Arimaa/Introduction to Strategy/Rabbit Advancement
The timing of rabbit advances is the most difficult strategic issue in Arimaa. While other pieces can retreat homeward, rabbits can only move forward, left, or right. The inability of rabbits to backtrack can cause a lot of problems for one who is quick to advance them. Even in the opening, however, advanced rabbits can help stronger pieces to gain control of enemy territory. In this game, Gold advanced several rabbits effectively even while the board was full. The pros and cons of rabbit advancement must be weighed on a case-by-case basis.
Exposed to capture
Apart from the elephant, any piece which crosses the midline is in danger of capture. Rabbits, however, are particularly vulnerable. Even a cat can push, pull, and freeze a rabbit, so it hardly taxes the defender's resources to hold a rabbit hostage. Moreover, not even the friendly elephant can rescue a rabbit hostage, because rabbits can't go home even if liberated. Friendly pieces can at best contest control of an opposing trap to prevent a hostage rabbit from being captured.
When nothing more important is happening in the position, advanced rabbits are a marker of strategic disadvantage. In this game, diagrammed at right, Silver to move has no way to protect all three advanced silver rabbits. Worse yet, Silver has no way to threaten a goal. Therefore, although material is still even, Silver is clearly losing.
It is extremely rare for the board to empty out enough that rabbits may advance unaided. Advancing a rabbit almost always entails a willingness to contest an opposing trap for the remainder of the game, or a willingness to abandon the advanced rabbit to be captured later on.
Weaken goal defense
Even when one intentionally advances a few rabbits, to threaten goal or for some other purpose, it can be dangerous to move all the rabbits off the back row. In the diagram at left (from this game), it is Silver's turn to move, but he is helpless to stop Gold's a6-rabbit from reaching goal in two turns. If even one silver rabbit remained on the home rank, it could slide over to unfreeze the dog on c8, which could then block the a6-rabbit. Instead, Silver is lost.
Contrast Silver's predicament to Gold's situation with four rabbits still at home. The three advanced gold rabbits indirectly threaten goal right, left, and center, yet the four gold rabbits at home ensure that Silver would have to fight through more than one line of defense to score a goal.
There is some merit to advancing two rabbits on the same wing. Two rabbits advancing one in front of the other are a greater goal threat than one rabbit advancing alone. The back rabbit prevents the front one from being pushed backwards, and also prevents the front rabbit from being frozen. However, advancing all four rabbits on one wing hardly increases the threat of goal, whereas it does strip the defense. The majority of rabbits should be kept at home for defense unless exchanges have denuded the board and a breakthrough is imminent, or one side wishes to cement a deadlock on an opposing trap.
Blockade friendly elephant
The ideal pieces for blockading an elephant are rabbits of the same color. An elephant can push away an opposing piece as long as there is an open square to push it into, but an elephant can't push away a friendly piece under any circumstance. Since rabbits can't retreat, they often can't get out of the way of friendly pieces even if they are not frozen.
In the position at right (from this game), Silver is utterly lost strategically. Silver's own rabbits hem in the silver elephant on e3 and horse on g2. If the silver elephant pushes to e2, it will only allow an even more efficient blockade. Because Silver can do nothing in the mean time, Gold can take several moves to rotate the gold camel out and up the a-file while bringing across another gold rabbit to seal the blockade. Once the gold camel makes it behind Silver's front line, it can rampage unimpeded, and in particular free from fear of the silver elephant.
The advanced silver rabbits don't threaten goal; they are actually worse than useless. Silver would gladly remove a rabbit or two as part of his move in order to regain mobility, but disappear is not among the moves available to rabbits, at least not Arimaa rabbits.
Once a significant amount of material has been traded, or several traps have been contested across the board, holes start to appear in each player's defenses. The emptier and more ragged the board becomes, the more an advanced rabbit is a direct threat to win the game. In the diagram at left, from this game, Gold to move is at an enormous material disadvantage, but the presence of an advanced gold rabbit on each wing generates significant winning chances, particularly with the gold elephant assisting and the silver elephant temporarily out of position. Indeed, with perfect play by both sides, Gold has a forced win in five moves, although the tactics are complex.
When the board is empty and ragged, the latent threat of an advanced rabbit to reach goal can be a strategic asset, even if there is no direct win available. However, in order to have a long-term impact the advanced rabbit must be as little exposed to capture as possible. Ideally it should be on the edge of the board on either the fourth or the seventh rank. A rabbit on the edge of the fourth rank forces opposing pieces to decentralize themselves in order to keep it off the seventh rank, but is still far enough from any trap that the opponent may not have time to capture it. A rabbit on the edge of the seventh rank is in greater danger, but is also a greater threat because a strong friendly piece may come usher it home at any time. The value of such a rabbit depends on the position being so sharp the opponent can't spare two moves to capture the intruder.
In live games advanced rabbits also place great psychological pressure on the opponent. Even when a correct defense exists, a blunder by the defender loses the game, whereas a blunder by the attacker only loses a rabbit.
