Nim游戏的一些限制和推广
发布时间：2019-03-02 12:07
【摘要】：Nim游戏是博弈论中最经典的游戏模型之一,可以被描述为：有若干堆石子,每堆各有若干个.游戏参与人为两人.移动方法是两名参与者交替从任意一堆中取出任意正整数个石子.在normal规则下,谁先取完谁赢.在misere规则下,谁先取完谁输.本文深入研究了Nim游戏的限制和扩展.分别通过对Nim游戏的移动方法中的从哪堆中取石子和取多少做动态限制得到了两种更加复杂有趣的新游戏模型,我们分别称其为：有指针限制的Nirn和有指针和Modular双重限制的Nim.通过增加一堆的Nim游戏的参与人数而得到了一种新的Nim游戏,我们称其为一堆有界带有联盟的Nim.本文还研究了在Nim游戏中添加新的移动方法所得到的(s,t)-Wythoff游戏与其P-位置之间的关系.本文共分四章：第一章,绪论.主要介绍公平组合游戏的历史与发展,阐述了国内外的研究现状.第二章,彻底解决了有指针限制的Nim和有指针和Modular双重限制的Nim这两种游戏模型在normal规则下的所有P-位置.第三章,主要研究一堆有界带有联盟的Nim,提出了任意联盟的结构形式.本文分别给出了在miscrc规则下,带有联盟-[1,1]和联盟-[2,1]的Nim游戏中各联盟的所有非安全位置.指出了Annela R. Kelly在文献[36,37]中观点的错误原因,并给出了正确答案.第四章,主要研究由Nim游戏推广所得到的(s,t)-Wythoff游戏与其P-位置之间的关系.我们将(s,t)-Wythoff的P-位置逐个添加为该游戏附加的合法移动形成新的游戏模型,并分析了这些新游戏模型的P-位置特征.我们的结果表明,对任意的整数s≥1,总存在数对(s,t)使得新游戏模型的P-位置不依赖于附加合法移动的选择.
[Abstract]:Nim game is one of the most classic game models in game theory, which can be described as: there are several stacks, each of which has several. There are two people involved in the game. The move method is for two participants to alternately remove any positive integer stone from any pile. Under the normal rule, whoever wins first. Under the misere rule, whoever wins first loses. In this paper, the limitations and extensions of Nim games are studied in depth. Two more complex and interesting new game models are obtained by using dynamic restriction on which heap of stones and how much of the moving method of Nim game, respectively. We call them Nirn with pointer restriction and Nim. with pointer and Modular restriction, respectively, which we call the two new game models, namely, Nirn with pointer restriction and Nim. with pointer and Modular restriction, respectively. By increasing the number of players in a bunch of Nim games, we get a new Nim game, which we call a bunch of bounded Nim. with alliances. This paper also studies the relationship between the (s, t)-Wythoff game and its P-position by adding a new mobile method to the Nim game. This paper is divided into four chapters: chapter one, introduction. This paper mainly introduces the history and development of fair combination game, and expounds the research status at home and abroad. In the second chapter, all P-positions of the two game models, Nim with pointer restriction and Nim with pointer and Modular restriction, are completely solved under the normal rule. In the third chapter, we mainly study a bunch of bounded Nim, with alliance, and propose the structure of arbitrary alliance. In this paper, we give all the non-secure positions of the leagues in Nim games with alliance-[1,1] and alliance-[2,1] under the miscrc rule. This paper points out the wrong reason of Annela R. Kelly's viewpoint in reference [36, 37] and gives the correct answer. In chapter 4, we mainly study the relationship between (s, t)-Wythoff game and its P-position, which is derived from the extension of Nim games. We add the P-position of (s, t)-Wythoff to the game one by one to form a new game model, and analyze the P-position features of these new game models. Our results show that for any integer s 鈮，
本文编号：2433037
[Abstract]:Nim game is one of the most classic game models in game theory, which can be described as: there are several stacks, each of which has several. There are two people involved in the game. The move method is for two participants to alternately remove any positive integer stone from any pile. Under the normal rule, whoever wins first. Under the misere rule, whoever wins first loses. In this paper, the limitations and extensions of Nim games are studied in depth. Two more complex and interesting new game models are obtained by using dynamic restriction on which heap of stones and how much of the moving method of Nim game, respectively. We call them Nirn with pointer restriction and Nim. with pointer and Modular restriction, respectively, which we call the two new game models, namely, Nirn with pointer restriction and Nim. with pointer and Modular restriction, respectively. By increasing the number of players in a bunch of Nim games, we get a new Nim game, which we call a bunch of bounded Nim. with alliances. This paper also studies the relationship between the (s, t)-Wythoff game and its P-position by adding a new mobile method to the Nim game. This paper is divided into four chapters: chapter one, introduction. This paper mainly introduces the history and development of fair combination game, and expounds the research status at home and abroad. In the second chapter, all P-positions of the two game models, Nim with pointer restriction and Nim with pointer and Modular restriction, are completely solved under the normal rule. In the third chapter, we mainly study a bunch of bounded Nim, with alliance, and propose the structure of arbitrary alliance. In this paper, we give all the non-secure positions of the leagues in Nim games with alliance-[1,1] and alliance-[2,1] under the miscrc rule. This paper points out the wrong reason of Annela R. Kelly's viewpoint in reference [36, 37] and gives the correct answer. In chapter 4, we mainly study the relationship between (s, t)-Wythoff game and its P-position, which is derived from the extension of Nim games. We add the P-position of (s, t)-Wythoff to the game one by one to form a new game model, and analyze the P-position features of these new game models. Our results show that for any integer s 鈮，
本文编号：2433037
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