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On the Sprague-Grundy values of games with a pass

Hikaru Manabe, Ryohei Miyadera, Koki Suetsugu

TL;DR

The paper addresses the challenge of SG-analysis for impartial games when a pass-move is allowed. It introduces an SG-homomorphism framework that, under the one-move game condition on components, reduces the SG-value of a pass-enabled disjunctive compound to the SG-value of nim with a pass, using component SG-values as nim-pile sizes. This yields bounds on the time to determine winning strategies and extends to SG-value computations for certain chocolate games via the hypergraph and one-move perspectives. The results provide concrete formulas and corollaries for stair chocolate games and show how the pass-move can have a minimal impact in some NS-property cases, offering practical tools for analyzing pass-enabled impartial games.

Abstract

In this paper, we consider two-player impartial games with a pass-move. A disjunctive compound of games is a position in which, on each turn, the current player chooses one of the components and makes a legal move in it. For disjunctive compounds, it is known that the time to determine which player has a winning strategy is bounded by the time to compute the SG-values of the components plus the time for their XOR. However, if we allow a pass-move during the play, the analysis of such games becomes much more difficult. A pass-move allows each player to skip exactly one turn in non-terminal positions during the game, after which neither player may use a pass-move again. We establish a homomorphism on the SG-values of games with a pass-move. That is, if every component satisfies a condition called one-move game, the SG-value of the disjunctive compound of the components with a pass-move is the same as the SG-value of nim with a pass-move where the size of every pile is the same as the SG-value of every component of the compound. This guarantees that the time to determine which player has a winning strategy in a disjunctive compound with a pass can be bounded by the sum of the time to determine SG-values of all components without a pass and a position in nim with a pass. We also show how the homomorphism is used for determining SG-values of some chocolate games.

On the Sprague-Grundy values of games with a pass

TL;DR

The paper addresses the challenge of SG-analysis for impartial games when a pass-move is allowed. It introduces an SG-homomorphism framework that, under the one-move game condition on components, reduces the SG-value of a pass-enabled disjunctive compound to the SG-value of nim with a pass, using component SG-values as nim-pile sizes. This yields bounds on the time to determine winning strategies and extends to SG-value computations for certain chocolate games via the hypergraph and one-move perspectives. The results provide concrete formulas and corollaries for stair chocolate games and show how the pass-move can have a minimal impact in some NS-property cases, offering practical tools for analyzing pass-enabled impartial games.

Abstract

In this paper, we consider two-player impartial games with a pass-move. A disjunctive compound of games is a position in which, on each turn, the current player chooses one of the components and makes a legal move in it. For disjunctive compounds, it is known that the time to determine which player has a winning strategy is bounded by the time to compute the SG-values of the components plus the time for their XOR. However, if we allow a pass-move during the play, the analysis of such games becomes much more difficult. A pass-move allows each player to skip exactly one turn in non-terminal positions during the game, after which neither player may use a pass-move again. We establish a homomorphism on the SG-values of games with a pass-move. That is, if every component satisfies a condition called one-move game, the SG-value of the disjunctive compound of the components with a pass-move is the same as the SG-value of nim with a pass-move where the size of every pile is the same as the SG-value of every component of the compound. This guarantees that the time to determine which player has a winning strategy in a disjunctive compound with a pass can be bounded by the sum of the time to determine SG-values of all components without a pass and a position in nim with a pass. We also show how the homomorphism is used for determining SG-values of some chocolate games.
Paper Structure (4 sections, 3 theorems, 7 equations, 2 figures, 1 table)

This paper contains 4 sections, 3 theorems, 7 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Disjunctive compound
  3. Hypergraph compound and SG-decreasing games
  4. Chocolate games

Key Result

theorem thmcountertheorem

For any position $g$ of an impartial game, $\mathcal{G}(g) =0$ if and only if $g$ is a $\mathcal{P}$-position and $\mathcal{G}(g) \neq 0$ if and only if $g$ is an $\mathcal{N}$-position.

Figures (2)

  • Figure 1:
  • Figure 3:

Theorems & Definitions (11)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem thmcountertheorem: spr, gru
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem thmcountertheorem: spr, gru
  • definition thmcounterdefinition
  • theorem thmcountertheorem: hyper
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • ...and 1 more