Arimaa/Relative Value of Pieces
Arimaa piece values can only be weighed within a given position; when a piece is captured, the value of other pieces may change. Even the alignment affects the relative value of each piece.
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, every capture slightly undermines the immense influence of elephants. As the board empties out, goal threats increasingly trump any other advantage. Elephants remain very powerful in the endgame, but have a somewhat reduced ability to dictate affairs.
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.
Value of Other Pieces
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.
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. One forced to prioritize goal defense could lose material simply because there isn't time to save it. Raw quantity may gain importance, since it can be hard for a depleted army to stop a goal.
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 player holding a blockade, frame, or hostage should make the most of such an advantage, which could be diminished by "even" exchanges.
- 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 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 whose elephant is defending a hostage will likely benefit from even trades, which will make it harder for the opponent to overload that elephant with a second threat.
- An even piece-for-piece trade weakens every stronger piece, but strengthens every weaker piece.
- When material is even and then a dog is captured outright, the side up a dog will benefit by exchanging 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). However, a player up by a camel and a rabbit might become even stronger by exchanging dogs, cats, or rabbits. While such an exchange would slightly weaken the camel, it would strengthen the friendly rabbits, which could begin to threaten goal against thinning enemy forces.
- A player with the only remaining camel shouldn't readily trade a horse, as there is a large advantage in having multiple pieces which only the enemy elephant can threaten.
- 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, "even" exchanges give the bot fewer possible positions to evaluate, which can be an advantage for the bot.
With all of this 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. By most reckonings, there is only a small increase in value from rabbit to cat, cat to dog, and dog to horse. The increase from horse to camel is slightly greater, since each side has only one camel. 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 horse exchange, for example, will strengthen dogs and weaken camels, altering the implications of a subsequent trade involving either. In the endgame, the value of any move relates closely to goal prospects; an otherwise bad exchange is good if it lets a friendly rabbit reach goal first.
Material vs. Position
A solid camel hostage is worth approximately 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. Holding a weak hostage could be the equivalent of losing material; one should not try to take a hostage, let alone give up anything to get it, unless it would decidedly improve one's overall position.
Likewise, a frame that ties up both elephants can sometimes create an advantage for the framer, but would likely not make up for a substantial loss of material, unless the framer's elephant could rotate out. In this game, Silver gave up a cat and a rabbit to secure a horse frame which the silver elephant then rotated out of, which allowed Silver to overcome a large material deficit. Any position that gives one the only free elephant is highly valuable.
A strong piece should be sacrificed only for short-term gain, such as a goal or actual captures that will outweigh the loss. At the end of this game, Silver gave up his camel but got a rabbit to goal. Both sides had much at stake around c6 and f6, but all that material quickly became irrelevant.
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.