Calculating Ultra-Strong and Extended Solutions for Nine Men's Morris, Morabaraba, and Lasker Morris

TL;DR

This work addresses computing extended and ultra-strong solutions for Nine Men’s Morris and its variants, including Morabaraba and Lasker Morris, by developing a partitioned, multi-valued retrograde analysis that can classify draws and optimize play against fallible opponents. It introduces a draw-classification scheme based on stable subspace heuristics and stores relative first-keys to propagate values across a partitioned state space, enabling ultra-strong play beyond traditional game-theoretic values. The authors solve Morabaraba (first-player win in 49 moves) and provide extended strong solutions for all three variants, along with extensive results, verification, and a scalable framework. The practical impact lies in significantly improving how endgame databases and ultra-strong gameplay can be computed for complex, cyclic, zero-sum games, with potential applicability to other board games as well.

Abstract

The strong solutions of Nine Men's Morris and its variant, Lasker Morris are well-known results (the starting positions are draws). We re-examined both of these games, and calculated extended strong solutions for them. By this we mean the game-theoretic values of all possible game states that could be reached from certain starting positions where the number of stones to be placed by the players is different from the standard rules. These were also calculated for a previously unsolved third variant, Morabaraba, with interesting results: most of the starting positions where the players can place an equal number of stones (including the standard starting position) are wins for the first player (as opposed to the above games, where these are usually draws). We also developed a multi-valued retrograde analysis, and used it as a basis for an algorithm for solving these games ultra-strongly. This means that when our program is playing against a fallible opponent, it has a greater chance of achieving a better result than the game-theoretic value, compared to randomly selecting between "just strongly" optimal moves. Previous attempts on ultra-strong solutions used local heuristics or learning during games, but we incorporated our algorithm into the retrograde analysis.

Figures (9)

• Figure 1: Pseudocode of retrograde analysis. Recall that the kinds of information recorded for a position (in the array $R$) are count($n$) and value(win/loss in $n$), but only one of these is stored at a time (union type). (Also, in Nine Men’s Morris and its variants, win/loss can be determined by the parity of DTW.) The priority queue is keyed with DTW, and in the simplest case, it can be implemented with a simple queue, because states are processed monotonically. But when end states can have different DTWs (because we take into account the partitioning of the state space, see Fig. \ref{['fig:RA2']}), then the method of two queues can be used as described at the end of Subsection \ref{['sec:RAImpl']}. Note that states in secondary subspaces are treated as end states. Also note that $e$ can be in either a primary or a secondary subspace, but the predecessors are restricted to the primary subspaces.
• Figure 2: Initialization of retrograde analysis tailored to Nine Men’s Morris and to the partitioning of the state space. This replaces Step 1 in Fig. \ref{['fig:RA1']}.
• Figure 3: Game-theoretic values with DTWs of all possible moves in the starting position of Morabaraba. The optimal moves are marked with an "!’’.
• Figure 4: A game-theoretical draw position in the standard or Lasker variant. White to move. Although White can preserve his material advantage, he cannot win. This is one of the only two such positions in the subspace of 8 white and 3 black stones. The only move here that can keep this stone count indefinitely is marked with an arrow.
• Figure 5: Distribution of DTWs in Morabaraba. The very first bar shows the number of draws. Note that every second bar corresponds to an even depth, which means that it represents the number of losses with that depth. These are usually smaller than neighbouring bars, because it is an advantage to have the right to move first.