Arimaa
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Arimaa is a two-player board game invented by Omar Syed, a computer engineer trained in artificial intelligence. Syed was inspired by Garry Kasparov's defeat at the hands of the chess computer Deep Blue to design a new game which would be difficult for computers to play well, but would have rules simple enough for his four-year-old son Aamir to understand. ("Arimaa" is "Aamir" spelled backwards plus an initial "a"). In 2002 Syed published the rules to Arimaa and announced a $10,000 prize, available yearly through 2020, for the first computer program (running on an inexpensive, off-the-shelf computer) able to defeat a top-ranked human player in a match six games or longer. The challenge match has happened twice so far. Both times David Fotland, who is also the developer of Many Faces of Go (http://www.smart-games.com/manyfaces.html), won the Arimaa computer championship and the right to play for the $10,000, only to see his program beaten 8-0 by Syed himself in 2004 and 7-1 by Frank Heinemann in 2005. Syed has applied for a patent on the Arimaa rules, and the name "Arimaa" is trademarked; see below for more.
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Rules
Arimaa is played on a chessboard with four squares distinguished as trap squares, namely c3, f3, c6, and f6 in algebraic chess notation. The two players, Gold and Silver, each control sixteen pieces: these are, in order from strongest to weakest, one elephant, one camel, two horses, two dogs, two cats, and eight rabbits. The pieces may be represented by the chess king, queen, rooks, bishops, knights, and pawns respectively; Gold can be represented by white, and Silver by black.
The objective of the game is to move a rabbit of one's own color onto the home rank of the opponent. Thus Gold wins by moving a gold rabbit to the eighth rank, and Silver wins by moving a silver rabbit to the first rank. However, as it is difficult to usher a rabbit to the goal line while the board is full of pieces, an intermediate objective is to capture opposing pieces by pushing or pulling them into the trap squares.
The game begins with an empty board. Gold places the sixteen gold pieces in any configuration on the first and second ranks. Silver then places the sixteen silver pieces in any configuration on seventh and eighth ranks. The diagram at right shows one possible initial placement.
After the pieces are placed on the board, the players alternate moves, starting with Gold. A move consists of one to four steps. With each step a friendly piece may shift into an unoccupied square one space left, right, forward, or backward, except that rabbits may not step backward. The steps of a move may be made by a single piece or distributed between several pieces in any order. A move must make a net change to the position, thus one may not, for example, take one step forward and one step back with the same piece, effectively passing the turn.
Arimaa_Tactics.jpg
The diagram at left helps illustrate the remaining rules of movement.
A player may use two steps of a move to dislodge an opposing piece with a stronger friendly piece which is adjacent. For example, a friendly dog may dislodge an opposing rabbit or cat, but not a dog, horse, camel, or elephant. The stronger piece may pull or push the adjacent weaker piece. When pulling, the stronger piece moves into an empty square, and the square it came from is occupied by the weaker piece. The silver elephant on d5 could move to d4 (or c5 or e5) and pull the gold horse from d6 to d5. When pushing, the weaker piece is moved to an adjacent empty square, and the square it came from is occupied by the stronger piece. The gold elephant on d3 could push the silver rabbit on d2 to e2 and then occupy d2. Note that the rabbit on d2 can't be pushed to d1, c2, or d3, as it must move into an empty square.
Friendly pieces may not be dislodged. Also, a piece may not push and pull simultaneously. For example the gold elephant on d3 could not simultaneously push the silver rabbit on d2 to e2 and pull the silver rabbit from c3 to d3. An elephant can never be dislodged, since there is nothing stronger.
A piece which is adjacent to a stronger opposing piece is frozen, unless it is also adjacent to a friendly piece. Frozen pieces may not move. A frozen piece can freeze another still weaker piece. The silver rabbit on a7 is frozen, but the one on d2 is able to move because it is adjacent to a silver piece. Similarly the gold rabbit on b7 is frozen, but the gold cat on c1 is not. The dogs on a6 and b6 do not freeze each other because they are of equal strength. Again, an elephant cannot be frozen, since there is nothing stronger, though an elephant can be blockaded by being surrounded by pieces two deep, so that it can't dislodge the encircling pieces.
A piece which enters a trap square is captured and removed from the game unless there is a friendly piece adjacent. Silver to move could capture the gold horse on d6 by pushing it to c6 with the elephant on d5. Also a piece on a trap square is captured if all adjacent friendly pieces move away. Thus if the silver rabbit on c4 and the silver horse on c2 move away, voluntarily or by being dislodged, the silver rabbit on c3 will be captured.
Note that a piece may voluntarily step into a trap square, even if it is captured thereby. Also, the second half of a pulling move may be completed, even if the piece doing the pulling is captured on the first step. For example, Silver to move could step the silver rabbit from f4 to g4, step the silver horse from f2 to f3, which captures the horse, and still pull the gold rabbit from f1 to f2 as part of the horse's move.
In the diagrammed position, if it were Gold's turn to move, Gold could win in three steps: The dog on a6 can push the rabbit on a7 to a8, and when the dog is on a7, it unfreezes the rabbit on b7, which can step to b8 for the victory.
There are several ways for the game to end apart from a rabbit reaching its goal, all of which are extremely rare:
- If all sixteen rabbits are captured, the game is a draw.
- If, at the beginning of a player's turn, no moves are possible because all friendly pieces are frozen or blockaded, the player whose move it is loses.
- If the same position occurs three times with the same player to move, the player whose move caused it to occur the third time loses. Only positions at the end of each move are considered by this rule, not positions at the end of each step. (This rule originally did not consider side to move, but Omar changed it to be more like the chess repetition rule on May 28, 2005.)
