Arimaa/Relative Value of Pieces
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[edit] Importance of Elephants
The Elephant is a unique and extraordinarily valuable piece that can fearlessly move around the board without any risk of capture. So intimidating are the Elephants that it is not uncommon for all thirty of the other pieces to huddle along the perimeter of the board during the opening phase of the game. If an Elephant is ever carelessly lost or even blockaded, it often leads to ruin, as in this game from the 2007 World Championships.
If a player threatens a single enemy piece with capture, the opposing Elephant can stand beside the trap square indefinitely in order to prevent the loss of material. As such, it is often necessary to overwork the enemy Elephant by simultaneoulsly threatening to capture pieces in two or more trap squares.
Because Elephants can't be pushed or pulled into traps, they are never traded, and it is moot to consider the exchange value of an Elephant relative to other pieces. Nevertheless, one can say that every capture slightly undermines the immense influence of Elephants relative to the other pieces. The more the board empties out, the more goal threats start to trump capture threats. Elephants remain very powerful in the endgame, but have a somewhat reduced ability to dictate affairs.
[edit] Value of Other Pieces
An interesting feature of Arimaa is that there are no fixed values for the pieces. During the opening phase, a player may be delighted to sacrifice a Dog and Rabbit in order to trap the enemy Camel but in an endgame consisting of a Gold Elephant, Dog and 2 Rabbits versus Silver Elephant, Camel and Rabbit, Silver will find that it is extremely disadvantageous to possess only three pieces rather than four, despite the presence of the Camel.
It requires considerable amounts of experience and judgement to learn how to evaluate positions with unbalanced material. Expert players may observe the same game yet disagree on which side has the stronger forces. Some guidelines to keep in mind are:
- Advanced Rabbits are often vulnerable to capture in the opening but highly valuable during the endgame
- Possessing strong pieces is important during the opening phase, but sheer quantity of pieces becomes increasingly important in the middlegame and, especially, the endgame
- Possessing large numbers of weak pieces can be advantageous in a blockade situation, while possessing fewer, stronger pieces can be advantageous in open tactical games with many contested or open traps
[edit] "Even" Exchanges
Often an even exchange (the capture of identical forces on each side) will benefit one player more than the other. Unlike the situation in chess, where the player with more valuable pieces usually benefits from even trades, in Arimaa the weaker side often benefits. For example, if Gold has the advantage because the Silver Elephant is blockaded, that advantage may disappear if each side captures three Rabbits. Similarly, if Gold has won a material advantage of a Camel for a Dog and a Cat in the opening, trading off both pairs of Horses will turn it into a material advantage for Silver. Some general rules of thumb:
- When material is identical, a player who has a strategic advantage due to holding a blockade, frame, or hostage normally attempts to win material outright, because even exchanges favor the other player:
- A player whose Elephant is blockaded will normally benefit from even trades, because the opponent will have fewer pieces to spare for blockade duty.
- A player whose Elephant is pinned to the defense of a framed piece will usually benefit from even trades, because the opponent will have fewer pieces to spare for maintaining the frame.
- Camel hostages generally become less advantageous to the player holding the hostage as pieces are exchanged, because the relative value of the Camel itself declines as the board empties out.
- When material is not identical, the exchange of a weak piece reduces the relative value of every stronger piece, whereas the exchange of a strong piece increases the relative value of every weaker piece.
- A player who has won a Dog for nothing in the opening would like to trade Camels, Horses, and Dogs but is indifferent to trades of Cats and Rabbits.
- A player who has won a Camel for nothing in the opening is indifferent to all trades, and will instead normally try to win further material outright. However, once he is ahead by a Rabbit or two, even trades become very favorable.
- If material is balanced in that one player has stronger pieces and the other player more numerous pieces, the player with more numerous pieces usually benefits from any even exchange.
- The side with the greater number of Rabbits will usually benefit from material exchanges, even non-identical exchanges, as long as they are roughly equal, such as two Dogs for a Horse and a Cat.
[edit] Opening Trades
Whether an uneven trade is favorable or not depends heavily on what other pieces have been traded beforehand. The only uneven trades about which there is any agreement are those which happen at the very beginning of the game, before anything else has been traded, and even there expert opinion is not unanimous. Some tenuous ideas about opening trades include:
- A Cat is worth more than a Rabbit, but not by much
- A Dog is worth about two Rabbits
- A Horse is worth about a Dog and a Rabbit
- A Camel is worth more than a Horse and a Cat, but less than a Horse and a Dog
These values must be treated with extreme caution. They correspond to opinions as of January 2007, which is significantly different than opinions as of January 2005. The value of two small pieces relative to one big piece has risen in the general estimation.
Three-for-one trades are extremely rare, and even harder to reach agreement about. For example, there used to be near total agreement that a Horse was worth more than three Rabbits in the opening, but now some players would have a clear preference for the three Rabbits.
[edit] Sacrifices
A player may decide to sacrifice material in order to secure a long-term strategic advantage. It is extremely difficult to evaluate these types of positions and the player must consider many factors such as Elephant mobility, number of pieces remaining, rabbit goal threats and many others before deciding whether to sacrifice the material.
This game was ended by a camel sacrifice in exchange for a strong goal threat. Later analysis determined that the goal was forced, so it might be considered a tactical sacrifice, but in a live game it is impossible to calculate with such precision, so the sacrifice was strategic at the time. It is a critical consideration for this goal-race sacrifice that the elephants are on opposite sides of the board: otherwise the stronger side would likely be able to defend and regroup.
In this game, Silver sacrificed a cat to gain a camel hostage. The soundness of this type of sacrifice is widely disputed: some players wouldn't sacrifice even a rabbit to gain a camel hostage, while others would, in some circumstances, sacrifice a full dog.
A relatively common material sacrifice in the opening is to sacrifice a rabbit in order to gain a horse held hostage by a camel. It is common in part because there is no consensus who has gained in the exchange, so players will still gladly take either side of the position. Also there is no agreement about the material value of a horse frame, perhaps because its value can in truth be anywhere from negative to nearly a full horse depending on the position of the other pieces.
Arimaa theory is still very limited in regard to what positional features are worth material sacrifice. Players must use their positional judgment in these situations rather than mathematical formulas.
