Arimaa/Relative Value of Pieces
A piece's value depends not just on its raw strength, but on its practical impact. The "weakest" pieces, rabbits, are the only ones which can score a goal; while a rabbit can never dislodge or freeze any piece, a strong goal threat may have a far-reaching impact. Other pieces may rise in value when stronger enemy pieces are captured or marginalized.
Importance of Elephants
As the strongest Arimaa piece, the elephant is unique in its value. Since it cannot be pushed or pulled, an elephant can be captured only when it has chosen to occupy a trap square. With a game still in progress, an elephant loss will almost always be devastating. A strong elephant blockade can also be ruinous. When only one side has a functional elephant, that is the ultimate strategic advantage.
Since it is normally easy to avoid losing one's elephant, it is moot to consider the exchange value of an elephant relative to other pieces. Nevertheless, an elephant loses influence when it can't easily move, whether due to a blockade, hostage, frame, trap control fight, or goal threat. An elephant mobility advantage can amount to a lot.
The one 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.
The practical value of each piece depends on its strength relative to the opponent's remaining forces. If the silver camel is captured, any remaining gold horse becomes stronger, as the silver elephant is now the only threat it faces. There is a similar effect on down the line; the fewer threats a piece faces, the stronger it becomes.
A strong goal threat is a valuable commodity; when one side has a path to goal, the opponent must stop that goal at any cost. One forced to prioritize goal defense could lose material simply because there isn't time to save it.
In an even exchange, each side captures identical material within a few turns. This is not always a wash; such an exchange might actually undermine a strong position. Suppose Gold has lost a dog while Silver has lost her camel, with all other pieces remaining. Each side retains two horses, but Gold's are now functionally stronger, as there is no silver camel to threaten them. An "even" horse trade would leave Gold fewer pieces with which to threaten the two silver dogs and to support the gold camel.
Some general rules of thumb:
- A strong blockade, frame, or hostage may be weakened by "even" exchanges.
- An even trade will leave a blockader fewer pieces to spare for blockade duty.
- An even trade will make it harder for a framer to accomplish anything else while still holding the frame.
- An even trade will make it harder for a hostage-holder to overload the opponent with a second threat.
- An even piece-for-piece trade strengthens every weaker piece.
- A player up by a dog will usually benefit from an exchange of camels or horses, as this will reduce the threats to dogs. Cat-for-cat or rabbit-for-rabbit trades, by contrast, might undermine a dog advantage, which makes one's own cats and rabbits stronger than the opponent's.
- A player up by exactly a camel has nothing to gain from any even exchange (elephants are never exchanged). Any other difference in forces, however, may make some even trades favorable.
- A player with the only remaining camel shouldn't readily trade a horse; a horse and camel should try to overload an opponent who has only one piece which could threaten either.
- Any exchange brings the endgame closer. If one side has a strong latent goal threat or is well up in rabbits, that side may benefit by moving the game along.
When a human plays a bot, any exchange gives the bot fewer possible positions to evaluate, which can be an advantage for the bot.
With all of this said, it is sometimes best just to keep things simple. If one has a material lead but is in a jam, even a "bad" exchange may blunt a comeback by the opponent.
When material is traded for non-identical material, the implications likewise depend on what has happened beforehand. When forces have thinned, the looming threat of a goal may greatly increase the value of each remaining rabbit. On a full board, however, a rabbit is thought to be worth slightly less than a cat. A dog is thought to be worth slightly more than a cat. With each step up, the difference is greater. In the opening, two-to-one balances might be figured thus:
- 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
In the middlegame, such balances depend on how individual piece values have shifted. An initial dog exchange, for example, will strengthen cats, perhaps making a horse and cat together worth more than a camel. In the endgame, goal prospects are what matter; any trade is good if it leads to a friendly goal.
When forces become uneven, subsequent gameplay must account for the asymmetry. If one has the only remaining camel but fewer total pieces than the opponent, "even" exchanges might render the camel useless. Conversely, one with more pieces but no camel might be happy to let the opponent reciprocate any capture. If both camels and three out of four horses remain, the player with two horses will likely benefit from a camel exchange even if he is well behind in weaker pieces; once camels are eliminated, two horses can overload the enemy elephant.
Material vs. Position
By most reckonings, a camel hostage is rarely worth more than a cat. In other words, it would usually be better to capture a dog or save a friendly dog than to take a long-term camel hostage. A weak hostage may be of negative value for the hostage-holder.
Likewise, a frame will only benefit the framer if it creates an advantage in free pieces. A strong frame is sometimes worth nearly the value of the framed piece itself, and might become an even greater advantage if the opponent refuses to cut his losses and abandon the piece. In this game, Silver gave up a cat and a rabbit to secure a horse frame which the silver elephant then rotated out of; this paid off for Silver. Any position that gives one the only free elephant is highly valuable.
Strong pieces are often sacrificed in the endgame, when the focus is on goal threats. In this game, Silver gave up his camel at home but soon got a rabbit to goal. This was a type of race position; Silver chose to pursue a southeastern goal attack rather than defend in the northwest.
In a given Arimaa position, it is not always clear which side has the ultimate advantage. For estimates regarding specific material balances, see these material evaluators, of which HarLog is considered the most accurate. This does not however account for positional features; a goal threat or stuck elephant might shift the advantage dramatically. The overall distribution of pieces is also important. Furthermore, any advantage depends on the player knowing how to capitalize; often, the real advantage is with the one who best understands the position at hand.