Finally, if an opposing rabbit is dislodged onto its goal line and dislodged off within the same move, the game continues.
Computer ineptitude
There are a number of facets of Arimaa which make it relatively difficult for computer programs to beat good human players. Because so much effort has gone into the development of strong chess-playing software, it is particularly relevant to understand why techniques effective for chess are ineffective or only partially effective for Arimaa.
The most important factor is that chess programs use brute-force searching coupled with static position evaluation that relies heavily on material considerations. Chess programs examine many, many possible moves, but they are not good (compared to humans) at determining who is winning at the end of a series of moves unless one side has more pieces than the other. The same is true for Arimaa programs, but their results are not as good in practice.
When brute-force searching is applied to Arimaa, the depth of the search is limited by the huge number of options each player has on each move. An Arimaa player has roughly as many legal choices for each step as a chess player has for each move. Thus a program which can search to depth of sixteen ply can look ahead eight moves for each player in chess, but only approximately two moves (eight steps) for each player in Arimaa.
Chess software also makes use of sophisticated pruning techniques, which allow the software to conclude that one move is better than another without examining every possible continuation of the weaker move. If the opponent can crush a certain move with one reply, it isn't necessary to examine other replies, which dramatically increases search speed. In Arimaa software, in contrast, the speedup provided by pruning is less, because it isn't obvious to a materialistic evaluation function that a strategically crushing reply is in fact crushing. If the detriments of a move are beyond the search horizon, it can't be quickly discarded from consideration.
In most Arimaa positions, particularly towards the beginning of the game when the board is still crowded, a competent player can avoid losing any pieces within the next two moves. Compared to chess, Arimaa has relatively few ways for either player to force captures or exchanges in the short term. Captures can often be pushed beyond the horizon of a brute-force search. The struggle is initially more positional, and revolves around making captures unavoidable at some point in the future. This sort of contest magnifies the importance of being able to judge who is gaining or losing ground in more subtle ways than by capturing or losing pieces. Thus the strength of computer programs (examining millions of positions) is not as significant as their weakness (judging the position apart from material considerations).
The weakness of Arimaa programs in the opening phases is further magnified by the setup phase. In chess every game starts from the same position. By compiling before the game a list of stock replies to all standard opening moves, chess programs may often make a dozen or more excellent moves before starting to "think". Humans do the same, but have a smaller and less reliable memory of openings, which puts them at a relative disadvantage. Arimaa, in contrast, has millions of possible ways to set up the pieces even before the first piece moves. This prevents programs from having any meaningful opening book.
As the game progresses, exchanges and the advancement of rabbits tend to make the position more open and tactical. Arimaa programs tend to play better in this sort of position, because they see tactical shots which humans overlook. However, it is usually possible for humans to avoid wide-open positions by conservative play, and to angle for strategic positions in which computers fare worse. Against a conservative opponent it is almost impossible to bust open the position in Arimaa, whereas in chess it is merely difficult. One must beat defensive play by the accumulation of small, long-term advantages, which programs do not do very well.
One additional technique from computer chess which does not apply to Arimaa is endgame tablebases. Master-level chess games sometimes trade down into unclear endgames with only a few pieces, for example a king and two knights vs. a king and a pawn. It is possible to build, by retrograde analysis, an exhaustive table of the correct move in all such positions. Programs have only to consult a pre-generated table in such positions, rather than "thinking" afresh, which gives them a relative advantage over humans. Arimaa, in contrast, seldom comes to an endgame of any sort, because it is almost always decided in the middlegame. Equal exchanges of pieces are less common than in chess, so it is rare for a game of Arimaa to "trade down" and still be unclear. An average game of Arimaa has only eight captures, and from time to time top humans can still defeat top programs without losing a single piece, for example the final game of the 2005 challenge match (http://arimaa.com/arimaa/gameroom/replayFlash.cgi?gid=12121&s=w).
Omar Syed hopes that, because traditional computer game-playing techniques are only marginally effective for Arimaa, programmers will be forced to use artificial intelligence techniques to create a strong Arimaa-playing program. The successful quest to build a world-championship-caliber chess program have produced many techniques to successfully play games, but has contributed essentially nothing to the more general reasoning; in fact, the techniques of chess playing programs have been excluded from some definitions of artificial intelligence; a goal for Arimaa is that the techniques involved in playing it will help the larger goals of artificial intelligence.
The structure of Omar Syed's man against machine challenge, however, is at odds with his professed confidence in the difficulty of creating a strong program. In the annual challenge, programs are run on machines chosen and provided by Omar Syed himself, under the criterion that it be a typical, inexpensive, off-the-shelf home computer. The challenge would not be open to anyone requiring expensive multi-processor machines such as those used to challenge top-level chess players, much less something like the custom-built supercomputer Deep Blue, even though it was the success of this hardware-intensive approach which inspired the Arimaa's invention.
Strategy and tactics
For beginning insights into good play, see Arimaa Tactics and Arimaa Strategy.
Patent and trademark
Omar Syed has applied for a patent on the rules of Arimaa; it is unclear if or when this will be granted, or what scope they will cover if granted. Syed has stated that he does not intend to restrict noncommercial use, though the meaning of noncommercial is not clear at this time.
Omar Syed has also applied for a trademark on the name "Arimaa".
See also
External links
- Official Arimaa Website (http://arimaa.com/arimaa/)
- David Fotland's Arimaa Program (http://www.smart-games.com/arimaa.html)