Arimaa/Introduction to Strategy/Trap Control
Each trap is either owned by one player or shared. To own a trap is to be safe from capture in it, and further to be able to capture opposing pieces in it. At the beginning of the game each player owns their two home traps (c3 and f3 for Gold, c6 and f6 for Silver), but this could change quickly as pieces advance. It is very valuable to take complete ownership of an enemy trap. An enemy trap usually has plenty of enemy pieces nearby, which might be captured in quick succession, in contrast to the laborious process of repeatedly prying loose a single enemy piece and dragging it all the way to a home trap for capture. When pieces are threatened in their own home trap, they might delay or avoid capture by scattering, but this is even better for an attacker, since it leaves a hole through which a rabbit might reach goal. Threatening to own an opposing trap is quite forcing: it limits the opponent's options to immediate defense or sharp counter-attack.
|Shared trap control in the opening. (game)|
Attacking usually refers to advancing a non-elephant piece with a threat to share control of an enemy trap. This is a common objective in the opening, often with a horse. There are several advantages to taking shared trap control:
- It might later be possible to convert shared control to complete control, depending on the whole-board fight.
- It reduces the number of traps in which the opponent can make a capture, giving an advantage in tactical fights. In particular, a player who owns no traps has not even the strategic threat of dragging a piece across the board to a place it can be captured, because there is no such place.
- Since friendly pieces won't be captured on this wing, they are free to advance. This can lead to a long-term space advantage and more freedom to maneuver than the opponent.
- Rabbits can advance on this wing, with the latent goal threat possibly giving a tactical advantage (see Arimaa/Introduction to Strategy/Rabbit Advancement#Cement deadlock).
- The opponent is restricted to defence, so has fewer options.
The drawbacks of attacking for trap control are:
- Attacking pieces may get hostaged, framed, or blockaded, particularly if attacking against a well-defended trap. If it is not possible to quickly gain complete control and capitalize on it, stronger defenders may arrive later, depending on the whole-board fight.
- The home traps may be left weak.
- If attackers cannot retreat, the loss of flexibility may be a problem. In particular, rabbit advances require care.
In the position shown, a silver horse has penetrated enemy lines in the opening, and Silver is ahead because of a long-term trap control advantage. Although Silver has no immediate way to take complete control of the c3 trap, Gold has no direct way to deal with the intruding horse: the gold camel is far away, and taking the horse hostage with the elephant would be bad. Silver is free to advance on the western wing.
A shared trap can be either deadlocked, if the strongest gold defender is equal to the strongest silver defender, or contested, when one player has the strongest adjacent piece but the other has several weaker pieces to protect one another from capture. At a deadlocked trap, neither player can hope to capture anything in the short term because the other player's strong piece keeps the trap safe.
When the two elephants are deadlocked at one trap and both are unwilling to leave, then a second and perhaps third trap might be deadlocked by the camels and horses, with positions superficially similar to those in an elephant deadlock. However, any local struggle which involves the elephants is fundamentally different from one which doesn't involve the elephants, because the latter will be radically altered by the arrival of an elephant. In the diagram, the elephants are deadlocked in a horse frame at the f3 trap, while the camels are deadlocked in a horse hostage at the c6 trap. The position around c6 is locally analogous to a camel hostage with an elephant deadlock. However, this position plays very differently: Silver to move can put her elephant on c4 and dog on d5, winning the gold camel in exchange for the framed horse.
Each successive trap deadlock adds a layer of strategic decision. In addition to maneuvering locally for increased control around each trap, there is a global question of which strong pieces to allocate to which quadrant. Crossing from one quadrant to another creates critical tactical timing questions of the form, "If I take ownership of that trap, will I gain enough to compensate for losing ownership of the trap I am leaving?"
When a deadlock is broken by the arrival of a stronger piece, the stronger side will often be able to own the trap. However, the course of this fight will depend on the former strength of each side's control. For example, if a camel is pinned in a camel deadlock, and the opposing elephant arrives, it will capture the framed piece on the first turn and the pinned camel on the second turn. In contrast, if a camel is pinned in a camel deadlock and a friendly elephant arrives, it will first relieve the pin then extract the framed piece from the trap, then perhaps gain ownership of the trap and start capturing. Even if the end result is the same, the amount of time taken can be important.
A trap is contested when one player has the strongest adjacent piece but the other has several weaker pieces to protect one another from capture and unfreeze one another when taken hostage. At the moment neither player can make a capture, but the player with the strongest local piece is guaranteed safety, and may be able to eventually own the trap if the attack is supported with at least one additional piece.
In this game, diagrammed at right, the players are contesting all four traps. Until one player can establish total control of some trap, no captures will be possible by either player. Nevertheless, this is a somewhat less stable situation than when multiple traps are deadlocked, because at each trap the player with the strongest piece can fight to own it. Whoever gets ownership of a trap first will start capturing pieces, quickly gaining a huge advantage.
This is a positional race, as opposed to a capturing race or goal race. Whichever player is going to lose the race to own a trap must globally re-allocate forces before disaster strikes. Yet each re-allocation presents new opportunities to the opponent. For example, if the silver elephant took ownership of f6 at right, the gold camel would cross and take ownership of c3, a poor exchange for Silver. It would perhaps be wiser for a silver dog to assist the silver elephant in owning f3, although that would make the gold horse near f6 all the more potent a threat.
In the position at left (from this game), Silver has deadlocked both of Gold's home traps. It is tempting for Gold to target the exposed silver camel on b3, since camel hostages are valuable, but in this context we see that camel hostages are merely part of a larger strategy of trap control. One takes a camel hostage because it allows one to eventually own a trap the other player can't contest. A camel hostage is useless if it isn't part of a grand trap ownership plan.
In the diagrammed position, it is most important for the gold camel to re-assert ownership of f3. Silver has stuck his neck out so far that Gold has no need (and no time!) to try to secure a more subtle positional advantage. The fight around the f3 trap is critical. If Silver can maintain shared control of both c3 and f3, then Gold's position is hopeless, but if Gold regains total control of at least one home trap, Silver must be careful lest some advanced piece be captured.
In modern Arimaa theory, one player often willingly concedes the disadvantage of a frame or hostage for the advantage of deadlocking the elephants around an opposing trap, with a long-range view to contesting the other opposing trap as well. The deadlock can then be tightened by rabbit advancement. The tighter a deadlock becomes, the greater the compulsion on the defender to contest one or both opposing traps in response. Such struggles are quite different from race games where each elephant is massacring pieces in the enemy camp. On the contrary, in a trap-control game it is possible that neither player will make any captures until one player has already gained a decisive positional advantage.