[Paper Review] When game comparison becomes play: Absolutely Categorical Game Theory

Urban Larsson, Richard J. Nowakowski+1 more|arXiv (Cornell University)|Sep 9, 2016

Game Theory and Applications1 references3 citations

TL;DR

This paper introduces the Left Provisonal Game (LPG), a normal-play combinatorial game construction that enables categorical game comparison across diverse combinatorial game universes—such as normal-play, misère, and scoring-play—by encoding game comparison as a second-player win condition in a derived normal-play game. The key contribution is proving that $ G \succcurlyeq H $ in any Absolute Universe if and only if Left wins the LPG $[G,H]$ when playing second, thereby establishing a category structure $\mathbf{LNP}(\mathbb{U})$ for all such universes.

ABSTRACT

Absolute Universes of combinatorial games, as defined in a recent paper by the same authors, include many standard short normal- misère- and scoring-play monoids. In this note we show that the class is categorical, by extending Joyal's construction of arrows in normal-play games. Given $G$ and $H$ in an Absolute Universe $U$, we study instead the Left Provisonal Game $[G, H]$, which is a normal-play game, independently of the particular Absolute Universe, and find that $G\longrightarrow H$ (implying $G\succcurlyeq H$) corresponds to the set of winning strategies for Left playing second in $[G,H]$. By this we define the category ${\bf LNP(U)}$.

Motivation & Objective

• To unify game comparison across diverse combinatorial game classes—normal-play, misère, scoring—by embedding them in a common categorical framework.
• To resolve the challenge of comparing games in non-group-like universes (e.g., misère or scoring-play) where inverse games are not generally available.
• To define a universal game construction—Left Provisonal Game (LPG)—that reduces game comparison to a standard normal-play outcome evaluation.
• To establish that $ G \succcurlyeq H $ holds if and only if Left wins the LPG $[G,H]$ as second player, thereby enabling categorical structure.
• To provide a computational implementation in CG-suit 0.7 for automated comparison of misère-dicot games using the LPG framework.

Proposed method

• Define the Left Provisonal Game $[G,H]$ as a normal-play game whose Left options are restricted to those satisfying the Proviso condition in the Absolute Universe $\mathbb{U}$, and Right options are unrestricted.
• Construct the LPG using standard combinatorial game notation: $[G,H] = \{[G^L,H] \in \text{Proviso}, [G,H^R] \in \text{Proviso} \mid [G^R,H], [G,H^L]\}$, ensuring it is always a valid normal-play game.
• Introduce the concept of 'Maintenance' in $\mathbb{U}$, where $[G,H] \in \text{Maintain}(\mathbb{U})$ if Left can always respond to Right’s moves with a winning reply.
• Prove that $ G \succcurlyeq H $ holds in $\mathbb{U}$ if and only if $[G,H] \in \text{Proviso}(\mathbb{U}) \cap \text{Maintain}(\mathbb{U})$, which is equivalent to $[G,H] \geq 0$ in normal-play.
• Implement the LPG construction in CG-suit 0.7 via a custom procedure `CompareDM` that evaluates game comparisons by recursively analyzing the outcome of $[G,H]$ and its dual.
• Use the outcome function `Moutcome` to classify positions as Left win (1), Right win (-1), or draw (0 or 11), and apply dual-game logic to determine comparative dominance.

Experimental results

Research questions

• RQ1 Can game comparison in non-normal-play universes (e.g., misère or scoring) be reduced to a standard normal-play outcome evaluation using a universal construction?
• RQ2 Is there a categorical structure underlying all Absolute Universes of combinatorial games, even when they lack group properties?
• RQ3 Does the Left Provisonal Game $[G,H]$ correctly encode the relation $ G \succcurlyeq H $ as a second-player win for Left in normal-play?
• RQ4 Can the LPG construction be implemented algorithmically to enable automated game comparison in tools like CG-suit?
• RQ5 What are the minimal conditions under which the LPG ensures that $ G \succcurlyeq H $ if and only if Left wins $[G,H]$ as second player?

Key findings

• The Left Provisonal Game $[G,H]$ is always a valid normal-play game, regardless of the underlying Absolute Universe $\mathbb{U}$, due to its recursive construction with Proviso-restricted options.
• The relation $ G \succcurlyeq H $ holds in any Absolute Universe $\mathbb{U}$ if and only if the LPG $[G,H]$ is a second-player win for Left, i.e., $[G,H] \geq 0$ in normal-play.
• The construction of $\mathbf{LNP}(\mathbb{U})$, the category of games in $\mathbb{U}$ with morphisms $ G \to H $ iff $ G \succcurlyeq H $, is established via the LPG, generalizing Joyal’s normal-play category to all Absolute Universes.
• The Proviso condition ensures that $ G \succcurlyeq H $ is preserved under all left- and right-atomic extensions, making it a sufficient and necessary condition for game comparison.
• The implementation in CG-suit 0.7 via `CompareDM` successfully evaluates game comparisons in misère-dicot games by analyzing the LPG and its dual, with correct outcome classification using `Moutcome`.
• The method correctly identifies $ G \succcurlyeq H $ in all standard universes: normal-play, dicot misère, free misère, and scoring-play, under the respective conditions stated in Corollary 3.21.