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 board is full and then the silver camel is captured, Gold suddenly has three pieces (the gold camel and both gold horses) which only the silver elephant can threaten; thus the gold horses have gotten much stronger. There is a similar effect on down the line; the fewer threats a piece faces, the stronger it becomes.
A rabbit gains strength as it becomes a greater potential goal threat. As the board clears, individual rabbits quickly rise in value. 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. 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; in fact, the weaker side often benefits from such an exchange. Suppose Gold has lost a dog and a cat 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 thus favor Silver. Two such trades, removing all horses from the board, would leave Silver with two powerful dogs and two powerful cats, against Gold's weakened camel, dog, and cat. Gold likely then couldn't mount an effective attack, as there wouldn't be enough pieces to support the gold camel, and the gold dog could be countered easily. In that situation, two "even" horse trades would have swung the advantage from Gold to Silver.
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 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 tend to be weakened by an even trade, which would diminish the value of that piece.
- An even piece-for-piece trade weakens every stronger piece, but strengthens every weaker piece.
- When the board is full and then a dog is captured outright, the side with two dogs will benefit by exchanging camels or horses, as this will make it easier for the two friendly dogs to overload stronger enemy forces. Another way to exploit such a dog advantage is to use one's cats and rabbits to overload the one remaining enemy dog; cat-for-cat or rabbit-for-rabbit trades are thus not ideal for one with a dog advantage.
- A player up by a camel and some rabbits would likely benefit by exchanging dogs or cats. Although such an exchange would slightly weaken the camel, it would strengthen the rabbits, which could soon create goal threats that would overload the remaining 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.
- 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 slightly more than a Rabbit
- 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 a trade's value depends on how individual piece values have shifted; when a piece becomes stronger, it is worth more. An initial horse trade, for example, will strengthen the dogs, each of which might then be worth more than two rabbits (although rabbits themselves may rise in value as things progress further). In the endgame, the value of any move relates closely to goal prospects. If a cat is crucial to goal defense, it may be worth more than two rabbits, since a cat-for-two-rabbits exchange might make way for a third rabbit to reach goal. Conversely, it may be worthwhile to trade one's camel for a rabbit, if that rabbit is the opponent's last best chance for a goal.
For estimates regarding specific material balances, see these material evaluators, of which HarLog is considered the most accurate.
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.
A position that gives one the only free elephant is highly valuable. A frame that ties up both elephants could be a wash, but if the framer's elephant can rotate out, the frame could be worth a lot. In this game, Silver gave up a cat and a rabbit to secure such a horse frame.
A strong piece should only be sacrificed for a tactical purpose, such as a goal or the actual capture of an even stronger piece. 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 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.