Playing the perfect Kriegspiel endgame

Abstract

Retrograde analysis is an algorithmic technique for reconstructing a game tree starting from its leaves; it is useful to solve some specific subsets of a complex game, for example a Chess endgame, achieving optimal play in these situations. Position values can then be stored in “tablebases” for instant access, in order to save analysis time, as is the norm in professional chess programs. This paper shows that a similar approach can be used to solve subsets of certain imperfect information games such as Kriegspiel (invisible Chess) endgames. Using a brute force retrograde analysis algorithm, a suitable data representation and a special lookup algorithm, one can achieve perfect play, with perfection meaning fastest checkmate in the worst case and without making any assumptions on the opponent. We investigate some Kriegspiel endgames (KRK, KQK, KBBK and KBNK), building the corresponding tablebases and casting light on some long standing open problems.