Categories of impartial rulegraphs and gamegraphs (2312.00650v2)
Abstract: The traditional mathematical model for an impartial combinatorial game is defined recursively as a set of the options of the game, where the options are games themselves. We propose a model called gamegraph, together with its generalization rulegraph, based on the natural description of a game as a digraph where the vertices are positions and the arrows represent possible moves. Such digraphs form a category where the morphisms are option preserving maps. We study several versions of this category. Our development includes congruence relations, quotients, and isomorphism theorems and is analogous to the corresponding notions in universal algebra. The quotient by the maximum congruence relation produces an object that is essentially equivalent to the traditional model. After the development of the general theory, we count the number of non-isomorphic gamegraphs and rulegraphs by formal birthday and the number of positions.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Explain it Like I'm 14
What this paper is about (big picture)
The paper studies a new, very visual way to describe two-player “take turns” puzzle games (like NIM) using diagrams called directed graphs (digraphs). In these diagrams:
- Each dot (vertex) is a position you can be in during the game.
- Each arrow shows a legal move from one position to another.
The authors build a full mathematical theory around these diagrams, explain how to make “translations” between different games while preserving the rules, and show how to systematically simplify a game without changing who wins. They then count how many essentially different small games there are.
What the authors wanted to figure out
- Can we model impartial games (games where both players have the same moves available) directly as digraphs in a clean, general way?
- How do “rule-respecting translations” between games work, and what do they preserve?
- How can we merge positions that are effectively the same (to simplify a game) without changing the outcome?
- How does this new diagram model connect to the classic “set of options” definition of games used in books?
- How many different games exist for small sizes, if we consider “different” to mean “not the same up to relabeling”?
How they approached it (with simple analogies)
Games as road maps
Think of a game as a city map:
- A place on the map is a “position.”
- A one-way street (arrow) is a legal move from one position to another.
- The starting position is the city gate (the unique “source”).
- The game ends when you reach a dead end (a “terminal” position with no arrows out).
They call:
- A rulegraph: any such map with no infinite one-way walks (so every game must end).
- A gamegraph: a rulegraph with exactly one starting position and every place reachable from it.
Translations that respect options
An “option-preserving map” is like a perfect tour guide that translates each position in one game to a position in another game, and exactly matches the sets of possible next moves. So after translating, all your options are still the same (just renamed).
These maps are powerful because they keep important game information intact.
Tags that are computed from future options (valuations)
A “valuation” is a tag assigned to each position that is computed from the tags of its options. Examples:
- Nim-number (Sprague–Grundy value): a number computed as the smallest nonnegative integer not found among the nim-numbers of ’s options (this “smallest missing number” is called mex). It tells you how that position behaves when combined with others.
- Outcome under normal play, : P or N (Previous-player win or Next-player win) computed from the outcomes of the options.
- Formal birthday: roughly how many “layers” away a position is from the end, counting upward from terminal positions.
Key idea: if a map preserves options, it preserves these valuations automatically. That means a good translation keeps who wins, the nim-number, and the “distance from ending.”
Combining games
They define a “sum” of two games where, on your turn, you play in exactly one of them. The nim-number of the sum is the bitwise XOR (nim-sum) of the nim-numbers. This matches classic results in impartial game theory.
Simplifying by merging “the-same” positions (quotients)
If two positions have the same future behavior (the same options up to translation), you can merge them. Doing this systematically produces a simpler version of the original game that keeps outcomes the same. The authors treat these merges using “congruence relations” (a formal way to say “these positions are equivalent”).
Their biggest simplification (quotient by the maximum congruence) gives you the classic set-of-options model of games from the textbooks. But you can also stop earlier to keep helpful human-friendly structure (like seeing specific board patterns), which can make strategy more intuitive.
Main results and why they matter
- Option-preserving maps keep important game information the same:
- They preserve nim-numbers, normal-play outcomes, misère outcomes, and formal birthdays.
- They send moves to moves and do not invent fake new moves inside the translated image.
- For gamegraphs, a rule-respecting translation that hits every position must send the start to the start. In short: surjective implies “start goes to start,” and vice versa.
- The image of a game under a rule-respecting translation is itself a valid game or rulegraph (so analyses remain inside the same world).
- They organize rulegraphs and gamegraphs into categories (collections with rule-respecting maps) and show:
- The “isomorphisms” (perfect two-way translations) are exactly the bijective option-preserving maps.
- Some classical category-theory properties hold here, but not all (for instance, a Schröder–Bernstein-like property fails for rulegraphs, but a good version holds for gamegraphs).
- They build a theory of congruences and quotients (merging equivalent positions), and prove isomorphism theorems similar to those in universal algebra. Important takeaways:
- There is no “collapse to a single point” quotient (you can’t squash a whole game vertically into one node, because moves have direction and end).
- Every rulegraph has a unique minimum simple quotient (a most reduced form with no nontrivial merges left).
- The largest sensible quotient matches the classic set-of-options definition of impartial games. This bridges the practical digraph picture with the textbook model.
- Examples illustrate the ideas:
- NIM as a ladder of positions; nim-number of one-pile NIM with stones is .
- Wythoff’s game mapped into a subtraction game via a sum-of-heaps translation.
- Grundy’s game: natural merging of positions by ignoring tiny heaps that don’t matter later, plus further merges that simplify even more.
- A geodetic achievement game on a grid: translating detailed board states into a simpler 3×3 “region count” matrix makes analysis easier while keeping options intact.
- A maze game with mixed terminal outcomes shows that not every outcome rule can be captured as a valuation, highlighting limits and pointing to future extensions.
- Counting games: After setting up the theory, they count how many non-isomorphic (essentially different) gamegraphs and rulegraphs exist for small sizes, organized by the number of positions and by “formal birthday” layers.
Why this matters
- Clearer models for real play: People think of games as moving pieces on boards, not as abstract sets. Gamegraphs capture that natural viewpoint while remaining mathematically precise.
- Guaranteed-safe simplifications: Their quotient method lets you simplify complex games (merge “the-same” positions) without changing who wins. This helps both human understanding and computer analysis.
- Strong connections to classic theory: Their biggest simplification gives exactly the standard set-of-options model, proving the new picture is not just intuitive but fully compatible with the traditional one.
- A foundation for tools and algorithms: Because valuations and outcomes are preserved under option-preserving maps, you can analyze a simpler translated game and trust the results for the original.
- Paths to new research: The paper builds a category-theoretic framework that can be extended, for example to mixed terminal outcomes (like the maze), and encourages classification and counting of small games.
Closing thought
By turning impartial games into graphs of positions and moves, and by focusing on maps that preserve options, the authors create a powerful bridge between intuitive gameplay and rigorous mathematics. Their framework makes it easier to:
- translate between games,
- simplify without losing important information,
- connect visual game play to classic theory,
- and systematically compare and count different games.
This helps both learners and researchers understand why certain strategies work and how different games relate beneath the surface.
Collections
Sign up for free to add this paper to one or more collections.