Arimaa/Relative Value of Pieces
Because Arimaa pieces are defined by their power over and vulnerability to other pieces, the relative value of each piece changes throughout the game.
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 (the d4 cat would then step to d3 and then e3 to unfreeze that rabbit and let it goal.) If it's Gold's turn, however, Gold can capture the silver elephant in two steps, taking an overwhelming material lead while still having time to stop 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 have the same offensive power; the only functional difference is that a gold horse could be threatened by the enemy camel. The absence of enemy horses also makes friendly dogs stronger, as they face fewer threats.
As the board clears, rabbits gain importance. Any rabbit lost is one less goal possibility, and one less piece to stand in the way of an enemy goal.
In the endgame, goal threats are often worth more than material; when one side has a path to goal, the opponent must stop that goal at any cost. At the end of this game, Silver gave up his camel but created a strong goal threat, which Gold was unable to stop. Both sides had much to gain or lose around c6 and f6, but Gold had too many strong pieces concentrated there.
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 taken firm control.
Some general rules of thumb:
- 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, which will make it harder for the opponent to accomplish anything else while still holding the frame.
- A player with the strongest free piece (such as having a free camel while holding the enemy camel hostage) will be weakened by any even trade.
- 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.
- When a player has the only remaining camel on the board, a friendly horse shouldn't be traded for any single piece, since there is a large advantage in having multiple pieces which only the enemy elephant can threaten.
- 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.
- 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 can be an advantage for the bot.
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
When even material has already been exchanged, such a two-for-one trade would likely favor the side capturing two pieces.
None of this is hard-and-fast; how the remaining pieces are configured is another important consideration.
Sometimes, material can be sacrificed to gain a positional advantage. In a common example, you may be able to 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.