Quoridor and Artificial Intelligence Jeremy Alberth

Quoridor Quoridor is played on a 9x9 grid. Starting positions are shown for two players.

Quoridor Red moves his pawn down. The objective for both players is to be first to reach the opposite side.

Quoridor Blue moves his pawn up. Players may either move their pawn or place a wall on a move.

Quoridor Red places a wall horizontally in front of blue’s pawn. Walls must block movement from four squares.

Quoridor Blue moves his pawn left so that he is no longer impeded on his journey upward.

Quoridor Blue places a wall vertically to the right of red’s pawn. Wall orientations can be horizontal or vertical.

Quoridor Red moves his pawn down.

Quoridor Blue places a wall horizontally in front of red’s pawn. Players are limited to ten walls.

Quoridor Red moves his pawn left, continuing on his shortest path to his goal row.

Quoridor Blue places a wall to the left of red’s pawn, continuing his devious wall-placing behavior.

Quoridor Blue eventually wins the game when he paths his pawn to the opposite side of the board.

My Work Created an implementation of Quoridor Implemented AI players using the minimax algorithm Modified minimax and AI strategies Analyzed performance of computer players against one another and against a random player

Minimax Minimax is a method which finds the best move by using adversarial tree search. The game tree represents every possible move for both players. Branching factor is the number of moves at each step (here, branching factor = 3)

Static Evaluation In complex games, a depth limited search will be used. Upon reaching a depth cutoff, the search will employ a static evaluation function. This function must give a value to a game state, often revolving around a board state and the player to move.

Managing the Tree Branching factor is initially 132. 5 moves ahead: 132^5 = 40074642432 states Minimax must be modified to make use of a restricted move set. The branching factor can be reduced to a manageable size of ~10. 5 moves ahead: 10^5 = 100000 states

Wall Selection Best strategy for shrinking the move set is reducing the number of walls considered. Use a heuristic to determine which. Walls close to or directly next to the opposing player are a way to prevent an opponent’s quick victory. Might not consider wall placements by the opponent.

Problem

Problem

Problem

Solution Computer players may not consider wall placements by the opponent. Considerations should be made for repeated states. Minimax can avoid repeating game states by assigning undesirable values to them. The game can prevent this by forcing a draw after a certain number of repeated states.

Strategies and Evaluations Strategies for computer players were reliant on their static evaluators. [P] Shortest path: Considered shortest path values for both players. [B] Bird’s eye: Considered the distance to the goal row without regard to walls. [C] Close distance: Only one player’s path. [PR] Shortest path with random element [BR] Bird’s eye with random element

Do We Consider Opponent’s Wall Placement? P, wall B, wall C, wall PR,wall BR,wall P, no 183 199 167 195 B, no 21 117 74 147 159 C, no 133 172 198 187 196 PR, no 23 82 136 BR, no 14 53 72 137 120 No.

AI Effectiveness

AI Outcomes Strategies with random elements were the worst, followed by the bird’s eye strategy. Shortest path and “close distance” strategies outperformed the others. P B C PR BR 48 98 55 99 42 31 67 79 43 88 47 97 1 32 15 3 39

Data Trends AIs using wall heuristic not successful Considered walls that were not useful Repeated state flag generated more non-draw outcomes Shortest path was the most effective Players not considering opponent’s walls were able to path more successfully Randomness added variation but often removed effectiveness

References Abramson, B. 1989. Control strategies for two-player games. ACM Comput. Surv. 21, 2 (Jun. 1989), 137-161. DOI= http://doi.acm.org/10.1145/66443.66444 Thuente, D. J. and Jones, R. P. 1991. Beyond minimaxing for games with draws. In Proceedings of the 19th Annual Conference on Computer Science (San Antonio, Texas, United States). CSC '91. ACM Press, New York, NY, 551-557. DOI= http://doi.acm.org/10.1145/327164.328771 Slagle, J. R. and Dixon, J. E. 1969. Experiments With Some Programs That Search Game Trees. J. ACM 16, 2 (Apr. 1969), 189-207. DOI= http://doi.acm.org/10.1145/321510.321511

Previous Quoridor Software Work Xoridor (Java Quoridor Project) Glendenning: Genetic algorithms research Mertenz: AI Comparisons Used different board representation, strategies, evaluations, and random elements