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This indeed is a great game. I have played it for a few times now and my favourite way of mating is leaving the Anti-King unchecked with the same move as I check the ordinary King. Sort of a double check wich, as I interpret the rules leads to a mate. Good game Tomas
The problem, if any, would be that it is always the Anti-King which gets checkmated, and that the King is here only to prevent the players from discarding all their pieces or to lose by double check. So, if you want to checkmate the King nearly as often as the Anti-King, it's no use weakening the Anti-King by allowing the enemy pieces to jump it. Stronger armies, say with a Cardinal and a Marshal on a 10x8 board - not 10x10 which also weakens the Anti-King, unless you post the Pawns on the third line as in Grand Chess -, make the King more vulnerable, but the setups of Capablanca Chess or Gothic Chess make it also more difficult for the Anti-King to avoid mate, because the Cardinal and Marshal have less difficulty in escaping the zone of the Anti-King than Rooks, Bishops or Knights, and it might be better to report them on the outer files.
Actually, when I play, the anti-King isn't always the one that gets checkmated. I think initiative plays a large role - if you can force the anti-King to move around, you can move your pieces in for a checkmate without obstruction, or at the very least, severely impede your opponent's development. I alternate fairly equally between which enemy king I checkmate at the end. An interesting case that can occur in Anti-King chess is a sort of checkmate of both the king and anti-king. In a game I played with the Java program, a pawn was checking the anti-king, but I moved it forward to check the king. The pawn wasn't protected, so the king could take the pawn, but that would leave the anti-king without check. Inversely, the anti-king could have moved into check, but the king would still remain checked as well. This, of course, demonstrates that a single pawn can force mate. Quite an interesting game.
Here is a CV with two Kings like Muller's example for negative-value piece. In Anti-King win is by checkmate of regular King or removing check from other's Anti-King. Two other CVs with two Kings are Two Kings Chess and Double Chess.
Both Aronson's Berolina Pawn version and Anti-King Chess II have strategy to keep the side's Anti-King in check. In AKC-I with Berolina note that Anti-King is initially attacked by four pieces checking, and it will take a while to get them "safely" out of the way. Anti-King Chess II may benefit from changing Anti-King move to Knight move only as subvariant.
How do these relate to negative values? That pieces may want to be removed, if possible, in end game in order to have no forces nearby to attack opponent Anti-King, but their over-all average value would be positive just taking on negative value at end. Player may just settle for checkmating regular King.
Fergus Duniho's insightful strategy for actual game played 13 years ago: Strategy, where few pieces were captured.
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