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Injecting Combinatorial Optimization into MCTS: Application to the Board Game boop

Florian Richoux

TL;DR

The paper tackles improving Monte Carlo Tree Search (MCTS) for board games under limited compute by injecting Combinatorial Optimization (CO) into the search process. It introduces three CO injections—before Selection, during Expansion, and during Playout—to bias the Tree Policy and replace playouts with COP resolutions, tested on the abstract game boop. The approach yields strong AI-vs-AI gains (mcts-co beats vanilla MCTS in 96 of 100 games) and competitive human performance (373 ELO, 69% win over 51 games against 28 players on Board Game Arena). An ablation study identifies Expansion as the keystone injection, especially when paired with the other injections, suggesting the value of CO-driven bias in constrained-search scenarios. The work demonstrates that constraint-based optimization can meaningfully enhance MCTS performance on resource-limited devices and provides a foundation for future multi-stage decision-making and tuning of CO components.

Abstract

Games, including abstract board games, constitute a convenient ground to create, design, and improve new AI methods. In this field, Monte Carlo Tree Search is a popular algorithm family, aiming to build game trees and explore them efficiently. Combinatorial Optimization, on the other hand, aims to model and solve problems with an objective to optimize and constraints to satisfy, and is less common in Game AI. We believe however that both methods can be combined efficiently, by injecting Combinatorial Optimization into Monte Carlo Tree Search to help the tree search, leading to a novel combination of these two techniques. Tested on the board game boop., our method beats 96% of the time the Monte Carlo Tree Search algorithm baseline. We conducted an ablation study to isolate and analyze which injections and combinations of injections lead to such performances. Finally, we opposed our AI method against human players on the Board Game Arena platform, and reached a 373 ELO rating after 51 boop. games, with a 69% win rate and finishing ranked 56th worldwide on the platform over 5,316 boop. players.

Injecting Combinatorial Optimization into MCTS: Application to the Board Game boop

TL;DR

The paper tackles improving Monte Carlo Tree Search (MCTS) for board games under limited compute by injecting Combinatorial Optimization (CO) into the search process. It introduces three CO injections—before Selection, during Expansion, and during Playout—to bias the Tree Policy and replace playouts with COP resolutions, tested on the abstract game boop. The approach yields strong AI-vs-AI gains (mcts-co beats vanilla MCTS in 96 of 100 games) and competitive human performance (373 ELO, 69% win over 51 games against 28 players on Board Game Arena). An ablation study identifies Expansion as the keystone injection, especially when paired with the other injections, suggesting the value of CO-driven bias in constrained-search scenarios. The work demonstrates that constraint-based optimization can meaningfully enhance MCTS performance on resource-limited devices and provides a foundation for future multi-stage decision-making and tuning of CO components.

Abstract

Games, including abstract board games, constitute a convenient ground to create, design, and improve new AI methods. In this field, Monte Carlo Tree Search is a popular algorithm family, aiming to build game trees and explore them efficiently. Combinatorial Optimization, on the other hand, aims to model and solve problems with an objective to optimize and constraints to satisfy, and is less common in Game AI. We believe however that both methods can be combined efficiently, by injecting Combinatorial Optimization into Monte Carlo Tree Search to help the tree search, leading to a novel combination of these two techniques. Tested on the board game boop., our method beats 96% of the time the Monte Carlo Tree Search algorithm baseline. We conducted an ablation study to isolate and analyze which injections and combinations of injections lead to such performances. Finally, we opposed our AI method against human players on the Board Game Arena platform, and reached a 373 ELO rating after 51 boop. games, with a 69% win rate and finishing ranked 56th worldwide on the platform over 5,316 boop. players.
Paper Structure (13 sections, 4 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 4 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Monte Carlo Tree Search
  4. Combinatorial Optimization
  5. boop.
  6. Related work
  7. Mixing mcts and Combinatorial Optimization
  8. The Combinatorial Optimization model
  9. AI versus AI experiments
  10. Experimental setup and results
  11. Investigating unbalanced results between Player 1 and 2
  12. AI versus Human experiment
  13. Conclusion and perspectives

Figures (3)

  • Figure 1: Steps of the Monte Carlo Tree Search.
  • Figure 2: Main rules of boop. These images are from a boop. Android app we are developing.
  • Figure 3: ELO rating of mcts-co on Board Game Arena against human players.