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Affine Normal Play

Urban Larsson, Richard J. Nowakowski, Carlos P. Santos

Abstract

There are many combinatorial games in which a move can terminate the game, such as a checkmate in chess. These moves give rise to diverse situations that fall outside the scope of the classical normal play structure. To analyze these games, an algebraic extension is necessary, including infinities as elements. In this work, affine normal play, the algebraic structure resulting from that extension, is analyzed. We prove that it is possible to compare two affine games using only their forms. Furthermore, affine games can still be reduced, although the reduced forms are not unique. We establish that the classical normal play is order-embedded in the extended structure, constituting its substructure of invertible elements. Additionally, as in classical theory, affine games born by day n form a lattice with respect to the partial order of games.

Affine Normal Play

Abstract

There are many combinatorial games in which a move can terminate the game, such as a checkmate in chess. These moves give rise to diverse situations that fall outside the scope of the classical normal play structure. To analyze these games, an algebraic extension is necessary, including infinities as elements. In this work, affine normal play, the algebraic structure resulting from that extension, is analyzed. We prove that it is possible to compare two affine games using only their forms. Furthermore, affine games can still be reduced, although the reduced forms are not unique. We establish that the classical normal play is order-embedded in the extended structure, constituting its substructure of invertible elements. Additionally, as in classical theory, affine games born by day n form a lattice with respect to the partial order of games.
Paper Structure (17 sections, 30 theorems, 23 equations, 24 figures)

This paper contains 17 sections, 30 theorems, 23 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Motivational prelude
  3. Terminating moves and infinite heat components
  4. Entailing moves
  5. Carry-on moves
  6. Mathematical structure $\mathbbm{N}\space\mathbbm{p}\space^\infty\!$ of affine game forms
  7. Fundamental Theorem of Normal Play and the partial order of games
  8. Options-only comparison
  9. A challenging example
  10. inquisitor game
  11. Strategies of responses and game trees
  12. Options-only comparison in $\mathbbm{N}\space\mathbbm{p}\space^\infty\!$
  13. Reductions and reduced forms
  14. Conway-embedding and invertibility
  15. Classification of affine games
  16. ...and 2 more sections

Key Result

Theorem 5

Let $G$, $H$, and $J$ be affine games. We have the following:

Figures (24)

  • Figure 1: A hot amazons position.
  • Figure 2: First things first.
  • Figure 3: Carlsen-Anand, World Championship, RUS, 15 Nov 2014, Round 6. An eventual capture of the white pawn at e5 by the black knight is simply bad since that knight is captured by the white rook at h5 after two "automatic moves" resulting from the exchange of a pair of rooks at g8. Due to that, Carlsen thought he had the position under control and made the very bad move 26.Kd2?. Incredibly, Anand missed the opportunity and replied also with the bad move 26...a4?. Anand could have explored the opportunity with a pair of zwischenzugs: 26.Kd2 Ne5! 27.Rg8 Nc4 (zwischenzug) 28.Kd3 Nb2 (one more zwischenzug) 29.Kd2 Rg8 (only now Black recaptures the white rook at g8, with a winning position).
  • Figure 4: Infinite heat atari go component.
  • Figure 5: The Left option $W^{L}$ is a check.
  • ...and 19 more figures

Theorems & Definitions (87)

  • Definition 1
  • Definition 2: Affine Games
  • Definition 3: Disjunctive Sum
  • Theorem 5
  • proof
  • Definition 6: Individualized Outcomes
  • Definition 8: Outcomes
  • Definition 9: Equivalence and Order
  • Theorem 11
  • proof
  • ...and 77 more