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; rabbits thus gain influence as goal paths open up. Other pieces gain influence when the opponent is less able to counter them.
As the one piece that cannot be pushed or pulled, the elephant has a unique role. 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 is forced to stay in one area; an elephant mobility advantage can amount to a lot.
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; as threats are eliminated, weaker pieces can do more.
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 blockader or framer should normally try to avoid even exchanges, which could leave his forces thin on the other wing. To a lesser extent, a hostage-holder's advantage might be undermined by an "even" trade, which could make it harder to utilize the strongest free piece.
- By contrast, even exchanges may strengthen a goal threat. Whether or not a forced goal is imminent, it becomes costly to defend goal as the board clears.
- One who is well ahead in quantity of pieces will usually benefit from even trades, as a material loss tends to be harder on an already depleted army.
- 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, as a horse can be quite powerful against a camel-less opponent.
- An even trade of strong pieces may favor one whose remaining forces are more balanced than the opponent's, as the opponent will then have a weak wing until he can move a strong piece across.
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. If one side is well ahead in both quantity and strength, anything close to an even trade might simply move the game along and make a turnaround less likely. Sometimes, one stronger piece is traded for two weaker pieces; such a two-for-one trade could favor either side, depending on the pieces and the position.
On a full board, a cat is thought to be worth slightly more than a rabbit, and a dog slightly more than a cat. With each step up, the difference is greater, though it may narrow as pieces are exchanged. 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
Such balances are never hard-and-fast, as every position is different. When space has cleared, 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 side has the only remaining camel but the other has more total pieces, "even" exchanges might render the camel useless. After an initial exchange of a camel for two weaker pieces, the side without a camel might be happy to let the opponent reciprocate any further capture. If both camels and three out of four horses remain, the player with two horses may 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, one who holds a camel hostage while down by a dog is probably losing, as he would be behind in material even after a horse-for-camel trade. A camel hostage usually shouldn't be taken at the expense of a dog, and sometimes shouldn't be taken at all; a long-term hostage will not benefit the hostage-holder if he has no advantage elsewhere.
The value of a frame likewise depends on the free pieces. A strong horse frame is sometimes nearly as good as an outright horse capture; if the opponent refuses to cut his losses and abandon the framed horse, the advantage could become greater still. 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. In this game, Gold gave up his camel in the northeast but soon forced a goal in the northwest. One must assess the strength of a goal threat before making such a sacrifice. If defending goal would require a material sacrifice, there is no choice, unless one can reach goal first.
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.