Arimaa/Relative Value of Pieces
Importance of Elephants
As the strongest Arimaa piece, the elephant is unique in its value. Although it cannot be pushed or pulled, an elephant is not entirely immune from capture—if it voluntarily enters a trap square, it needs an adjacent friendly piece to keep it alive. With a game still in progress, an elephant loss will almost always be devastating. A strong elephant blockade can also be ruinous.
The only thing worth more than an elephant is a guaranteed goal—an elephant may need to be sacrificed to help a friendly rabbit reach goal, but since the game has been won there is no problem. In rare circumstances, such as move 38s of this game, giving up one's elephant may be the only way to stop an enemy goal.
In the position at right, Silver to move could goal in four steps, even though the silver elephant would be lost when the e3 rabbit stepped to e2. If it's Gold's turn, however, Gold can capture the silver elephant and, with care, prevent any silver goal.
Value of Other Pieces
The practical value of each piece depends on its strength relative to the opponent's remaining forces. If both silver horses are gone, a gold camel and horse are nearly equal in strength, since a silver dog is the strongest piece which either can threaten. The absence of enemy horses also makes friendly dogs stronger, as they face few threats. Without silver horses, the only difference between a gold camel and dog is that the gold camel can threaten a silver dog, and a gold dog can be threatened by the silver camel.
As the board clears, the functional difference between pieces diminishes, and thus quantity of material becomes increasingly important. Try not to trade two pieces for one moderately stronger piece.
In the endgame, goal threats are often worth more than material—when an enemy rabbit has a path to goal, goal defense must take priority over captures. A strong goal threat of your own may also be preferable to a material lead. At the end of this game, Silver gave up his camel but created a strong goal threat, which Gold was unable to stop. By holding a gold horse hostage, the silver camel kept most of Gold's strong pieces away from the emerging goal threat, thus it didn't matter that Gold won this material fight.
Often an even exchange (the capture of identical forces on each side) will benefit one player more than the other. In chess, the player with stronger pieces usually benefits from even trades, but in Arimaa the weaker side often benefits. Consider a situation in which Gold has lost a dog and a cat while Silver has lost his camel, with all other pieces remaining. Gold is poised to overload the silver elephant, now the only piece which can threaten a gold horse. Gold's horses are now functionally stronger than Silver's, and thus an "even" horse trade would favor Silver. Two such trades, removing all horses from the board, put Silver in the driver's seat; Silver is still up two-to-one in dogs and cats, pieces which now face fewer threats. As long as the silver elephant keeps the gold camel in check, Silver has a large advantage in free pieces despite his own camel's absence. After falling behind, Silver has by two "even" exchanges gained a decisive advantage.
Some general rules of thumb:
- When material is identical, a player holding a blockade, frame, or hostage usually attempts to win material outright, because even exchanges can diminish such an advantage:
- A player whose elephant is blockaded will normally benefit from even trades, which will leave the opponent fewer pieces to spare for blockade duty.
- A player whose elephant is pinned to the defense of a framed piece will usually benefit from even trades, because with fewer pieces the opponent will have a harder time maintaining the frame.
- A player holding a hostage should avoid even trades of pieces weaker than the hostaged piece, as such trades would diminish the value of the hostage.
- When material is not identical, the exchange of a weak piece reduces the relative value of every stronger piece, whereas the exchange of a strong piece increases the relative value of every weaker piece.
- A player who has won a dog for nothing in the opening, and thus retains two dogs to the opponent's one, will then benefit by exchanging camels or horses, as this will leave fewer pieces which can threaten a dog. Cat-for-cat or rabbit-for-rabbit trades, however, will weaken the dog advantage, as one has given up equal material for pieces which a dog might have captured outright.
- A player who has won a camel for nothing in the opening, and thus has the only remaining camel, should try to win further material outright; elephants are never traded, and any other "even" trade would weaken the camel. However, if he can take another piece or two without losing anything himself, he has then gained a significant edge in quantity of material. Even trades then become more favorable, since they maintain the quantity advantage while diminishing the functional difference between pieces.
- Whenever a player has the only remaining camel on the board, a friendly horse is almost a second camel, and thus generally shouldn't be traded for any single piece.
- The fewer the total number of pieces, the more important quantity becomes.
- If one player has stronger pieces and the other has more numerous pieces, the player with more numerous pieces usually benefits from an even exchange.
- On a mostly equitable board, the side with more Rabbits will usually benefit from any exchange which is roughly equal, such as two Dogs for a Horse and a Cat.
Finally, when a human plays a bot, "even" exchanges give the bot fewer possible positions to evaluate, which is an advantage for the bot on an otherwise equitable board.
With all of this having been said, a trade would always be better than losing a piece outright. If a friendly piece cannot be rescued, get whatever you can in return.
When material is traded for non-identical material, the implications likewise depend on what has happened beforehand. Some rules of thumb when the board is full:
- A Cat is worth more than a Rabbit, but not by much
- A Dog is worth approximately two Rabbits
- A Horse is worth approximately a Dog and a Rabbit
- A Camel is worth approximately a Horse and a Cat
These values correspond to opinions as of April 2011, and are subject to change as strategy is further refined.
When even material has already been exchanged, such a two-for-one trade would likely favor the side capturing two pieces.
Sometimes, material can be sacrificed to gain an overall advantage in the position. In a common example, you may find that you can take a camel hostage, but using your turn to that end would leave a friendly piece vulnerable to capture. Holding the enemy camel hostage does not automatically put one in control; in general, a hostage is only of benefit when one has an advantage in free pieces, which will be somewhat diminished by any material sacrifice. Moreover, if the enemy elephant chooses to abandon its camel, it may get something in return, having a head start while you make the capture. Add that to an initial sacrifice, and your opponent could come out fully compensated for his camel loss. A camel hostage is rarely worth sacrificing more than a cat for, and may not even be worth a rabbit. Such a dilemma requires you to carefully consider the entire position, thinking ahead to how you could capitalize and how your opponent could counter.
In this game, Silver sacrificed a cat to gain a camel hostage.
In Arimaa, it is not always clear which side has the ultimate advantage. Even an apparently favorable position depends on the player knowing how to capitalize; the real advantage is with the one who best understands the situation at hand.