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 stronger enemy pieces are captured or marginalized.
As the one piece that cannot be pushed or pulled, the elephant can roam the board freely, at least until it gets blockaded or needs to defend against a threat. An elephant mobility advantage can amount to a lot.
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. Strong pieces might get tied up defending goal, or might not have time to move as other pieces defend 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.
If a friendly piece cannot be saved, getting compensation for it is the best that one can do. When one can save a piece or give it up for equal material, here are some considerations:
- 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 free pieces.
- By contrast, even exchanges may strengthen a goal threat, by clearing space and reducing threats to advanced rabbits.
- 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 will strengthen weaker pieces, but may indirectly weaken stronger pieces.
- 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 with the only remaining camel shouldn't readily trade a horse, as a horse can be quite powerful against a camel-less opponent.
- A player with the only remaining camel should be wary of "even" trades in general, unless he is well up in quantity of pieces.
- 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.
If none of the above clearly apply, one contemplating an exchange should consider simplicity versus complexity. Removing pieces from the board will leave fewer possible moves, perhaps making it harder to surprise or confuse the opponent. One who is behind can sometimes benefit by complicating the position, and might thus want to retain as many pieces as possible, if there is no clear upside to an exchange.
On 29g of this game, Gold defended c3 rather than exchanging camels, as Silver would have had a strong goal attack once the gold camel was gone.
When material is traded for non-identical material, the implications likewise depend on what has happened beforehand. Capturing a camel for a horse may not help a player who is well behind in quantity of pieces, as his dwindling army might be overloaded even by a camel-less opponent. 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.
Since an elephant can't be pushed or pulled, it is almost never traded for material. It is thus moot to consider an exchange value of an elephant relative to other pieces. On a full board, a cat faces almost as many threats as does a rabbit, so is worth only slightly more than a rabbit. The difference accelerates slightly with each respective stronger piece. For a while, 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. A piece may greatly rise in value when a just-stronger enemy piece is captured or marginalized; after a horse exchange, for example, each dog may be worth more than two rabbits, even though rabbits themselves have been slightly strengthened also. When space has cleared, goal prospects matter most; 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, any exchange which deprives the camel of a friendly horse will likely render the camel useless, as the enemy elephant might then neutralize the camel at little cost elsewhere. The implications of a trade depend on what pieces it strengthens. If one side has lost a horse and the other has lost multiple weaker pieces, a camel exchange may allow the side with two horses to overload the side that has more pieces.
Material vs. PositionEdit
Taking a hostage might sometimes require one to forgo a capture or leave a friendly piece hanging. A hostage is usually worth less than a dog. If the defender is up in material, he might give up the hostaged piece and still come out even or ahead, by capturing an attacking enemy piece. This does not mean that one shouldn't take a hostage while down in material, just that one should seriously consider other moves. If the defender is already well-developed on the hostage wing, a hostage may be of negative value, as the defender's elephant may soon be free while the hostage-holding elephant is stuck defending against a trap attack.
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; strong threats might outweigh a material deficit, especially if strong enemy pieces are passive. 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.