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• W2184328455 abstract "In this paper, we explore poset games, a large class of combinatorial games which includes Nim, Chomp, Hackendot, Subset-Takeaway, and others. We prove a general theorem about poset games, which we call the Poset Game Periodicity Theorem: as a poset expands along two chains, losing positions and positions with any fixed g-value have a periodic pattern. We use the theorem to (1) find polynomial-time winning strategies for a new, large class of poset games, (2) resolve two open conjectures about the game of Chomp, and (3) prove several important results about the computational complexity of calculating g-values in poset games. 1 Summary of Motivation I found poset games a particularly appealing topic for mathematics research for four reasons: (1) I was intrigued by the way that poset games with simple rules could have very complex structure; (2) the literature of poset games includes many problems that have long remained unsolved, which makes it an exciting area; (3) combinatorial game theory is relevant to a number of real-world problems, with particular applications in computer science and economics; and (4) finding a winning strategy to a combinatorial game is an especially tangible and satisfying result. 2 Introduction This paper analyzes a class of two-player games known as poset games. A poset (partiallyordered set) is a mathematical object satisfying a few simple properties, and any poset can be turned into a two-player game in a straightforward way. Poset games, like posets themselves, vary widely, and although a few specific types of poset games have been studied over the past hundred years, very few results are known about poset games in general. This paper offers a major new theorem about general poset games: as a poset expands in two directions, periodic patterns emerge in the associated poset game not only in losing positions, but also in positions with any fixed g-value (g-values are an important, general classification of game positions). Using this theorem, which we name the Poset Game Periodicity Theorem, we further prove several more specific results: (1) we resolve two open conjectures about a specific poset game called Chomp; (2) we prove several results about the computational complexity of calculating g-values in poset games; and (3) we give an efficient (i.e., polynomial-time) winning strategy for a large class of poset games. 2.1 Posets and Poset Games A partially-ordered set (poset) is a set X and a partial ordering of its elements — some elements are smaller than other elements, but not every pair of elements can necessarily be" @default.
• W2184328455 created "2016-06-24" @default.
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• W2184328455 date "2002-01-01" @default.
• W2184328455 modified "2023-09-27" @default.
• W2184328455 title "Poset-Game Periodicity" @default.
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