Mate with the two Bishops in Kriegspiel
Thomas S. Ferguson · eScholarship (California Digital Library) · 2011
MATE WITH THE TWO BISHOPS IN KRIEGSPIEL T. S. Ferguson, 03/08/95 UCLA It is generally known that the kriegspiel endgame with a king and two bishops versus a king alone is a win for the player with the two bishops. This assumes, of course that the bishops are on opposite colored squares and that the king is initially guarding the two bishops, say king on d4, and bishops on d5 and e5 as in the diagram below. In the following, we take the player with the two bishops to be white and his opponent to be black. White wins. Interestingly, there does not exist a strategy for white that wins with probability one in this position when nothing is known as to the whereabouts of the black king. What we can say is that for every > 0 there is a strategy for white that wins with probability at least 1 − no matter where black starts and what strategy black uses. In the terminology of game theory, there exists an -optimal strategy for white for every > 0 and the value of the position is a win for white. The reason there does not exist an optimal strategy for white is that white cannot reach a position in which he need guard the bishops only on one side. In particular, he cannot reach the side of the board without risking losing one of the bishops or allowing a possible stalemate. If he does reach the side of the board successfully, he has a strategy that mates surely in a finite number of moves without using randomization as will be seen later. He can safely move towards the edge, for example in the above position, by Bc4, Kd5, Bd4, Kc5, reaching the following critical position. White wins. To bring the king and bishops to the edge while still guarding them, white must play Bb5 which risks stalemate at a5. Therefore in an attempt to reach the edge, he may randomize