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. The loss of one's elephant is likely to be devastating, and is not worth any material which might be captured in exchange. A strong elephant blockade can be nearly as 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.

Value of Other Pieces

The practical value of each piece depends on its strength relative to the opponent's remaining forces. If all horses are gone, a camel is itself of no more consequence than a horse, since a dog is the strongest piece which it can threaten. Conversely, the absence of enemy horses makes dogs stronger, as they face fewer threats.

As the board clears, quantity of material becomes increasingly important, as you'll need to block enemy goals and at the same time create goal threats of your own. Rabbits are of more consequence than they may seem, as each rabbit lost is one less goal possibility, and one less piece to stand in the way of an enemy counterpart. You should be careful about giving up two weaker pieces for a stronger piece, as that can come back to hurt you later, even if the trade was materially favorable at the time.

In the endgame, goal threats are often worth more than material—when the enemy threatens to force a goal, goal defense must take priority over captures. Furthering a goal threat of your own may also be more advantageous than having 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. In taking the silver camel, the gold elephant stayed too far away from where it really needed to be.

"Even" Exchanges

Often an even exchange (the capture of identical forces on each side) will benefit one player more than the other. Unlike the situation in chess, where the player with stronger pieces usually benefits from even trades, in Arimaa the weaker side often benefits. For example, if Gold has lost a dog and a cat while Silver has lost its camel, with all other pieces remaining, Gold is poised to overload the silver elephant, now the only piece which can threaten a gold horse. In this situation, Gold's horses are of more value than Silver's, and thus an "even" horse trade would favor Silver. Two such trades, removing all horses from the board, shift the balance decisively toward Silver, who is still up two-to-one in dogs and cats, pieces which now face fewer threats. Gold has few options for attacking, as his camel is now highly vulnerable to being hostaged, which would give Silver a large advantage in free pieces despite his own camel's absence. After losing his camel in a significant setback, Silver has by two "even" exchanges gained a decisive advantage. Some general rules of thumb:

• When material is identical, a player who has a strategic advantage due to holding a blockade, frame, or hostage usually attempts to win material outright, because even exchanges can diminish the advantage:
  • A player whose Elephant is blockaded will normally benefit from even trades, which will leave the opponent with 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 camel hostage should try to maximize his advantage in free pieces, otherwise the hostage may lose its value.
• 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 is best served by trying to win further material outright. However, if he can take another piece or two without losing anything himself, he has then gained a significant edge in quantity of material, and even trades become very favorable, as they in effect increase that edge.
  • When a player has the only remaining camel on the board but not a significant edge in quantity of material, he should do his best to avoid "even" horse trades, especially with only one friendly horse left. When the enemy lacks a camel of his own, only his elephant can threaten a horse; the camel and the horse can thus force the enemy elephant to choose which to defend against. With no horse on its side, however, the camel can more easily be kept in check by the enemy elephant.
  • If material is balanced in that one player has stronger pieces and the other player more numerous pieces, the player with more numerous pieces usually benefits from any even exchange.
  • The side with the greater number of Rabbits will usually benefit from material exchanges, even non-identical exchanges, as long as they are roughly equal, such as two Dogs for a Horse and a Cat.

Finally, when a human plays a bot, "even" exchanges can work in the bot's favor simply by reducing the number of possible positions for the bot to evaluate.

Uneven Trades

When a single piece is traded for two weaker pieces, the implications of the exchange 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.

Once even material is exchanged, quantity becomes more of an issue, and thus a two-for-one trade would increasingly tend to favor the player capturing two pieces.

Sacrifices

Sometimes, a player will have an opportunity to secure a long-term advantage, but at the cost of some material. 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. Such strategies can be difficult to evaluate, as the entire board must be taken into account. One must consider factors such as Elephant mobility, total pieces remaining, rabbit goal threats, and threats of pieces to one another before deciding whether such a sacrifice is worthwhile.

In this game, Silver sacrificed a cat to gain a camel hostage. The soundness of this strategy is questionable, as a camel hostage is of benefit only when one has an advantage in free pieces, which any material sacrifice will diminish. Moreover, even if you eventually capture the camel, the enemy elephant will do something when it abandons the hostage—add that to an initial sacrifice, and your opponent may well come out even or ahead. Some experienced players would not even sacrifice a rabbit to gain a camel hostage, and few would sacrifice more than a cat for this purpose.

Conclusion

Arimaa theory is still limited in regard to determining which side has the advantage at any given position. Note also that any positional advantage depends on the player knowing how to capitalize. Different positions call for different strategies, and the real advantage is with the player who best understands the situation.