Partizan subtraction games
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Eric Duchêne
Abstract
Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with n ∈ Sℒ (resp., n ∈ Sℛ) when it is Left’s (resp., Right’s) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i. e., an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate other kinds of behaviors for the outcome sequence. In addition to dominance, three other disjoint behaviors are defined, weak dominance, fairness, and ultimate impartiality. We consider the problem of computing this behavior with respect to Sℒ and Sℛ, which is connected to the well-known Frobenius coin problem. General results are given, together with arithmetic and geometric characterizations when the sets Sℒ and Sℛ have size at most 2.
Abstract
Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with n ∈ Sℒ (resp., n ∈ Sℛ) when it is Left’s (resp., Right’s) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i. e., an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate other kinds of behaviors for the outcome sequence. In addition to dominance, three other disjoint behaviors are defined, weak dominance, fairness, and ultimate impartiality. We consider the problem of computing this behavior with respect to Sℒ and Sℛ, which is connected to the well-known Frobenius coin problem. General results are given, together with arithmetic and geometric characterizations when the sets Sℒ and Sℛ have size at most 2.
Chapters in this book
- 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
Chapters in this book
- 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
