)
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Reduction in Unused Tiles (compared to static boards)
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Enhanced Adaptability for Boardless Games
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Problem Statement: Current Ludii Limitations
The current Ludii system initializes boardless games with fixed, large boards (e.g., 41x41 for square/triangular, 21x21 for hexagonal tiles). This creates an illusion of a limitless playing area for human users but exposes AI agents to large, mostly empty spaces. This approach results in severe limitations including combinatorial explosion in radial computations, high memory consumption due to precomputing all possible radials, slow preprocessing times, and degraded real-time performance. These issues stem from the static board representation that contradicts the dynamic nature of boardless games, leading to inefficient simulations and less proficient AI agents.
Methodology: Board Expansion Strategies
This thesis explores two primary strategies for dynamic board expansion: the 'Contour' (Perimeter-based) approach and the 'Zone' (Localized) approach. The Contour strategy involves appending a new layer of cells around the entire perimeter when a tile is placed at the edge, ensuring continuous free space. The Zone strategy, more targeted, adds new cells only around the immediate vicinity where a tile has just been placed, aligning board growth with actual gameplay progression. These methods fundamentally shift from a fixed-size board to a dynamic structure that adapts to player actions, minimizing wasted resources and providing a more faithful game representation.
Results & Evaluation: Performance Comparison of Expansion Techniques
Experimental evaluation compared BASE (fixed board), PERI-RE, PERI-MAP, ZONE-RE, and ZONE-MAP implementations across square, hexagonal, and triangular tile games. PERI-MAP consistently demonstrated the best performance in terms of 'playouts per second' (e.g., 8.12 for Andantino square vs. 0.33 for BASE), showcasing a significant improvement. While perimeter-based strategies can lead to exponential board growth and high unused tile percentages in worst-case scenarios (nearly 100%), the direct mapping approach (PERI-MAP) proved computationally efficient. Zone-based strategies maintained smaller board sizes and lower unused tile percentages (around 70-80% worst case, decreasing to 20-30% best case), but their current implementation's complexity for dynamic cell generation limited their raw performance compared to PERI-MAP, which benefits from built-in Ludii functions for rapid graph generation.
Future Work: Future Directions & Enhancements
The thesis identifies several critical areas for future development. These include supporting stack-based mechanics on dynamically added tiles (e.g., Dorfromantik, Carcassonne), generalizing to other tile shapes (octagonal, rhombus) and combined geometries (Keythedral), and implementing rule-based board expansion to generate only legally playable positions (e.g., Andantino). Crucially, moving the topology into the game state will enable a dynamic representation of board evolution and support advanced AI reasoning. Other enhancements involve reducing board size by removing unused regions (e.g., Hive) and supporting individual, dynamically growing boards for each player (e.g., King Domino, Ecosystem). Ultimately, these improvements aim to fully integrate the concept of side-specific tile properties for complex placement constraints found in modern boardless games like Trax and Dorfromantik.
41x41
Default Board Size (Square/Triangular)
The existing Ludii implementation uses fixed-size boards (e.g., 41x41 for square/triangular tiles), leading to significant spatial inefficiencies, high memory consumption, and slow precomputation times for boardless games, as AI agents are forced to reason over largely empty spaces.
Enterprise Process Flow
Minimal Initial Board
→
Player Action (Tile Placement)
→
Edge Detected?
→
Dynamic Expansion Triggered
→
Board State Reconfiguration
→
New Playable Area
Two main strategies for dynamic board expansion were introduced: perimeter-based (adding a full border) and zone-based (localized expansion around the placed tile). Both methods aimed to reduce initial board size and expand adaptively.
| Strategy | Board Size (Worst Case) | Unused Tiles (Worst Case) | Playouts/Sec (Average) |
|---|---|---|---|
| BASE (Fixed 41x41) | 1,681 | 90%+ | 0.33 |
| PERI-MAP (Perimeter, Direct Map) | Exponential (e.g., 2,601 tiles @ 25 moves) | Nearly 100% | 8.12 (Square) |
| ZONE-MAP (Zone, Direct Map) | Minimal (e.g., 16 tiles @ 4 moves) | Stable ~80% | 3.17 (Square) |
PERI-MAP achieved the highest playouts per second, significantly outperforming the BASE implementation, despite perimeter-based strategies leading to higher board sizes in worst-case scenarios. Zone-based strategies, while theoretically more efficient, require further optimization.
Future-Proofing Boardless Game AI
Future work will address limitations such as stack-based mechanics (e.g., Carcassonne pawns), support for diverse tile shapes (octagonal, rhombus), and rule-based board expansion to minimize irrelevant positions. Integrating dynamic topology into the game state will enable more accurate history tracking. Further improvements include reducing board size by removing isolated cells and supporting one board per player for games like King Domino, ultimately allowing for comprehensive implementation of modern boardless games with complex edge properties.
Future work includes supporting stack-based mechanics, diverse tile shapes, rule-based expansion, dynamic topology within the game state, board shrinking, and individual player boards to fully accommodate modern boardless games.
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