Impartial games with entailing moves
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Urban Larsson
Abstract
Combinatorial game theory has also been called “additive game theory” whenever the analysis involves sums of independent game components. Such disjunctive sums invoke comparison between games, which allows abstract values to be assigned to them. However, there are rulesets with entailing moves that break the alternating play axiom and/or restrict the other player’s options within the disjunctive sum components. These situations are exemplified in the literature by a ruleset such as nimstring, a normal play variation of the classical children’s game dots & boxes, and top entails, an elegant ruleset introduced in the classical work Winning Ways by Berlekamp, Conway, and Guy. Such rulesets fall outside the scope of the established normal play theory. Here we axiomatize normal play via two new terminating games, ∞(Left wins) and∞(Right wins), and achieve a more general theory. We define affine impartial, which extends classical impartial games, and we analyze their algebra by extending the established Sprague-Grundy theory with an accompanying minimum excluded rule. Solutions of nimstring and top entails are given to illustrate the theory.
Abstract
Combinatorial game theory has also been called “additive game theory” whenever the analysis involves sums of independent game components. Such disjunctive sums invoke comparison between games, which allows abstract values to be assigned to them. However, there are rulesets with entailing moves that break the alternating play axiom and/or restrict the other player’s options within the disjunctive sum components. These situations are exemplified in the literature by a ruleset such as nimstring, a normal play variation of the classical children’s game dots & boxes, and top entails, an elegant ruleset introduced in the classical work Winning Ways by Berlekamp, Conway, and Guy. Such rulesets fall outside the scope of the established normal play theory. Here we axiomatize normal play via two new terminating games, ∞(Left wins) and∞(Right wins), and achieve a more general theory. We define affine impartial, which extends classical impartial games, and we analyze their algebra by extending the established Sprague-Grundy theory with an accompanying minimum excluded rule. Solutions of nimstring and top entails are given to illustrate the theory.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents XIII
- The game of flipping coins 1
- The game of blocking pebbles 17
- Transverse Wave: an impartial color-propagation game inspired by social influence and Quantum Nim 39
- A note on numbers 67
- Ordinal sums, clockwise hackenbush, and domino shave 77
- Advances in finding ideal play on poset games 99
- Strings-and-Coins and Nimstring are PSPACE-complete 109
- Partizan subtraction games 121
- Circular Nim games CN(7, 4) 139
- Misère domineering on 2 × n boards 157
- Relator games on groups 171
- Playing Bynum’s game cautiously 201
- Genetically modified games 229
- Game values of arithmetic functions 245
- A base-p Sprague–Grundy-type theorem for p-calm subtraction games: Welter’s game and representations of generalized symmetric groups 281
- Recursive comparison tests for dicot and dead-ending games under misère play 309
- Impartial games with entailing moves 323
- Extended Sprague–Grundy theory for locally finite games, and applications to random game-trees 343
- Grundy numbers of impartial three-dimensional chocolate-bar games 367
- On the structure of misère impartial games 389
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents XIII
- The game of flipping coins 1
- The game of blocking pebbles 17
- Transverse Wave: an impartial color-propagation game inspired by social influence and Quantum Nim 39
- A note on numbers 67
- Ordinal sums, clockwise hackenbush, and domino shave 77
- Advances in finding ideal play on poset games 99
- Strings-and-Coins and Nimstring are PSPACE-complete 109
- Partizan subtraction games 121
- Circular Nim games CN(7, 4) 139
- Misère domineering on 2 × n boards 157
- Relator games on groups 171
- Playing Bynum’s game cautiously 201
- Genetically modified games 229
- Game values of arithmetic functions 245
- A base-p Sprague–Grundy-type theorem for p-calm subtraction games: Welter’s game and representations of generalized symmetric groups 281
- Recursive comparison tests for dicot and dead-ending games under misère play 309
- Impartial games with entailing moves 323
- Extended Sprague–Grundy theory for locally finite games, and applications to random game-trees 343
- Grundy numbers of impartial three-dimensional chocolate-bar games 367
- On the structure of misère impartial games 389
