Arimaa/Introduction to Strategy/Elephant Blockade
Strongest Free Piece Revisited
Not long after discovering the camel hostage strategy, human players accidentally discovered that some computer programs could be lured into an elephant blockade by the offer of a free piece for capture. Much later it was further discovered that the program may squirm to try to free its elephant, and in the process get jammed all the way against an edge of the board.
In this game, diagrammed at right, Gold has lost a cat while Silver has a full army, but Silver is nonetheless completely lost. Not only is there no empty square for the silver elephant to step into, there is no empty square into which it can push its tormentors. The gold elephant and camel will keep an eye on the silver camel and horses respectively. Gold can ward off any attempts to break the silver elephant out of its prison, while herding silver pieces at will into the f3 trap. This is a much more decisive advantage than a camel hostage. Silver has no plausible lines of play in this case.
Most elephant blockades, however, are not as hugely advantageous as that one. It is a rare opponent who will voluntarily move his elephant to the edge of the board when a blockade is looming. The diagram below left, from this game, shows a slightly less advantageous situation with the blockaded elephant one square away from the edge of the board.
Here the gold elephant can't move, true, but nine silver pieces are required to maintain the blockade, including both the silver elephant and camel. Indeed, if all the pieces involved in the blockade stayed put, the strongest free piece would actually be the gold camel.
As it happens, however, Silver can undertake a rotation (or replacement) of the pieces participating in the blockade. When it comes to being in the way, a weak piece serves just as well as a strong one. Silver to move can free his camel for duty in only four steps, while maintaining the blockade: camel h6 south, rabbit h7 south, rabbit h8 south, and rabbit g8 east. The gold elephant can't make use of g8 to dig its way to freedom without getting smothered against the edge, so Silver can fill in that hole next move. Thus Silver needs only one turn to equal Gold for having the strongest free piece.
Furthermore, if Gold plays passively, Silver can continue to rotate pieces, freeing his elephant as well in two or three more turns. Because of this threat, it is very important that Gold not remain passive. Gold must immediately begin preparing a rescue mission to erode the blockade from the side, or even from the front if the silver elephant tries to leave. This will necessarily expose gold pieces to danger, but at least it puts some play into the position. For Gold to hang back is to await execution.
Note that even if Silver manages to rotate the elephant out of the blockade, it will require a few more pieces to maintain than a blockade on the edge. Furthermore those pieces will protrude one square further, making them slightly easier targets for would-be blockade busters. Still, the blockade is quite advantageous to Silver.
When Rotation Is Impossible
The diagram at right, from this game in the 2006 Arimaa World Championship, features an elephant blockaded one step further from the edge of the board, which is correspondingly less advantageous. Indeed, it is no longer realistic for Silver to expect to be able to free his elephant by rotating blockaders appropriately. True, the f6 trap is participating in the blockade at the moment, but Gold might bring a piece to f5 or e6, allowing the gold elephant to step to freedom, so Silver must soon occupy at least the latter squares.
An elephantless blockade would require silver pieces on g8, f7, g7, h7, e6, h6, f5, g5, h5, and g4. Not only are ten pieces necessary, but the bubble of blockaders also presents a large surface area for Gold to assail, extending within two steps of Gold's home trap at f3. Silver would be too busy warding off threats to the blockade to ever start capturing pieces with his freed elephant.
Since Silver can't rotate his elephant out of this blockade in practice, it does not give him the strongest free piece like an ideal blockade does. Yet the blockade is not worthless. Its value is that, although both Gold and Silver have a free camel, the silver camel is more free.
Suppose that Silver, while maintaining the blockade, were to use his camel to attack the c3 trap. Gold could defend c3 with his own camel, but couldn't endanger the attacking silver camel. In contrast, if Gold were to attack c6 with his camel, Silver would have the option of giving up the blockade to cross wings and take the gold camel hostage. (This principle recurs again and again in the study of elephant mobility.)
This difference of freedom pegs the value of the blockade to Silver at somewhat less than a camel hostage. Silver certainly can't expect to get more out of the position, because if Gold is willing to give up his camel as a hostage, he can frustrate anything else Silver might undertake. Indeed, the gold camel can probably break the blockade at any time if it is willing to expose itself. On the other hand, Silver can't necessarily force Gold to expose his camel. Gold can play in the west as well as hovering in the east making threats to break the blockade. If Silver has trouble generating a threat in the west while maintaining the blockade in the east, he may be forced to give up the blockade for an advantage smaller than a camel hostage.
Worthless Elephant Blockades
All the top Arimaa computer programs are now aware of elephant blockades, but most are still confused by the huge disparity in value from one blockade to the next. In the diagram at left, from this game, Gold's blockade of the silver elephant actually has negative value, for several reasons.
- Gold will never be able to rotate the gold elephant out of d4.
- The gold camel is so buried on c2 that it has less freedom of movement than the silver camel, despite the theoretical difference in elephant mobility.
- The gold pieces on e4 and f3 (at the edge of the blockade bubble) are in jeopardy.
- The blockade is unmaintainable. Silver doesn't actually need to break the blockade in order to make progress, but could break it if necessary.