It is often preferable to advance a rabbit rather than endure the risk that a stronger piece will be dragged into enemy territory and cut off from hope of rescue.
In the diagram to the right, the silver elephant could step left to b4, then return to c4 while pulling the gold camel onto b4. In reply, Gold could return the camel to b3 after unfreezing it by advancing the a1-rabbit three squares. In the following move, Silver may decide to pull the gold rabbit further along the a-file in the hopes of eventually capturing it on the c6 trap.
Notice that neither player is making a mistake in the sequence of events described above. Gold knows perfectly well that the advanced rabbit will be vulnerable to capture, but prefers to expose it rather than give up a camel hostage. Silver, on the other hand, was correct to attack the exposed camel even though it could be freed, because Silver gained a rabbit as a target in the process. Even if it had been Silver's intention all along to pull the a1-rabbit to c6, directly attacking the a1-rabbit would have been much less effective than encouraging it to take three steps forward voluntarily.
Blockade opposing elephant
If a defending elephant becomes decentralized while holding a hostage near a home trap, there may be an opportunity for the attacking side to blockade it with a swarm of weak pieces, i.e. rabbits, cats, and dogs. In this 2005 Arimaa Challenge game at left, Silver has exploited the computer's susceptibility to elephant blockades.
Notice that the immobilized gold cat has become part of the blockade holding in the gold elephant. While it appears that the gold elephant can escape by pushing the a3-horse south and then using its newly mobilized cat to push the b4-rabbit south, the escape attempt would not succeed. Silver would have at least two methods of re-establishing the blockade, this time without any possibility for movement by the gold elephant. Silver has the pleasant option of continuing the blockade or, as in the actual game, capturing the cat.
Often in such situations the gold elephant is not absolutely blockaded, because it may still escape to the center of the board via the first two ranks. Even if the gold elephant could so escape, however, the blockading silver pieces would surge forward in its wake, ensuring long-term trap control of c3 for Silver, and consequent indirect goal threats.
Silver has stripped his goal defense by advancing so many rabbits, but this is of little consequence as long as no gold pieces can advance in the west. With the west fully blockaded, the center and east are dominated by the silver elephant and camel, which will make it very difficult for Gold to organize an effective goal attack. If the gold elephant had sidestepped the blockade and retained mobility, Silver's thin goal defense might be more of an issue.
The two diagrams at right illustrate the effect of rabbit advancement on a deadlocked trap. It is important to recall that, as long as both players commit an elephant to the trap at c6, neither player will ever be able to capture anything in c6. The advanced rabbits are not in immediate danger; they have only a strategic effect on the mobility of both elephants.
In the first diagram, the gold elephant does not want to abandon c6, because Silver would capture the b7-horse. Likewise the silver elephant does not want to abandon c6, because Gold would start capturing a series of small silver pieces in c6. This is the essence of an elephant deadlock: either side will suffer for leaving.
In the second diagram, the gold elephant does not want to abandon c6, because Silver would capture not only the b7-horse, but also all the advanced gold rabbits. The advanced gold rabbits make it harder for the gold elephant to leave than it was before. On the other hand, the silver elephant is even more reluctant to abandon c6, because Gold would not only capture a few small silver pieces, Gold would also very soon have an unstoppable goal threat. Thus the advanced gold rabbits make it catastrophic for the silver elephant to leave. Since both elephants will suffer for leaving even more than they would have before, the deadlock has been considerably tightened.
Generally the player advancing rabbits gains from cementing the deadlock, because the threat of eventually scoring a goal weighs more heavily than the threat of eventually losing a few more rabbits. Also, advanced rabbits can threaten to accumulate into a blockade, as above. Nevertheless, one should not unthinkingly push rabbits forward into any deadlock, because it also robs the advancing player of the option of retreating. If the rest of the board were such that Gold might profit from abandoning c6, or indeed might be forced to abandon c6, then advancing rabbits amounts to senselessly sacrificing them.
Endgame trap control
After a few pieces have been exchanged, rabbits often need to participate actively in contesting trap control, simply because there might not be enough pieces remaining to do the job without rabbits. In the diagram at left, from this game, Silver has just advanced his dog and rabbit to contest control of the c3 trap.
The silver rabbit on b3 is not only a threat to reach goal, it is a critical asset in the fight for control of the southwest. It prevents the gold elephant from immediately capturing the c4-dog, and will delay the capture of the dog should the gold camel switch quadrants. The b3-rabbit also allows the c4-dog to threaten the c2-cat with capture through the trap. Finally, the b3-rabbit prevents a gold rabbit from occupying b3, i.e. it frustrates Gold's simplest plan to make the gold cat temporarily safe.
Gold had correctly advanced rabbits to defend the f3 trap. Had there been a Gold rabbit on b3 as well, it would have forestalled the myriad of threats listed above, and at a minimum delayed Silver in gaining control of c3. Taking rabbits off the back rank to fight for trap control does weaken one's goal defense somewhat, but in an endgame the additional power in contesting traps is usually worth it.