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首页> 外文期刊>Computational Intelligence and AI in Games, IEEE Transactions on >Calculating Ultrastrong and Extended Solutions for Nine Men’s Morris, Morabaraba, and Lasker Morris
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Calculating Ultrastrong and Extended Solutions for Nine Men’s Morris, Morabaraba, and Lasker Morris

机译：为九个人的莫里斯，莫拉巴拉巴和拉斯克·莫里斯计算超强和扩展解决方案

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摘要

The strong solutions of Nine Men's Morris and its variant, Lasker Morris, are well-known results (the starting positions are draws). We reexamined 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 multivalued 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 ultrastrong solutions used local heuristics or learning during games, but we incorporated our algorithm into the retrograde analysis.
机译：九名男子莫里斯及其变种拉斯克·莫里斯（Lasker Morris）的有力解决方案是众所周知的结果（起始位置为平局）。我们重新检查了这两个游戏，并为它们计算了扩展的强大解决方案。这样，我们的意思是从某些开始位置可以达到的所有可能游戏状态的游戏理论值，在这些开始位置，玩家放置的石头数量与标准规则不同。这些也是针对以前未解决的第三种变式Morabaraba计算得出的，结果很有趣：玩家可以放置相等数量的石头（包括标准起始位置）的大多数起始位置都是第一位玩家的获胜（相对于以上游戏，通常是平局）。我们还开发了多值逆向分析，并将其用作超强求解这些游戏的算法的基础。这意味着，当我们的程序与易犯错误的对手对抗时，与在“刚好”的最佳移动之间随机选择相比，与游戏理论值相比，它更有可能获得更好的结果。先前对超强解决方案的尝试是在游戏过程中使用局部启发式或学习的方法，但我们将算法纳入了逆行分析。

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