Recursive comparison tests for dicot and dead-ending games under misère play
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Urban Larsson
Abstract
In partizan games, where players Left and Right may have different options, there is a partial order defined as preference by Left: G ⩾ H if Left wins G + X whenever she wins H + X for any game position X. In normal play, there is an easy test for comparison: G ⩾ H if and only if Left wins G−H playing second. In misère play, where the last player to move loses, the same test does not apply-for one thing, there are no additive inverses-and very few games are comparable. If we restrict the arbitrary game X to a subset of games u, then we may have G ⩾ H “modulo U”; but without the easy test from normal play, we must give a general argument about the outcomes of G + X and H + X for all X ∈ U. In this paper, we use the novel theory of absolute combinatorial games to develop recursive comparison tests for the well-studied universes of dicots and dead-ending games. This is the first constructive test for comparison of dead-ending games under misère play using a new family of end-games called perfect murders.
Abstract
In partizan games, where players Left and Right may have different options, there is a partial order defined as preference by Left: G ⩾ H if Left wins G + X whenever she wins H + X for any game position X. In normal play, there is an easy test for comparison: G ⩾ H if and only if Left wins G−H playing second. In misère play, where the last player to move loses, the same test does not apply-for one thing, there are no additive inverses-and very few games are comparable. If we restrict the arbitrary game X to a subset of games u, then we may have G ⩾ H “modulo U”; but without the easy test from normal play, we must give a general argument about the outcomes of G + X and H + X for all X ∈ U. In this paper, we use the novel theory of absolute combinatorial games to develop recursive comparison tests for the well-studied universes of dicots and dead-ending games. This is the first constructive test for comparison of dead-ending games under misère play using a new family of end-games called perfect murders.
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
