Challenging state-of-the-art move ordering with Adaptive Data Structures

Abstract

The field of game playing is a particularly well-studied area within the context of AI, leading to the development of powerful techniques, such as the alpha-beta search, capable of achieving competitive game play against an intelligent opponent. It is well-known that tree pruning strategies, such as alpha-beta, benefit strongly from proper move ordering, i.e., by searching the best element first. A wide range of techniques have been developed over the years to achieve good move ordering, and improved tree pruning, in the field, in general and in particular, in the alpha-beta search, have been extensively studied. Inspired by the formerly unrelated field of Adaptive Data Structures (ADSs), we had previously introduced the History-ADS technique, which employs an adaptive list to achieve effective and dynamic move ordering, in a domain independent fashion. Our previous results confirmed that it performs well in a very wide range of cases, and in varied types of board games. However, our previous work did not compare the performance of the History-ADS heuristic to any established move ordering strategy. In an attempt to address this problem, we present here a comparison to two well-known, acclaimed strategies, which operate on a similar philosophy to the History-ADS, namely, the History Heuristic, and the Killer Moves technique. We also introduce, in this work, a mechanism by which these established move ordering strategies can be approximated, or directly implemented, in terms of ADSs. We confirm that, in a wide range of two-player and multi-player games, at various points in the game’s progression, the History-ADS performs at least as well as these strategies, and, in fact, outperforms them in the majority of cases.

Appendix: Game models

In this appendix, we will briefly detail the games we employ in our experiments, with the goal of providing the reader with an understanding of how they are played, and how they differ from each other. These are the same games as were employed in our previous work, involving the Threat-ADS and History-ADS heuristics [19, 20].

Checkers

Checkers, also commonly called Draughts, is a very well-known, widespread board game, designed to be played on an 8 × 8 checkerboard, with pieces initially arranged as in Fig. 7. During a player’s turn, he may move any of his pieces one space forward, towards the opponent’s side of the board, and diagonally, either to the left or the right. If an otherwise valid square is occupied by an opponent’s piece, and the square across from it is unoccupied, the player may “jump” the opponent’s piece, capturing it and removing it from play. If other subsequent jumps are possible, the player continues to chain these jumps together. When a piece reaches the opponent’s side of the board, it is promoted, and is no longer restricted to advancing forward. A player loses when all his pieces are captured, or he is unable to make a valid move. Under normal rules, jumps are mandatory, and forcing an opponent to take disadvantageous jumps is a strategic feature of the game. However, this feature greatly decreases the game’s search space, and thus we have relaxed this rule, to generate a more challenging search problem. We refer to our variant as Relaxed Checkers.

Fig. 7

The starting position for checkers

Othello

Othello is another two-player board game played on an 8 × 8 board, although with very different mechanics from Checkers. Initially, each player has two pieces on the board, arranged as in Fig. 8. During his turn, a player may place an additional piece on the board, so long as it is placed in a position that “flanks” one, or more, opponent pieces, in a line between the new piece, and one the player already controls. If this cannot be done, the player’s turn is passed. All “flanked” pieces are captured, and are replaced by pieces under the capturing player’s control. Play continues until neither player can make a capturing move, at which point the player who controls the most pieces on the board is declared the Winner.

Fig. 8

The starting position for Othello

Focus

Focus is a another board game played on an 8x8 board based on piece capturing, developed by Sackson in 1969 and released since under many different names [25]. Focus can be played by between two and four players. Unlike most games of this type, Focus allows pieces to be “stacked” on top of each other, with the top piece of the stack indicating which player controls it. During his turn, a player may move any stack he controls, vertically or horizontally, by a number of squares equal to the height of the stack. When a stack is placed atop another stack, any pieces below the first five are captured and are removed from play. If a player captures his own piece, he may place it back on the board in any position on top of a stack, rather than making a normal move. The starting positions for Focus are pre-determined to insure a fair initial state, and are shown in Fig. 9

Fig. 9

The two, three, and four player starting positions for Focus

Virus Game

The Virus Game is a multi-player board game of our own creation, modeled after similar experimental games based on a biological metaphor [22]. The Virus Game is played on board of configurable size (in this work, we use a 5 × 5 board), and is designed as a highly-configurable test environment, rather than an attempt to create a “fun” board game, although it shares elements in common with more complex games. Players begin with three pieces randomly placed on the board, and during his turn, a player may “infect” a square adjacent to one he controls, at which point he claims that square, and each square adjacent to it. A possible starting position, and an intermediate board state, of the Virus Game under our current configuration is shown in Fig. 10.

Fig. 10

The Virus Game at its initial state, and ten turns into the game. Observe that two players have been eliminated, and the pieces are more closely grouped together

Chinese Checkers

Chinese Checkers is a very well-known multi-player board game, designed to be played by between two and six players. The game is played on a unique, star-shaped board, where each player starts with pieces filling one “corner” of the star, and each player’s objective is to move all of his pieces to the corner opposite of his starting position. On his turn, a player may move one of his pieces to any of the adjacent positions, or “jump” an adjacent piece (including his own), to move a greater distance. As in Checkers, jumps may be chained together, allowing for great distances to be covered in a single move. The possible starting positions for Chinese Checkers are shown in Fig. 11.

Fig. 11

The two, three, four, and six player starting positions for Chinese Checkers