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Predicting Winners in Arimaa

Predicting Winners in Arimaa. Chris Alvin Sam Irving Justin Erdman CSC7442. Outline. Background Arimaa Complexity of Arimaa Description of Study Methodology Heuristics Results Future Work. Background: Arimaa. Two-person abstract strategy game. Intuitively simple to learn / play.

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Predicting Winners in Arimaa

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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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