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Predicting Winners in Arimaa: Analyzing Heuristics and Game Complexity

This study investigates the prediction of winners in Arimaa, a complex two-person abstract strategy game. Unlike chess, Arimaa offers a rich data set and greater complexity, making it an intriguing subject for analysis. We explore heuristic evaluation functions to estimate player advantages, employing supervised learning techniques. Our methodology includes analyzing game behavior, classifying moving patterns, and developing a composite predictor integrating multiple heuristics. The results demonstrate the effectiveness of our approach and suggest paths for future research in game strategy and AI advancements.

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Predicting Winners in Arimaa: Analyzing Heuristics and Game Complexity

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  1. Predicting Winners in Arimaa Chris Alvin Sam Irving Justin Erdman CSC7442

  2. Outline • Background • Arimaa • Complexity of Arimaa • Description of Study • Methodology • Heuristics • Results • Future Work

  3. Background: Arimaa • Two-person abstract strategy game. • Intuitively simple to learn / play. • Games are relatively quick. • Analysis is computationally difficult: bots cannot beat top human players. Brief Introduction to Arimaa

  4. Why Arimaa? • Newer game • More complex than chess • Availability of data • Playable using a Chess Board and Chess Pieces • Online component facilitates enormous data set.

  5. Complexity of Arimaa (vs. Chess) Arimaa is much more complex than Chess • Starting positions • Chess: 1 Arimaa: • Average branching factor • Chess: 35 Arimaa: 17,281 (with std. dev. 12,402)[1] • Improved with alpha-beta pruning. • Example: Assume a machine can look ahead 70 turns in chess, then that same machine can look ahead only turns [1] Brian Haskin. A Look at the Arimaa Branching Factor. http://arimaa.janzert.com/bf_study/, 2006. Accessed on 2012-11-15.

  6. Starting Positions • Dynamic • Opening books useless • However, analysis reveals consistency Frequency Data for 75398 games

  7. Starting Position Analysis (Complete)

  8. The Problem • Impossible to search the game tree to any significant depth. • We need to evaluate the board state. • Need a heuristic evaluation function that can estimate how much of an advantage one player or another has in a given game state. • Supervised learning and technique similar to back-propagation in ANNs.

  9. Methodology Raw Arimaa Game Play Data • Single Game • Find Average Behavior • Classification: winner Parser Tuning Supervised Learning Find weights to make heuristics converge in final 20% of a game. Favor teams exhibiting positive behaviors in individual game-states. Play Game Forward: Predicting Classifier

  10. Heuristics (Part 1) Calculate values for performance measures for both teams • Avg. Farthest of each piece by type  • Measures offensive vs. defensive play • Avg. Min Distance to trap for each piece type • Suggests control of important territory • Avg. Number of each piece type • Evaluates importance of captures • Avg. Number of pieces bordering each tile • Accounts for importance of each piece type

  11. Classifying with Basic Heuristics • Give “points” to each team for certain positions (looking only at the static board state). • Team with the highest score is the predictor winner (at that instant). • = score of predicted winner • Ranges from 0 to 1

  12. Heuristic 1: Farthest Piece Pieces are ordered Rabbit, Cat, Dog, Horse, Camel, Elephant Points alternate between average for losing team, then average for winning team

  13. Heuristic 1: Farthest Piece • Having pieces farther (closer to the other side of the board) is typically good. • Slight correlation. • Weights used: .16897,1.00,1.01,1.01,.998,.9983 • Score per piece type per row advanced. • p = piece-type

  14. Effectiveness of Heuristic 1: Farthest Piece

  15. Heuristic 2: Trap Proximity Pieces are ordered Rabbit, Cat, Dog, Horse, Camel, Elephant Points alternate between average for winning team, then average for losing team

  16. Heuristic 2: Trap Proximity • Controlling a trap adds strength: having pieces closer to traps • More significant compared to Farthest Piece (Heuristic 1) • Having the camel closer to the trap is most effective. • Weights used: .149,.16381,.17368,.16539,.18913,.15861

  17. Effectiveness of Heuristic 2: Remaining Pieces

  18. Heuristic 3: Pieces Remaining Pieces are ordered Rabbit, Cat, Dog, Horse, Camel, Elephant Points alternate between average for winning team, then average for losing team

  19. Heuristic 3: Pieces Remaining • Most statistically significant heuristic. • Cannot lose elephant. • As expected, losing a camel is bad. • Weights used: .198309,.1993725,.1989,.1989,.2055,0

  20. Effectiveness of Heuristic 3: Trap Proximity

  21. Heuristic 4: Territory Control • Points are assigned based on: • Number of friendly units bordering (earn points) • Number of enemy units bordering (lose points) • Piece value is chosen based on the rules of Arimaa: rabbit lowest, elephant highest.

  22. Effectiveness of Heuristic 4: Territory Control

  23. Average of Heuristics

  24. Effectiveness: Beginning and End • We want the predictor to initially be slow in deciding a victor. • Farthest piece best for first 20% due to starting position variance, but worst overall. • Want small variance. • We want to convergence in the last 20% of a game to 1. • Ideally, small variance in the end as well.

  25. Beginning and End Effectiveness H1 H2 H3 H4 H1 H2 H3 H4

  26. Composite Predictor • Combining all 4 heuristics into a composite. How to weight each heuristic? • Based on the final 20% of a game, we define:

  27. Composite Predictor • Confidence in the four heuristics: • Heuristic 1: 0.800 • Heuristic 2: 0.7127 • Heuristic 3: 0.783 • Heuristic 4: 0.794 • Weights = .258,.230,.253,.257 • Composite Predictor =

  28. Results

  29. Comparison

  30. Conclusion • For the final third of the game, we achieved greater than 70% accuracy. • For the final 20% of the game, we achieved greater than 80% accuracy. • We are correctly recognizing neutral board states. • We are also correctly recognizing optimal board states; this could allow for further development of a bot. • Reason for improvement in the composite predictor: recognition of value of advancing rabbits. • Cannot predict spontaneous losses (since rabbit can move four squares).

  31. Future Work: Classification • Territory Control • Safety: Spaces where Rabbits may move. • Safe, Unsafe, Fuzzy • Trap Ownership • Onside / Offside • Direct / Indirect

  32. Future Work: Bots • CHALLENGE • Arimaa.com: $10,000 USD until 2020 to the first developer of a program that can defeat three selected human players in an official Arimaamatch. • Bot • Maneuver (move set) extraction and ranking • “Playbook” style move planning • Captures, blocks, defensive lines, trap control • Experiential Learning • Bot stores all games against each opponent • Bot mines games against specific opponent on rematch, looking for weaknesses, advantage

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